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Hassan al-Turabi

Hassan al-Turabi (حسن الترابي; 1 February 1932 – 5 March 2016) was a Sudanese politician and scholar. He was the alleged architect of the 1989 Sudanese military coup that overthrew Sadiq al-Mahdi and installed Omar al-Bashir as president. He has been called "one of the most influential figures in modern Sudanese politics" and a "longtime hard-line ideological leader". He was instrumental in institutionalizing Sharia (Islamic law) in the northern part of the country and was frequently imprisoned in Sudan, but these "periods of detention" were "interspersed with periods of high political office".

al-Turabi was leader of the National Islamic Front (NIF) (which later changed its name to National Congress in the late 1990s ), a political movement that developed considerable political power in Sudan while never obtaining significant popularity among Sudanese voters. It embraced a "top down" approach to Islamisation by placing party members in high posts in government and security services. al-Turabi and the NIF reached the peak of their power from 1989 following a military coup d'état, until 2001, as what Human Rights Watch have called "the power behind the throne", head of the only Sunni Islamist movement to take control of a state.

al-Turabi oversaw highly controversial policies such as the creation of the "NIF police state" and associated NIF militias that consolidated Islamist power and prevented a popular uprising, but according to Human Rights Watch committed many human rights abuses, including "summary executions, torture, ill treatment, arbitrary detentions, denial of freedoms of speech, assembly, and religion, and violations of the rules of war, particularly in the south". Turabi was a leader of opposition to the American–Saudi "coalition forces" in the Gulf War, establishing in 1990–1991 the Popular Arab and Islamic Congress (PAIC), a regional umbrella for political Islamist militants, headquartered in Khartoum.

After 1996, al-Turabi and his party's "internationalist and ideological wing" saw a decline in influence in favor of more pragmatic leaders, brought on by the imposition of UN sanctions on Sudan in punishment for Sudan's assistance to Egyptian Islamic Jihad in their attempt to assassinate Egyptian President Hosni Mubarak. al-Turabi was out of power beginning in 1999, leading a splinter group of the National Congress known as the Popular National Congress. He was imprisoned by Omar Al-Bashir on 17 January 2011 for nine days, following civil unrest across the Arab world. He died in 2016 without facing trial for his role in the 1989 coup.

Early life and education
al-Turabi was born on 1 February 1932 in Kassala, eastern Sudan, to a Sufi Muslim sheikh, and received an Islamic education, before coming to Khartoum in 1951 to study law and joined the Muslim Brotherhood as a student. He graduated from Khartoum University School of Law in 1955 and also studied in London and at the Sorbonne in Paris in 1962, where he gained a PhD. He became a leader of the Sudanese Muslim Brotherhood in the early 1960s. He is a descendant of a famous 17th-century religious sheikh, Hamad al-Turabi.

Religious and political beliefs
Turabi's writings, rhetoric, sermons, and public pronouncements have often been described as progressive, theologically liberal, "moderate and thoughtful", but his time in power was notable for harsh human rights violations. The diplomat Andrew Natsios explained these contradictions by claiming that he took more moderate positions when being interviewed by English and French-speaking interviewers when compared to his speeches to his fellow Islamists.

al-Turabi's philosophy drew selectively from Sudanese, Islamic, and Western political thought to fashion an ideology for the pursuit of power. al-Turabi supported Sharia and the concept of an Islamic state, but his vision was not Wahhabi or Salafi. He appreciated that the majority of Sudanese followed Sufi Islam, which he set out to change with new ideas. He did not extend legitimacy to Sufis, Mahdists, and Islamic scholars, whom he saw as incapable of addressing the challenges of modern life. One of the strengths of his vision was to consider different trends in Islam. Although the political base for his ideas was probably relatively small, he had an important influence on Sudanese politics and religion.

His views on the role of women in society were relatively progressive. al-Turabi had his greatest success in recruiting supporters from the educated and professional classes in urban areas. He attached fundamental importance to the concept of shura (consultation) and ibtila, his view of modernity, which he believed should lead to a more profound worship of God. Religion was regularly tested by the reality of ibtila.

As a Sunni Islamist, Turabi's ideas differed in some ways from traditional Islamic ideas, such as in his lack of reverence for professional Islamic scholars. Rather than the ulama (class of Islamic scholars) being restricted to educated Islamic scholars, he stated that "because all knowledge is divine and religious, a chemist, an engineer, an economist or a jurist are all ulamas." In fact, in an Islamic democracy, which Turabi maintained he was working towards, "ideally there is no clerical ulama class, which prevents an elitist or theocratic government. Whether termed a religious, a theocratic, or even a secular theocracy, an Islamic state is not a government of the ulama."

al-Turabi originally espoused progressive Islamist ideas, such as the embrace of democracy, healing the breach and expanding the rights of women, where he noted:"The Prophet himself used to visit women, not men, for counseling and advice. They could lead prayer. Even in his battles, they are there! In the election between Othman and Ali to determine who will be the successor to the Prophet, they voted!"

He told another interviewer, "I want women to work and become part of public life" because "the home doesn't require much work anymore, what with all the appliances". During an interview on al-Arabiya TV in 2006, al-Turabi describes the word hijab as not a face veil but a cover or diaphragm put in a room to separate between men and the Prophet's wives, whereas niqab is just an old Arab habit. Hijab literally means "barrier" and he said it was "a curtain in the Prophet's room. Naturally, it was impossible for the Prophet's wife to sit there when people entered the room". The Prophet's wives sat behind it when talking to males because they were not allowed to show their faces. He opposed the death penalty for apostasy from Islam and opposed Ayatollah Khomeini's death sentence fatwa against Salman Rushdie. He declared Islamist organizations "too focused on narrow historical debates and behavioral issues of what should be forbidden, at the expense of economic and social development".

al-Turabi also laid out his vision for Sharia law that would be applied gradually instead of forcefully and would apply only to Muslims, who would share power with the Christians in a federal system.

In contrast Natsios writes that when in power, one of the pieces of national legislation he pressed for was that apostasy be punished by the death penalty, a position he has since disavowed. When he talks about women's rights, he is referring exclusively to Muslim women, whose honor and virtue will be protected within the context of Sharia law, ... Christian or non-Muslim women may be treated as property without rights or protection.

In 2006, out-of-power again, al-Turabi made international headlines issuing a fatwa allowing Muslim women to marry non-Muslim men, and allowing alcohol consumption in certain situations, in contradiction to historical Sharia law. He also embraced human rights and democracy in "a hundred-and-eighty-degree turn" of his views. One critic of Turabi complained to an American journalist in Khartoum of Turabi's ideological reversal, saying, "it is said in the daily papers and in the discussion centers here in the university that Turabi killed Ustazh Mahmoud", (i.e. liberal Sudanese cleric Mahmoud Mohammed Taha, hanged in 1985), "and now he’s stealing his ideas."

Political career
Early in his career, al-Turabi took control of Islamic Liberation Movement under the name of the Sudanese Muslim Brotherhood. In 1964 he became secretary-general of the Islamic Charter Front (ICF), an activist movement that served as the political arm of the Muslim Brotherhood, and was elected to parliament in the mid-1960s. He headed the Front of the Islamic Pact and the Party of the Islamic Bloc from 1964 50 1969. Following the military coup in 1969, Gaafar Nimeiry jailed al-Turabi for seven years. Released in 1977, al-Turabi became attorney general in 1979 and then Nimeiry’s foreign affairs adviser in 1983. Nimeiry put him back in prison in 1985; the new military government released him later the same year. al-Turabi established the National Islamic Front (NIF) in 1985 as a replacement for the ICF. al-Turabi held several ministerial positions in government of the democratically elected Sadiq al-Mahdi, which the NIF joined in 1988 as a coalition partner, but he was never comfortable with this arrangement.

National Islamic Front rule
On 30 June 1989, a coup d'état by General Omar Hassan al-Bashir began a process of severe repression, including purges and executions in the upper ranks of the army, the banning of associations, political parties, and independent newspapers and the imprisonment of leading political figures and journalists. From 1989 until 2001, Turabi served as what observers have called the "intellectual architect", or "the power behind the throne", sometimes officially as leader of the NIF and sometimes as speaker of the parliamentary assembly.

1989 coup
While there is a "pervasive belief" in Sudan that Turabi and the NIF actively collaborated with the coup-makers who called themselves the "Revolutionary Command Council for National Salvation", in fact the RCC-NS banned all political parties following the 1989 coup and arrested Turabi, as well as the leaders of other political parties, and held him in solitary confinement for several months. Before long however, NIF influence within the government was evident in its policies and in the presence of several NIF members in the cabinet.

Alleged abuses
Alleged human rights abuses by the NIF regime included war crimes, ethnic cleansing, a revival of slavery, torture of opponents, and an unprecedented number of refugees fleeing into Uganda, Kenya, Eritrea, Egypt, Europe and North America.

In 1994 a report issued by Human Rights Watch/Africa, conducted by Gáspár Bíró, a Hungarian law professor and the United Nations' special envoy to Sudan in 1993 found the Sudanese government to be practicing "widespread and systematic torture" of political detainees."Once uncommon in the Sudan, torture was now widespread, especially in the south. Non-Muslim women were raped, their children taken from them; paper bags filled with chili powder were placed over men's heads, and some were tied to anthills; testicles were crushed and burned by cigarettes and electric current, according to a 1994 report by Human Rights Watch/Africa."

Karate attack on al-Turabi in Ottawa
On 26 May 1992 al-Turabi was attacked at the Ottawa Macdonald–Cartier International Airport in Canada by Sudanese Karate Black belt master, Hashim Bedreddin Mohammed. He attacked al-Turabi by using two knifehand strikes to knock down Turabi, whose head struck the pavement putting him into a coma. Hashim was a Sufi in exile and an opponent of the National Islamic Front Islamist regime in Sudan and had won a karate world championship in 1983. He attacked al-Turabi in a rage when he saw him. A Somali man pushed Hashim away with a trolley which weakened the blow that could have otherwise killed al-Turabi. al-Turabi was hospitalized for 4 weeks with constant black outs. After al-Turabi was knocked out, Hashim assumed he was dead and departed. al-Turabi suffered from severe injuries, the use of his right arm was lost for a while, he had slurred speech and required the use of a cane. Hashim was supported by exiled Sudanese in Canada who launched the "Friends of Hashim Campaign" to support his attack on al-Turabi. One said "most Sudanese would appreciate what Hashim did". They called for an end to Islamist policies and a return to secularism.

Links to militant groups
Al-Qaeda's leader Osama bin Laden moved his base of operations from Saudi Arabia to Sudan around 1991 reportedly at the personal invitation of al-Turabi, and stayed until 1996 when he moved to Afghanistan. Bin Laden moved to Sudan after conflict with the Saudi government over their granting of permission to the United States to station troops in Saudi Arabia during the Gulf War against Saddam Hussein. Bin Laden believed he should lead the fight against Saddam using Afghan Arab forces. al-Turabi granted bin Laden a safe place from which to conduct jihadist activities; in return, bin Laden agreed to help the Sudanese government in roadbuilding and to fight animist and Christian separatists in Southern Sudan. While in Sudan, bin Laden is reported to have married one of al-Turabi's nieces.

Other violent groups al-Turabi invited and allowed to operate freely included Abu Nidal Organization, (which reportedly had killed more than 900 people in 20 different countries) Hezbollah, and Carlos the Jackal, (posing as a French arms dealer at the time). Carlos had converted from Marxism to Islamic extremism. (Sudanese sanctuary was not unconditional as it later allowed French intelligence to kidnap Carlos the Jackal while he was undergoing an operation on his right testicle.")

al-Turabi founded the Popular Arab and Islamic Congress in 1990–1991. Meeting here were several Islamic groups from around the world, including representatives from the Palestine Liberation Organization, Hamas, the Egyptian Islamic Jihad, the Algerian Islamic Jihad, and Hezbollah. In late 1991 or 1992 al-Turabi sought to persuade Shiites and Sunnis to put aside their divisions and join against the common enemy.

In August 1993, Sudan was placed on the United States' list of State Sponsors of Terrorism following the 1993 World Trade Center bombing in February. The U.S. State Department notes that "five of 15 suspects arrested" following the bombing were Sudanese.

Mubarak assassination attempt
Two years later an assassination attempt was made on then Egyptian President Hosni Mubarak by Egyptian Islamic Jihad organization, many of whose members were living in exile in Sudan. Evidence from the Egyptian and Ethiopian governments implicated the Sudanese government "The debacle led to a unanimous vote in the United Nations to impose stiff economic sanctions on Sudan. The Sudanese representative denied the charges, but the Sudanese delegation was already in disfavor, having been implicated only two years earlier in a plot to blow up UN headquarters."

Rather than disassociate himself from the plot, al-Turabi praised the attempted killing and called Mubarak stupid:"The sons of the Prophet Moses, the Muslims, rose up against him confounded his plans, and sent him back to his country...I found the man to be very far below my level of thinking and my view, and too stupid to understand my pronouncements."

Decline of influence
International sanctions took effect in April 1996 and were accompanied by a "general withdrawal of the diplomatic community" from Khartoum. At the same time Sudan worked to appease the United States and other international critics by expelling members of the Egyptian Islamic Jihad and encouraging bin Laden to leave.

In March 1996, national elections were held for the first time since the coup, and al-Turabi was elected to a seat in the National Assembly, where he served as speaker during the 1990s. This was his first instance of holding a political position with some consistency. During the "last few years of the 1990s", his influence and that of his party's "'internationalist' and ideological wing" waned "in favor of the 'nationalist' or more pragmatic leaders who focus on trying to recover from Sudan's disastrous international isolation and economic damage that resulted from ideological adventurism".

Imprisonment and later years
After a political falling out with President Omar al-Bashir in 1999, al-Turabi was imprisoned based on allegations of conspiracy before being released in October 2003. Arrested again in 2004 for allegedly being part of a plot to overthrow the government, he was released on 28 June 2005.

In 2004 he was reported to have been associated with the Justice and Equality Movement (JEM), an Islamist armed rebel group involved in the War in Darfur, whose leader, Khalil Ibrahim used to be one of Mr Turabi's followers. al-Turabi himself has denied any association. Turabi spent 16 months in prison from 2004 until 28 June 2005, during which time he spent several weeks on a hunger strike.

After the JEM attacked Khartoum and Omdurman on 10 May 2008, al-Turabi was arrested on the morning of 12 May 2008, along with other members of his Popular Congress Party (PCP). He said that he had expected the arrest, which occurred while he was returning to Khartoum from a PCP gathering in Sennar. He was questioned and released without charge later in the day, after about 12 hours in detention.

Presidential advisor Mustafa Osman Ismail said that al-Turabi's name had been found on JEM documents, but he denied that al-Turabi had been arrested, asserting that he had merely been "summoned" for questioning. al-Turabi, however, said that it was an arrest and that he had been held at Kobar Prison. According to al-Turabi, he was questioned regarding the relationship between the PCP and JEM, but he did not answer this question, although he denied that there was a relationship after his release; he also said that he was asked why he did not condemn the rebel attack. He said that the security officers questioning him had "terrified" him and that, although they claimed to have proof against him, they did not show him this proof when he asked to see it.

Salva Kiir Mayardit, the First Vice-President of Sudan and President of the Government of Southern Sudan, said that there had been no discussion about arresting al-Turabi at a presidency meeting on the previous day and that there was no security report implicating him. He alleged that al-Turabi was being used as a scapegoat.

In an interview on 17 May 2008, al-Turabi described the JEM's attack on Khartoum as "positive" and said that there was "so much misery in Darfur, genocidal measures actually". He also said that the JEM attack could spark more unrest.

On 12 January 2009, al-Turabi called on Bashir to surrender himself to the International Criminal Court for the sake of the country, while holding Bashir politically responsible for war crimes in Darfur. (He later changed his position and opposed Bashir turning himself in. ) He was then arrested on 14 January and held in prison for two months (until 8 March) at the Kobar Prison before being moved to Port Sudan prison. During this time members of his family expressed concern about his health and his being held in solitary confinement at least some of the time. Amnesty International also released a statement about al-Turabi's arrest on 16 January, describing it as "arbitrary" and politically motivated. Noting al-Turabi's advanced age and his need for medication and a special diet. The Sudanese Media Centre reported on 19 January that al-Turabi would be put on trial for his alleged assistance to the JEM.

On 8 March, he was released only days after the International Criminal Court issued an arrest warrant against Omar al-Bashir. On 11 April 2009, the PCP called for the creation of a transitional government to lead Sudan to the planned 2010 election, and al-Turabi suggested that he would not stand as a candidate due to his advanced age; he emphasized the importance of leadership coming from younger generations and said that he did not have enough energy to run. In April al-Turabi was stopped at Khartoum International Airport and prevented from travelling to Paris for medical tests despite having obtained permission to travel from the interior ministry.

al-Turabi announced on 2 January 2010 that the PCP had designated his deputy, Abdallah Deng Nhial, as its candidate for the 2010 presidential election. al-Turabi was again arrested in mid May 2010, but was released on 1 July 2010. On 18 January 2011, security forces arrested al-Turabi in Khartoum, presumably due to the recent instability of Sudan’s politics. al-Turabi commented on the recent price rises in Sudan stating it could result at a "popular uprising" if the unrealistic rises were not reversed. He added that the government should take lessons from the recent Tunisian Revolution.

Death
On 5 March 2016, Turabi died at a hospital in Khartoum at the age of 84. He was believed to have died of a heart attack. Turabi's funeral was held the next day, with several thousand mourners in attendance, and he was buried at Burri Al-Lamab, a cemetery in eastern Khartoum.

title: Hassan al-Turabi


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Talk:Audioslave

Audioslave in Cuba
Audioslave was the FIRST band to visit Cuba - not the second band. The were there probably 10 years before the Stones. I don't know how to fix this, if someone does, please do so They were there in 2005.Joanniegiannettidemko (talk) 02:20, 6 March 2018 (UTC)

Supergroup term misuse
This is not a supergroup. A popular band got a new singer and changed their name. That new singer was from a different popular band that was on hiatus. That’s a fairly common occurrence.

Good examples of a supergroup would be Cream, Blind Faith, and The Traveling Wilburburys. In those bands, the members had successful careers in different bands or as solo artists. 2600:1011:B042:4D75:F561:E652:557F:BBDD (talk) 19:06, 6 April 2023 (UTC)

 * Audioslave was an entirely new band formed by the singer of Soundgarden and the rhythm section of Rage Against the Machine. RATM did not "change their name", nor did Soundgarden. Both bands did not go on hiatus either; they both disbanded prior to Audioslave's formation, and reformed after Audioslave's disbandment. Thusly, the term 'supergroup' applies to Audioslave. HPSR (talk) 19:30, 6 April 2023 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:Audioslave


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User talk:Netsrik

Hello and welcome to Wikipedia! Hope you like it here, and stick around.

Here are some tips to help you get started:

 * Try the Tutorial, and feel free to experiment in the test area.
 * If you need help, post a question at the Help Desk
 * Eventually, you might want to read the Manual of Style and Policies and Guidelines.
 * Remember Neutral point of view
 * Explore, be bold in editing pages, and, most importantly, have fun!

Good luck!

[[User:Meelar|Meelar (talk)]]

P.S. One last helpful hint. To sign your posts like I did above (on talk pages, for example) use the '~' symbol. To insert just your name, type ~~~ (3 tildes), or, to insert your name and timestamp, use ~~~~ (4 tildes).

title: User talk:Netsrik


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.co

.co is the Internet country code top-level domain (ccTLD) assigned to Colombia.

It is administered by .CO Internet S.A.S., a subsidiary of Neustar since 2014. , there were no registration restrictions on second-level .co domains; any individual or entity in the world can register a .co domain.

.CO Internet S.A.S from Bogotá, Colombia, was appointed as the manager for the .co TLD through a public procurement process that took place in early 2009. .CO Internet received the re-delegation approval as the manager of the .co TLD by ICANN on December 9, 2009, and received formal confirmation of the request by the United States Department of Commerce on December 23, 2009.

Google treats .co as a generic top-level domain (gTLD) because "users and website owners frequently see [the domain] as being more generic than country-targeted".

Second-level domain names
When they took over administration of the .CO domain, .CO Internet S.A.S. implemented new domain policies that were more flexible than the historic ones that had been administered by the University of the Andes. The new policies were adjusted to international best practices and defined in consultation with local and international communities. With the new policies, Colombia would be able to sell second-level domain names to the world, such as widgets.co, where previously only third-level domain-names were available, such as widgets.com.co.

Notable single letter .CO domain names that have been allocated include:

On September 15, 2010, .CO Internet S.A.S. had taken registrations for over 500,000 .CO domain names. , more than 1 million .CO domains had been registered by people in over 200 countries and territories worldwide. As of January 2014, that number has grown to over 1.6 million .CO domains registered. As of December 2018, there were 2.2 million.

With respect to search engine optimization, Google confirmed that "it will rank .co domains appropriately if the content is globally targeted".

Summary of policies since 2010

 * Any person or entity in the world can register .co domain names
 * There are no domicile or burdensome documentation requirements
 * Registration period is between 1 and 5 years, subject to renewal
 * Registrants can easily transfer domain names

.CO domains became available via the following timeline:
 * April 1, 2010 – April 20, 2010: Sunrise A allowed registered local trademarks to apply for exact match domains.
 * April 26, 2010 – June 10, 2010: Sunrise B allowed trademarks of national effect to apply for exact match domains.
 * June 21, 2010 – July 13, 2010: Landrush allowed anyone to apply for domain names of high commercial value.
 * July 20, 2010: .co domains became generally available.

Third-level domain registrations
The third-level domain registrations closely mirror the "traditional" IANA .com/.net/.org/.gov/.edu/.mil-hierarchy, with the addition of a national equivalent of .name. Different from registrations directly under .co, which are used to signal globally-relevant interests, third-level domains are used to signal locally-relevant business, organizations, academic institutions, and government.

 * com.co – commercial
 * org.co – organizations
 * edu.co – educational
 * gov.co – government
 * net.co – network infrastructure
 * mil.co – military
 * nom.co – private person

History
IANA delegates ISO 3166-1 alpha-2 codes as country code top-level domains, and on December 24, 1991, the .co top-level domain was assigned to Colombia and delegated to the Universidad de los Andes.

In 2001, the university began to consider the possibility of marketing the domain as an alternative to the generic top-level domains. The government of Colombia objected on the basis that the university, a private entity, did not have regulatory oversight of the TLD and the Minister of Communications, Angela Montoya Holguín, wrote to them requesting that they not continue. In turn the university wrote to ICANN, rejecting the government's objections and stating their intention to appoint a subcontractor to handle the commercialization of the domain.

At a meeting on December 11, 2001, Holguín asked the Consultative Chamber and Civil Service of the Council of State to consider three issues:

 * 1) whether the .co domain is a public resource
 * 2) if the domain is public resource, whether it is intrinsically-linked with telecommunications
 * 3) if the domain is linked with telecommunications, who should profit from its commercialization

In relation to these three issues, the meeting concluded that:

 * 1) the .co domain, having been assigned to Colombia, is of public interest
 * 2) the administration of the domain is intrinsically-related to telecommunications, and hence falls under the purview of the Ministry of Communications, with the exception of those functions assigned to the ICFES by the Ministry of National Education
 * 3) unless the Congress of Colombia adopts an act allowing tax to be collected in relation to the registration of domain names, no amount can be charged for such a service

In response to the Council of State meeting, the university wrote to ICANN on 12 February 2002 stating that it had abandoned plans to commercialize the domain, and that as it could "no longer bear the administrative and operational responsibilities" it wished to discontinue its responsibility for operating the domain.

Finally, with the enactment of Law 1065 of 2006, the Ministry of Communications of Colombia initiated a public consultation process involving local and international participants, including members of the ICANN community, with the objective of defining the future of the .CO TLD. As a result of that process, through Resolution 001652 of 2008, the Ministry approved new policies that would govern the administration of the .CO TLD. A public procurement process began which resulted in the award of the administration contract to .CO Internet SAS. Finally, on February 7, 2010, the administration of the TLD was transitioned from the University of Andes to .CO Internet SAS, under the regulatory and policy supervision of the Ministry of Communications of Colombia. On July 20, 2010, second level .co domains became available to the rest of the world on a first-come, first-served basis. In 2014, .CO Internet S.A.S was acquired by Neustar for US$109 Million, and became a wholly owned subsidiary of Neustar. It is responsible for the promotion, administration, and technical operation of the .co TLD.

Accredited registrars
Only accredited registrars are able to sell .co domain names directly; other registrars selling .co domain names are acting as resellers. The list of accredited registrars is available on the .CO Internet website, and as of October 2011 there are 20 accredited registrars. Some of the 20 registrars operate under multiple brands.

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Islamic Army in Iraq

The Islamic Army in Iraq (الجيش الإسلامي في العراق, abbr. IAI) was an underground Islamist militant organization formed in Iraq following the 2003 invasion of Iraq by U.S.-led Coalition forces, and the subsequent collapse of the Ba'athist regime headed by Saddam Hussein. IAI was regarded as one of the largest, sophisticated and most influential Sunni insurgent groups in Iraq that led an asymmetrical military insurgency against Coalition forces. The group became known for its grisly videos of kidnappings and attacks on U.S. and Iraqi troops.

Although it carries an Islamic title, the group combines Sunni Islamism with Iraqi nationalism, and has been labelled as "resistance" by Iraq's Sunni Vice president Tariq al-Hashimi (who was sentenced to death in 2012) despite al-Hashimi's close relations with the U.S. government.

Following the withdrawal of American forces from Iraq in late 2011, the IAI largely demobilized and turned towards political activism, setting up the Sunni Popular Movement. The group’s turn away from armed opposition towards activism was criticised by other militant groups, including groups that the IAI had previously allied with such as the Mujahideen Army.

In the beginning of 2014, however, the group returned to armed militancy and was active anti-government violence in Anbar and Northern Iraq during the first phases of the War in Iraq (2013-17). The group was primarily active in the Diyala and Saladin Governorates. Most of its fighters have renounced fighting against the Iraqi state, although some have joined ISIS. Islamic Army in Iraq has not claimed any attacks since late 2014.

Roots and ideology
The precise details about the emergence of the IAI are unclear, although it is generally assumed that the group was established in the late summer of 2003 to fight and expel Coalition forces from Iraq. Former officers of Saddam Hussein's army from Sunni strongholds such as Ramadi, Fallujah, Tikrit and Baqubah who were skilled ex-soldiers from the disbanded Special Republican Guard, Fedayeen Saddam, and the Iraqi Intelligence Service formed and joined the Islamic Army in Iraq (IAI).

When the IAI first formed, it used kidnapping as a means of pursuing its goals. The group also threatened to target the January 2005 elections, although it didn't carry out any such attack. Unlike most terrorist organizations today, the IAI does not have Salafist tendencies, its primary focus and goal being the expulsion of foreign troops from Iraq. A November 2004 Washington Post interview with the group's leader, Ishmael Jubouri, stated that the IAI was predominantly composed of Iraqis (Sunnis, Shias, Kurds, and Arabs) trying to force foreign troops out of Iraq. The Terrorism Monitor put out by The Jamestown Foundation confirms some of what Jubouri was claiming. In a March 2005 article, the monitor said the group was composed primarily of Sunnis with a small Shiite congregation and, in general, was "[an] inclusive Islamic organization with Iraqi nationalist tendencies."

In a November 2006 Al Jazeera interview, spokesman Ibrahim al-Shamary expanded on who the IAI considers foreign troops, "There are two occupations in Iraq. Iran on one side through the militias which they control and through direct involvement with the national guard and the intelligence services, that causes the killing and destruction of the Sunnis. ... And then there is the American occupation which destroys the Iraqi people."

The group has released several joint statements with other groups such as Islamic Resistance Movement and the Islamic Front for the Iraqi Resistance, which are known to be of an ikhwan background. In one of these joint statements, six groups (including the IAI) called for Iraqis to participate in the referendum on the October 2005 constitution by voting against it. (This was in conspicuous contrast to al-Qaeda in Iraq, which said that simply participating in voting is a compromise of the fundamentals of Islam, even if one were to vote against it.)

When rumours spread in Iraq of the alleged demolition of the al-Aqsa Mosque, in April 2005, the IAI announced the formation of the "al-Aqsa Support Division." This group was to support Palestine in the armed struggle against Israel. The current status of the al-Aqsa Support Division is unknown, leading people to believe that the statement was merely rhetoric.

The group has shown support for the Free Syrian Army (FSA) and its fight against the Syrian government and allied Shiite paramilitary groups like Hezbollah. In June 2013, the IAI released a statement advising the FSA in methods in fighting.

Foreign hostages
The group was responsible for the abduction of the following people who were released unharmed:
 * Fereidoun Jahani; Iranian Consul.
 * Georges Malbrunot (41) and Christian Chesnot (37); French journalists.
 * Marwan Ibrahim al-Kassar and Mohammed Jawdat Hussein; Lebanese electrical workers.
 * Angelo dela Cruz; Filipino truck driver.
 * Rosidah Anom and Rafikan Binti Amin; female Indonesian nationals.

The IAI is also believed responsible for the execution of the following foreigners:
 * Enzo Baldoni; Italian journalist killed on or about August 26, 2004.
 * Raja Azad (49), engineer, and Sajad Naeem (29), his driver; Pakistani nationals working in Iraq for a Kuwaiti-based firm killed on or about July 28, 2004.
 * Dalibor Lazarevski, Dragan Marković, and Zoran Naskovski; nationals of Republic of Macedonia, working for United Arab Emirates-based Soufan Engineering on contracts and subcontracts for the U.S. military and its private contractors. The three were seized in August 2004 and the Macedonian government confirmed their execution by October 21, 2004; receipt of videos depicting two beheadings were announced, but not broadcast, on al-Jazeera TV on October 17, 2004.
 * Ronald Schulz; American contract electrician, killed around December 8, 2005.

Other activities
On 22 November 2003, the Islamic Army in Iraq organized and videotaped a shoulder-fired surface-to-air missile attack on a DHL Airbus A300 leaving from Baghdad International Airport.

The Islamic Army in Iraq claimed responsibility for the 1 September 2004, assassination attempt against Iraqi politician Ahmed Chalabi, leader of the Iraqi National Congress, in which two of his bodyguards were killed, two were wounded and two went missing (the IAI admitted capturing one of Chalabi's bodyguards and executing the other), and Chalabi escaped unharmed.

On 24 March 2005, the IAI claimed responsibility for a vehicle-borne improvised explosive device (VBIED) that detonated at a city entrance checkpoint in Ramadi. The attack killed 2 U.S. Marines, 11 Iraqi police commandos and wounded 3 Iraqi civilians.

On 22 April 2005, the IAI released a video of their members executing Lyubomir Kostov, a Bulgarian civilian contractor, who survived after the downing of his helicopter. He was helped to his feet and then shot with 27 rounds of ammunition. The group also claims to have shot down a commercial airliner in Iraq, although officials maintain the accident was caused by fog. The crash killed 34 people.

In 2006, videos were released of their snipers killing Coalition forces. The alleged name of the IAI sniper is "Juba". These sniper videos were distributed for free to Iraqi citizens on CDs as part of a propaganda, recruiting campaign and as a means of waging psychological warfare on Coalition forces. Islamic Army videos of attacks on U.S.-led Coalition forces were aired on Al-Zawraa TV channel, which is now banned in Iraq.

As of 2011, the group's content was distributed online by the Jihad Media Battalion (subtitled in English) and the Media Division of the Islamic Army in Iraq (subtitled in Arabic). These groups were considered distinct from al-Qaeda and the linked groups As-Sahab, Ansar ul-Mujahideen, and al-Fida Islamic Network and also distinct from the Global Islamic Media Front, Islamic Media Center, and the media center of the Islamic Jihad Army.

In the beginning of 2014, however, the group was active in the anti-government violence in Anbar and Northern Iraq and after the outbreak of the War in Iraq (2013-17). Most of its fighters have renounced fighting against the Iraqi state, although some have joined ISIS. Islamic Army in Iraq has not claimed any attacks since late 2014.

War with al-Qaeda in Iraq
In early 2007, the Islamic Army engaged in an armed conflict against Al-Qaeda in Iraq. In June, this ended in a ceasefire between the two rival groups. The IAI was quoted saying "The most important thing is that it's our common duty to fight the Americans;" nevertheless, the groups never adopted al-Qaeda's philosophy and refused to sign on to the al-Qaeda-led Islamic State of Iraq.

According to Iraqi sources, fighters from the Islamic Army battled Al-Qaeda gunmen around Samarra at least twice in October and November 2007, a possible indication that the cease-fire brokered earlier this year had collapsed (however, coalition officials later issued a statement claiming that Iraqi policemen and coalition troops, not Islamic Army fighters, had carried out the latter operation). Furthermore, although the Islamic Army denied that it had joined forces with the U.S. military, several news outlets reported that many Islamic Army commanders in and around Baghdad were now working together with the U.S.-led coalition to counter Al-Qaeda in Iraq.

title: Islamic Army in Iraq


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Reductive group

In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group G over a perfect field is reductive if it has a representation that has a finite kernel and is a direct sum of irreducible representations. Reductive groups include some of the most important groups in mathematics, such as the general linear group GL(n) of invertible matrices, the special orthogonal group SO(n), and the symplectic group Sp(2n). Simple algebraic groups and (more generally) semisimple algebraic groups are reductive.

Claude Chevalley showed that the classification of reductive groups is the same over any algebraically closed field. In particular, the simple algebraic groups are classified by Dynkin diagrams, as in the theory of compact Lie groups or complex semisimple Lie algebras. Reductive groups over an arbitrary field are harder to classify, but for many fields such as the real numbers R or a number field, the classification is well understood. The classification of finite simple groups says that most finite simple groups arise as the group G(k) of k-rational points of a simple algebraic group G over a finite field k, or as minor variants of that construction.

Reductive groups have a rich representation theory in various contexts. First, one can study the representations of a reductive group G over a field k as an algebraic group, which are actions of G on k-vector spaces. But also, one can study the complex representations of the group G(k) when k is a finite field, or the infinite-dimensional unitary representations of a real reductive group, or the automorphic representations of an adelic algebraic group. The structure theory of reductive groups is used in all these areas.

Definitions
A linear algebraic group over a field k is defined as a smooth closed subgroup scheme of GL(n) over k, for some positive integer n. Equivalently, a linear algebraic group over k is a smooth affine group scheme over k.

With the unipotent radical
A connected linear algebraic group $$G$$ over an algebraically closed field is called semisimple if every smooth connected solvable normal subgroup of $$G$$ is trivial. More generally, a connected linear algebraic group $$G$$ over an algebraically closed field is called reductive if the largest smooth connected unipotent normal subgroup of $$G$$ is trivial. This normal subgroup is called the unipotent radical and is denoted $$R_u(G)$$. (Some authors do not require reductive groups to be connected.) A group $$G$$ over an arbitrary field k is called semisimple or reductive if the base change $$G_{\overline k}$$ is semisimple or reductive, where $$\overline k$$ is an algebraic closure of k. (This is equivalent to the definition of reductive groups in the introduction when k is perfect. ) Any torus over k, such as the multiplicative group Gm, is reductive.

With representation theory
Over fields of characteristic zero another equivalent definition of a reductive group is a connected group $$G$$ admitting a faithful semisimple representation which remains semisimple over its algebraic closure $$k^{al}$$ page 424.

Simple reductive groups
A linear algebraic group G over a field k is called simple (or k-simple) if it is semisimple, nontrivial, and every smooth connected normal subgroup of G over k is trivial or equal to G. (Some authors call this property "almost simple".) This differs slightly from the terminology for abstract groups, in that a simple algebraic group may have nontrivial center (although the center must be finite). For example, for any integer n at least 2 and any field k, the group SL(n) over k is simple, and its center is the group scheme μn of nth roots of unity.

A central isogeny of reductive groups is a surjective homomorphism with kernel a finite central subgroup scheme. Every reductive group over a field admits a central isogeny from the product of a torus and some simple groups. For example, over any field k,
 * $$GL(n)\cong (G_m\times SL(n))/\mu_n.$$

It is slightly awkward that the definition of a reductive group over a field involves passage to the algebraic closure. For a perfect field k, that can be avoided: a linear algebraic group G over k is reductive if and only if every smooth connected unipotent normal k-subgroup of G is trivial. For an arbitrary field, the latter property defines a pseudo-reductive group, which is somewhat more general.

Split-reductive groups
A reductive group G over a field k is called split if it contains a split maximal torus T over k (that is, a split torus in G whose base change to $$\overline k$$ is a maximal torus in $$G_{\overline k}$$). It is equivalent to say that T is a split torus in G that is maximal among all k-tori in G. These kinds of groups are useful because their classification can be described through combinatorical data called root data.

GLn and SLn
A fundamental example of a reductive group is the general linear group $$\text{GL}_n$$ of invertible n × n matrices over a field k, for a natural number n. In particular, the multiplicative group Gm is the group GL(1), and so its group Gm(k) of k-rational points is the group k* of nonzero elements of k under multiplication. Another reductive group is the special linear group SL(n) over a field k, the subgroup of matrices with determinant 1. In fact, SL(n) is a simple algebraic group for n at least 2.

O(n), SO(n), and Sp(n)
An important simple group is the symplectic group Sp(2n) over a field k, the subgroup of GL(2n) that preserves a nondegenerate alternating bilinear form on the vector space k2n. Likewise, the orthogonal group O(q) is the subgroup of the general linear group that preserves a nondegenerate quadratic form q on a vector space over a field k. The algebraic group O(q) has two connected components, and its identity component SO(q) is reductive, in fact simple for q of dimension n at least 3. (For k of characteristic 2 and n odd, the group scheme O(q) is in fact connected but not smooth over k. The simple group SO(q) can always be defined as the maximal smooth connected subgroup of O(q) over k.) When k is algebraically closed, any two (nondegenerate) quadratic forms of the same dimension are isomorphic, and so it is reasonable to call this group SO(n). For a general field k, different quadratic forms of dimension n can yield non-isomorphic simple groups SO(q) over k, although they all have the same base change to the algebraic closure $$\overline k$$.

Tori
The group $$\mathbb{G}_m$$ and products of it are called the algebraic tori. They are examples of reductive groups since they embed in $$\text{GL}_n$$ through the diagonal, and from this representation, their unipotent radical is trivial. For example, $$\mathbb{G}_m\times \mathbb {G}_m$$ embeds in $$\text{GL}_2$$ from the map $$(a_1,a_2) \mapsto \begin{bmatrix} a_1 & 0 \\ 0 & a_2 \end{bmatrix}.$$

Non-examples

 * Any unipotent group is not reductive since its unipotent radical is itself. This includes the additive group $$\mathbb{G}_a$$.
 * The Borel group $$B_n$$ of $$\text{GL}_n$$ has a non-trivial unipotent radical $$\mathbb{U}_n$$ of upper-triangular matrices with $$1$$ on the diagonal. This is an example of a non-reductive group which is not unipotent.

Associated reductive group
Note that the normality of the unipotent radical $$R_u(G)$$ implies that the quotient group $$G/R_u(G)$$ is reductive. For example,"$B_n/(R_u(B_n)) \cong \prod^n_{i=1} \mathbb{G}_m.$"

Other characterizations of reductive groups
Every compact connected Lie group has a complexification, which is a complex reductive algebraic group. In fact, this construction gives a one-to-one correspondence between compact connected Lie groups and complex reductive groups, up to isomorphism. For a compact Lie group K with complexification G, the inclusion from K into the complex reductive group G(C) is a homotopy equivalence, with respect to the classical topology on G(C). For example, the inclusion from the unitary group U(n) to GL(n,C) is a homotopy equivalence.

For a reductive group G over a field of characteristic zero, all finite-dimensional representations of G (as an algebraic group) are completely reducible, that is, they are direct sums of irreducible representations. That is the source of the name "reductive". Note, however, that complete reducibility fails for reductive groups in positive characteristic (apart from tori). In more detail: an affine group scheme G of finite type over a field k is called linearly reductive if its finite-dimensional representations are completely reducible. For k of characteristic zero, G is linearly reductive if and only if the identity component Go of G is reductive. For k of characteristic p>0, however, Masayoshi Nagata showed that G is linearly reductive if and only if Go is of multiplicative type and G/Go has order prime to p.

Roots
The classification of reductive algebraic groups is in terms of the associated root system, as in the theories of complex semisimple Lie algebras or compact Lie groups. Here is the way roots appear for reductive groups.

Let G be a split reductive group over a field k, and let T be a split maximal torus in G; so T is isomorphic to (Gm)n for some n, with n called the rank of G. Every representation of T (as an algebraic group) is a direct sum of 1-dimensional representations. A weight for G means an isomorphism class of 1-dimensional representations of T, or equivalently a homomorphism T → Gm. The weights form a group X(T) under tensor product of representations, with X(T) isomorphic to the product of n copies of the integers, Zn.

The adjoint representation is the action of G by conjugation on its Lie algebra $$\mathfrak g$$. A root of G means a nonzero weight that occurs in the action of T ⊂ G on $$\mathfrak g$$. The subspace of $$\mathfrak g$$ corresponding to each root is 1-dimensional, and the subspace of $$\mathfrak g$$ fixed by T is exactly the Lie algebra $$\mathfrak t$$ of T. Therefore, the Lie algebra of G decomposes into $$\mathfrak t$$ together with 1-dimensional subspaces indexed by the set Φ of roots:
 * $${\mathfrak g} = {\mathfrak t}\oplus \bigoplus_{\alpha\in\Phi} {\mathfrak g}_{\alpha}.$$

For example, when G is the group GL(n), its Lie algebra $${\mathfrak gl}(n)$$ is the vector space of all n × n matrices over k. Let T be the subgroup of diagonal matrices in G. Then the root-space decomposition expresses $${\mathfrak gl}(n)$$ as the direct sum of the diagonal matrices and the 1-dimensional subspaces indexed by the off-diagonal positions (i, j). Writing L1,...,Ln for the standard basis for the weight lattice X(T) ≅ Zn, the roots are the elements Li − Lj for all i ≠ j from 1 to n.

The roots of a semisimple group form a root system; this is a combinatorial structure which can be completely classified. More generally, the roots of a reductive group form a root datum, a slight variation. The Weyl group of a reductive group G means the quotient group of the normalizer of a maximal torus by the torus, W = NG(T)/T. The Weyl group is in fact a finite group generated by reflections. For example, for the group GL(n) (or SL(n)), the Weyl group is the symmetric group Sn.

There are finitely many Borel subgroups containing a given maximal torus, and they are permuted simply transitively by the Weyl group (acting by conjugation). A choice of Borel subgroup determines a set of positive roots Φ+ ⊂ Φ, with the property that Φ is the disjoint union of Φ+ and −Φ+. Explicitly, the Lie algebra of B is the direct sum of the Lie algebra of T and the positive root spaces:
 * $${\mathfrak b}={\mathfrak t}\oplus \bigoplus_{\alpha\in\Phi^{+}} {\mathfrak g}_{\alpha}.$$

For example, if B is the Borel subgroup of upper-triangular matrices in GL(n), then this is the obvious decomposition of the subspace $$\mathfrak b$$ of upper-triangular matrices in $${\mathfrak gl}(n)$$. The positive roots are Li − Lj for 1 ≤ i < j ≤ n.

A simple root means a positive root that is not a sum of two other positive roots. Write Δ for the set of simple roots. The number r of simple roots is equal to the rank of the commutator subgroup of G, called the semisimple rank of G (which is simply the rank of G if G is semisimple). For example, the simple roots for GL(n) (or SL(n)) are Li − Li+1 for 1 ≤ i ≤ n − 1.

Root systems are classified by the corresponding Dynkin diagram, which is a finite graph (with some edges directed or multiple). The set of vertices of the Dynkin diagram is the set of simple roots. In short, the Dynkin diagram describes the angles between the simple roots and their relative lengths, with respect to a Weyl group-invariant inner product on the weight lattice. The connected Dynkin diagrams (corresponding to simple groups) are pictured below.

For a split reductive group G over a field k, an important point is that a root α determines not just a 1-dimensional subspace of the Lie algebra of G, but also a copy of the additive group Ga in G with the given Lie algebra, called a root subgroup Uα. The root subgroup is the unique copy of the additive group in G which is normalized by T and which has the given Lie algebra. The whole group G is generated (as an algebraic group) by T and the root subgroups, while the Borel subgroup B is generated by T and the positive root subgroups. In fact, a split semisimple group G is generated by the root subgroups alone.

Parabolic subgroups
For a split reductive group G over a field k, the smooth connected subgroups of G that contain a given Borel subgroup B of G are in one-to-one correspondence with the subsets of the set Δ of simple roots (or equivalently, the subsets of the set of vertices of the Dynkin diagram). Let r be the order of Δ, the semisimple rank of G. Every parabolic subgroup of G is conjugate to a subgroup containing B by some element of G(k). As a result, there are exactly 2r conjugacy classes of parabolic subgroups in G over k. Explicitly, the parabolic subgroup corresponding to a given subset S of Δ is the group generated by B together with the root subgroups U−α for α in S. For example, the parabolic subgroups of GL(n) that contain the Borel subgroup B above are the groups of invertible matrices with zero entries below a given set of squares along the diagonal, such as:
 * $$\left \{ \begin{bmatrix}

* & * & * & *\\ * & * & * & *\\ 0 & 0 & * & *\\ 0 & 0 & 0 & * \end{bmatrix} \right \}$$

By definition, a parabolic subgroup P of a reductive group G over a field k is a smooth k-subgroup such that the quotient variety G/P is proper over k, or equivalently projective over k. Thus the classification of parabolic subgroups amounts to a classification of the projective homogeneous varieties for G (with smooth stabilizer group; that is no restriction for k of characteristic zero). For GL(n), these are the flag varieties, parametrizing sequences of linear subspaces of given dimensions a1,...,ai contained in a fixed vector space V of dimension n:
 * $$0\subset S_{a_1}\subset \cdots \subset S_{a_i}\subset V.$$

For the orthogonal group or the symplectic group, the projective homogeneous varieties have a similar description as varieties of isotropic flags with respect to a given quadratic form or symplectic form. For any reductive group G with a Borel subgroup B, G/B is called the flag variety or flag manifold of G.

Classification of split reductive groups

Chevalley showed in 1958 that the reductive groups over any algebraically closed field are classified up to isomorphism by root data. In particular, the semisimple groups over an algebraically closed field are classified up to central isogenies by their Dynkin diagram, and the simple groups correspond to the connected diagrams. Thus there are simple groups of types An, Bn, Cn,  Dn,  E6, E7, E8, F4, G2. This result is essentially identical to the classifications of compact Lie groups or complex semisimple Lie algebras, by Wilhelm Killing and Élie Cartan in the 1880s and 1890s. In particular, the dimensions, centers, and other properties of the simple algebraic groups can be read from the list of simple Lie groups. It is remarkable that the classification of reductive groups is independent of the characteristic. For comparison, there are many more simple Lie algebras in positive characteristic than in characteristic zero.

The exceptional groups G of type G2 and E6 had been constructed earlier, at least in the form of the abstract group G(k), by L. E. Dickson. For example, the group G2 is the automorphism group of an octonion algebra over k. By contrast, the Chevalley groups of type F4, E7, E8 over a field of positive characteristic were completely new.

More generally, the classification of split reductive groups is the same over any field. A semisimple group G over a field k is called simply connected if every central isogeny from a semisimple group to G is an isomorphism. (For G semisimple over the complex numbers, being simply connected in this sense is equivalent to G(C) being simply connected in the classical topology.) Chevalley's classification gives that, over any field k, there is a unique simply connected split semisimple group G with a given Dynkin diagram, with simple groups corresponding to the connected diagrams. At the other extreme, a semisimple group is of adjoint type if its center is trivial. The split semisimple groups over k with given Dynkin diagram are exactly the groups G/A, where G is the simply connected group and A is a k-subgroup scheme of the center of G.

For example, the simply connected split simple groups over a field k corresponding to the "classical" Dynkin diagrams are as follows:
 * An: SL(n+1) over k;
 * Bn: the spin group Spin(2n+1) associated to a quadratic form of dimension 2n+1 over k with Witt index n, for example the form
 * $$q(x_1,\ldots,x_{2n+1})=x_1x_2+x_3x_4+\cdots+x_{2n-1}x_{2n}+x_{2n+1}^2;$$

 * Cn: the symplectic group Sp(2n) over k;
 * Dn: the spin group Spin(2n) associated to a quadratic form of dimension 2n over k with Witt index n, which can be written as:
 * $$q(x_1,\ldots,x_{2n})=x_1x_2+x_3x_4+\cdots+x_{2n-1}x_{2n}.$$

The outer automorphism group of a split reductive group G over a field k is isomorphic to the automorphism group of the root datum of G. Moreover, the automorphism group of G splits as a semidirect product:
 * $$\operatorname{Aut}(G)\cong \operatorname{Out}(G)\ltimes (G/Z)(k),$$

where Z is the center of G. For a split semisimple simply connected group G over a field, the outer automorphism group of G has a simpler description: it is the automorphism group of the Dynkin diagram of G.

Reductive group schemes
A group scheme G over a scheme S is called reductive if the morphism G → S is smooth and affine, and every geometric fiber $$G_{\overline k}$$ is reductive. (For a point p in S, the corresponding geometric fiber means the base change of G to an algebraic closure $$\overline k$$ of the residue field of p.) Extending Chevalley's work, Michel Demazure and Grothendieck showed that split reductive group schemes over any nonempty scheme S are classified by root data. This statement includes the existence of Chevalley groups as group schemes over Z, and it says that every split reductive group over a scheme S is isomorphic to the base change of a Chevalley group from Z to S.

Real reductive groups
In the context of Lie groups rather than algebraic groups, a real reductive group is a Lie group G such that there is a linear algebraic group L over R whose identity component (in the Zariski topology) is reductive, and a homomorphism G → L(R) whose kernel is finite and whose image is open in L(R) (in the classical topology). It is also standard to assume that the image of the adjoint representation Ad(G) is contained in Int(gC) = Ad(L0(C)) (which is automatic for G connected).

In particular, every connected semisimple Lie group (meaning that its Lie algebra is semisimple) is reductive. Also, the Lie group R is reductive in this sense, since it can be viewed as the identity component of GL(1,R) ≅ R*. The problem of classifying the real reductive groups largely reduces to classifying the simple Lie groups. These are classified by their Satake diagram; or one can just refer to the list of simple Lie groups (up to finite coverings).

Useful theories of admissible representations and unitary representations have been developed for real reductive groups in this generality. The main differences between this definition and the definition of a reductive algebraic group have to do with the fact that an algebraic group G over R may be connected as an algebraic group while the Lie group G(R) is not connected, and likewise for simply connected groups.

For example, the projective linear group PGL(2) is connected as an algebraic group over any field, but its group of real points PGL(2,R) has two connected components. The identity component of PGL(2,R) (sometimes called PSL(2,R)) is a real reductive group that cannot be viewed as an algebraic group. Similarly, SL(2) is simply connected as an algebraic group over any field, but the Lie group SL(2,R) has fundamental group isomorphic to the integers Z, and so SL(2,R) has nontrivial covering spaces. By definition, all finite coverings of SL(2,R) (such as the metaplectic group) are real reductive groups. On the other hand, the universal cover of SL(2,R) is not a real reductive group, even though its Lie algebra is reductive, that is, the product of a semisimple Lie algebra and an abelian Lie algebra.

For a connected real reductive group G, the quotient manifold G/K of G by a maximal compact subgroup K is a symmetric space of non-compact type. In fact, every symmetric space of non-compact type arises this way. These are central examples in Riemannian geometry of manifolds with nonpositive sectional curvature. For example, SL(2,R)/SO(2) is the hyperbolic plane, and SL(2,C)/SU(2) is hyperbolic 3-space.

For a reductive group G over a field k that is complete with respect to a discrete valuation (such as the p-adic numbers Qp), the affine building X of G plays the role of the symmetric space. Namely, X is a simplicial complex with an action of G(k), and G(k) preserves a CAT(0) metric on X, the analog of a metric with nonpositive curvature. The dimension of the affine building is the k-rank of G. For example, the building of SL(2,Qp) is a tree.

Representations of reductive groups
For a split reductive group G over a field k, the irreducible representations of G (as an algebraic group) are parametrized by the dominant weights, which are defined as the intersection of the weight lattice X(T) ≅ Zn with a convex cone (a Weyl chamber) in Rn. In particular, this parametrization is independent of the characteristic of k. In more detail, fix a split maximal torus and a Borel subgroup, T ⊂ B ⊂ G. Then B is the semidirect product of T with a smooth connected unipotent subgroup U. Define a highest weight vector in a representation V of G over k to be a nonzero vector v such that B maps the line spanned by v into itself. Then B acts on that line through its quotient group T, by some element λ of the weight lattice X(T). Chevalley showed that every irreducible representation of G has a unique highest weight vector up to scalars; the corresponding "highest weight" λ is dominant; and every dominant weight λ is the highest weight of a unique irreducible representation L(λ) of G, up to isomorphism.

There remains the problem of describing the irreducible representation with given highest weight. For k of characteristic zero, there are essentially complete answers. For a dominant weight λ, define the Schur module ∇(λ) as the k-vector space of sections of the G-equivariant line bundle on the flag manifold G/B associated to λ; this is a representation of G. For k of characteristic zero, the Borel–Weil theorem says that the irreducible representation L(λ) is isomorphic to the Schur module ∇(λ). Furthermore, the Weyl character formula gives the character (and in particular the dimension) of this representation.

For a split reductive group G over a field k of positive characteristic, the situation is far more subtle, because representations of G are typically not direct sums of irreducibles. For a dominant weight λ, the irreducible representation L(λ) is the unique simple submodule (the socle) of the Schur module ∇(λ), but it need not be equal to the Schur module. The dimension and character of the Schur module are given by the Weyl character formula (as in characteristic zero), by George Kempf. The dimensions and characters of the irreducible representations L(λ) are in general unknown, although a large body of theory has been developed to analyze these representations. One important result is that the dimension and character of L(λ) are known when the characteristic p of k is much bigger than the Coxeter number of G, by Henning Andersen, Jens Jantzen, and Wolfgang Soergel (proving Lusztig's conjecture in that case). Their character formula for p large is based on the Kazhdan–Lusztig polynomials, which are combinatorially complex. For any prime p, Simon Riche and Geordie Williamson conjectured the irreducible characters of a reductive group in terms of the p-Kazhdan-Lusztig polynomials, which are even more complex, but at least are computable.

Non-split reductive groups
As discussed above, the classification of split reductive groups is the same over any field. By contrast, the classification of arbitrary reductive groups can be hard, depending on the base field. Some examples among the classical groups are:
 * Every nondegenerate quadratic form q over a field k determines a reductive group G = SO(q). Here G is simple if q has dimension n at least 3, since $$G_{\overline k}$$ is isomorphic to SO(n) over an algebraic closure $$\overline k$$. The k-rank of G is equal to the Witt index of q (the maximum dimension of an isotropic subspace over k). So the simple group G is split over k if and only if q has the maximum possible Witt index, $$\lfloor n/2\rfloor$$.
 * Every central simple algebra A over k determines a reductive group G = SL(1,A), the kernel of the reduced norm on the group of units A* (as an algebraic group over k). The degree of A means the square root of the dimension of A as a k-vector space. Here G is simple if A has degree n at least 2, since $$G_{\overline k}$$ is isomorphic to SL(n) over $$\overline k$$. If A has index r (meaning that A is isomorphic to the matrix algebra Mn/r(D) for a division algebra D of degree r over k), then the k-rank of G is (n/r) − 1. So the simple group G is split over k if and only if A is a matrix algebra over k.

As a result, the problem of classifying reductive groups over k essentially includes the problem of classifying all quadratic forms over k or all central simple algebras over k. These problems are easy for k algebraically closed, and they are understood for some other fields such as number fields, but for arbitrary fields there are many open questions.

A reductive group over a field k is called isotropic if it has k-rank greater than 0 (that is, if it contains a nontrivial split torus), and otherwise anisotropic. For a semisimple group G over a field k, the following conditions are equivalent: For k perfect, it is also equivalent to say that G(k) contains a unipotent element other than 1.
 * G is isotropic (that is, G contains a copy of the multiplicative group Gm over k);
 * G contains a parabolic subgroup over k not equal to G;
 * G contains a copy of the additive group Ga over k.

For a connected linear algebraic group G over a local field k of characteristic zero (such as the real numbers), the group G(k) is compact in the classical topology (based on the topology of k) if and only if G is reductive and anisotropic. Example: the orthogonal group SO(p,q) over R has real rank min(p,q), and so it is anisotropic if and only if p or q is zero.

A reductive group G over a field k is called quasi-split if it contains a Borel subgroup over k. A split reductive group is quasi-split. If G is quasi-split over k, then any two Borel subgroups of G are conjugate by some element of G(k). Example: the orthogonal group SO(p,q) over R is split if and only if |p−q| ≤ 1, and it is quasi-split if and only if |p−q| ≤ 2.

Structure of semisimple groups as abstract groups
For a simply connected split semisimple group G over a field k, Robert Steinberg gave an explicit presentation of the abstract group G(k). It is generated by copies of the additive group of k indexed by the roots of G (the root subgroups), with relations determined by the Dynkin diagram of G.

For a simply connected split semisimple group G over a perfect field k, Steinberg also determined the automorphism group of the abstract group G(k). Every automorphism is the product of an inner automorphism, a diagonal automorphism (meaning conjugation by a suitable $$\overline k$$-point of a maximal torus), a graph automorphism (corresponding to an automorphism of the Dynkin diagram), and a field automorphism (coming from an automorphism of the field k).

For a k-simple algebraic group G, Tits's simplicity theorem says that the abstract group G(k) is close to being simple, under mild assumptions. Namely, suppose that G is isotropic over k, and suppose that the field k has at least 4 elements. Let G(k)+ be the subgroup of the abstract group G(k) generated by k-points of copies of the additive group Ga over k contained in G. (By the assumption that G is isotropic over k, the group G(k)+ is nontrivial, and even Zariski dense in G if k is infinite.) Then the quotient group of G(k)+ by its center is simple (as an abstract group). The proof uses Jacques Tits's machinery of BN-pairs.

The exceptions for fields of order 2 or 3 are well understood. For k = F2, Tits's simplicity theorem remains valid except when G is split of type A1, B2, or G2, or non-split (that is, unitary) of type A2. For k = F3, the theorem holds except for G of type A1.

For a k-simple group G, in order to understand the whole group G(k), one can consider the Whitehead group W(k,G)=G(k)/G(k)+. For G simply connected and quasi-split, the Whitehead group is trivial, and so the whole group G(k) is simple modulo its center. More generally, the Kneser–Tits problem asks for which isotropic k-simple groups the Whitehead group is trivial. In all known examples, W(k,G) is abelian.

For an anisotropic k-simple group G, the abstract group G(k) can be far from simple. For example, let D be a division algebra with center a p-adic field k. Suppose that the dimension of D over k is finite and greater than 1. Then G = SL(1,D) is an anisotropic k-simple group. As mentioned above, G(k) is compact in the classical topology. Since it is also totally disconnected, G(k) is a profinite group (but not finite). As a result, G(k) contains infinitely many normal subgroups of finite index.

Lattices and arithmetic groups
Let G be a linear algebraic group over the rational numbers Q. Then G can be extended to an affine group scheme G over Z, and this determines an abstract group G(Z). An arithmetic group means any subgroup of G(Q) that is commensurable with G(Z). (Arithmeticity of a subgroup of G(Q) is independent of the choice of Z-structure.) For example, SL(n,Z) is an arithmetic subgroup of SL(n,Q).

For a Lie group G, a lattice in G means a discrete subgroup Γ of G such that the manifold G/Γ has finite volume (with respect to a G-invariant measure). For example, a discrete subgroup Γ is a lattice if G/Γ is compact. The Margulis arithmeticity theorem says, in particular: for a simple Lie group G of real rank at least 2, every lattice in G is an arithmetic group.

The Galois action on the Dynkin diagram
In seeking to classify reductive groups which need not be split, one step is the Tits index, which reduces the problem to the case of anisotropic groups. This reduction generalizes several fundamental theorems in algebra. For example, Witt's decomposition theorem says that a nondegenerate quadratic form over a field is determined up to isomorphism by its Witt index together with its anisotropic kernel. Likewise, the Artin–Wedderburn theorem reduces the classification of central simple algebras over a field to the case of division algebras. Generalizing these results, Tits showed that a reductive group over a field k is determined up to isomorphism by its Tits index together with its anisotropic kernel, an associated anisotropic semisimple k-group.

For a reductive group G over a field k, the absolute Galois group Gal(ks/k) acts (continuously) on the "absolute" Dynkin diagram of G, that is, the Dynkin diagram of G over a separable closure ks (which is also the Dynkin diagram of G over an algebraic closure $${\overline k}$$). The Tits index of G consists of the root datum of Gk s, the Galois action on its Dynkin diagram, and a Galois-invariant subset of the vertices of the Dynkin diagram. Traditionally, the Tits index is drawn by circling the Galois orbits in the given subset.

There is a full classification of quasi-split groups in these terms. Namely, for each action of the absolute Galois group of a field k on a Dynkin diagram, there is a unique simply connected semisimple quasi-split group H over k with the given action. (For a quasi-split group, every Galois orbit in the Dynkin diagram is circled.) Moreover, any other simply connected semisimple group G over k with the given action is an inner form of the quasi-split group H, meaning that G is the group associated to an element of the Galois cohomology set H1(k,H/Z), where Z is the center of H. In other words, G is the twist of H associated to some H/Z-torsor over k, as discussed in the next section.

Example: Let q be a nondegenerate quadratic form of even dimension 2n over a field k of characteristic not 2, with n ≥ 5. (These restrictions can be avoided.) Let G be the simple group SO(q) over k. The absolute Dynkin diagram of G is of type Dn, and so its automorphism group is of order 2, switching the two "legs" of the Dn diagram. The action of the absolute Galois group of k on the Dynkin diagram is trivial if and only if the signed discriminant d of q in k*/(k*)2 is trivial. If d is nontrivial, then it is encoded in the Galois action on the Dynkin diagram: the index-2 subgroup of the Galois group that acts as the identity is $$\operatorname{Gal}(k_s/k(\sqrt{d}))\subset \operatorname{Gal}(k_s/k)$$. The group G is split if and only if q has Witt index n, the maximum possible, and G is quasi-split if and only if q has Witt index at least n − 1.

Torsors and the Hasse principle
A torsor for an affine group scheme G over a field k means an affine scheme X over k with an action of G such that $$X_{\overline k}$$ is isomorphic to $$G_{\overline k}$$ with the action of $$G_{\overline k}$$ on itself by left translation. A torsor can also be viewed as a principal G-bundle over k with respect to the fppf topology on k, or the étale topology if G is smooth over k. The pointed set of isomorphism classes of G-torsors over k is called H1(k,G), in the language of Galois cohomology.

Torsors arise whenever one seeks to classify forms of a given algebraic object Y over a field k, meaning objects X over k which become isomorphic to Y over the algebraic closure of k. Namely, such forms (up to isomorphism) are in one-to-one correspondence with the set H1(k,Aut(Y)). For example, (nondegenerate) quadratic forms of dimension n over k are classified by H1(k,O(n)), and central simple algebras of degree n over k are classified by H1(k,PGL(n)). Also, k-forms of a given algebraic group G (sometimes called "twists" of G) are classified by H1(k,Aut(G)). These problems motivate the systematic study of G-torsors, especially for reductive groups G.

When possible, one hopes to classify G-torsors using cohomological invariants, which are invariants taking values in Galois cohomology with abelian coefficient groups M, Ha(k,M). In this direction, Steinberg proved Serre's "Conjecture I": for a connected linear algebraic group G over a perfect field of cohomological dimension at most 1, H1(k,G) = 1. (The case of a finite field was known earlier, as Lang's theorem.) It follows, for example, that every reductive group over a finite field is quasi-split.

Serre's Conjecture II predicts that for a simply connected semisimple group G over a field of cohomological dimension at most 2, H1(k,G) = 1. The conjecture is known for a totally imaginary number field (which has cohomological dimension 2). More generally, for any number field k, Martin Kneser, Günter Harder and Vladimir Chernousov (1989) proved the Hasse principle: for a simply connected semisimple group G over k, the map
 * $$H^1(k,G)\to \prod_{v} H^1(k_v,G)$$

is bijective. Here v runs over all places of k, and kv is the corresponding local field (possibly R or C). Moreover, the pointed set H1(kv,G) is trivial for every nonarchimidean local field kv, and so only the real places of k matter. The analogous result for a global field k of positive characteristic was proved earlier by Harder (1975): for every simply connected semisimple group G over k, H1(k,G) is trivial (since k has no real places).

In the slightly different case of an adjoint group G over a number field k, the Hasse principle holds in a weaker form: the natural map
 * $$H^1(k,G)\to \prod_{v} H^1(k_v,G)$$

is injective. For G = PGL(n), this amounts to the Albert–Brauer–Hasse–Noether theorem, saying that a central simple algebra over a number field is determined by its local invariants.

Building on the Hasse principle, the classification of semisimple groups over number fields is well understood. For example, there are exactly three Q-forms of the exceptional group E8, corresponding to the three real forms of E8.

title: Reductive group


text:
User talk:Bmh ca

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ClockworkTroll 04:17, 21 Oct 2004 (UTC)

title: User talk:Bmh ca


text:
User talk:Trace (usurped)

Welcome to the Wikipedia
Welcome, newcomer!

Here are some useful tips to ease you into the Wikipedia experience:

 * First, take a look at the Wikipedia Tutorial, and perhaps dabble a bit in the test area.
 * When you have some free time, take a look at the Manual of Style and Policies and Guidelines. They can come in very handy!
 * Remember to use a neutral point of view!
 * If you need any help, feel free to post a question at the Help Desk
 * Explore, be bold in editing pages, and, most importantly, have fun!

Also, here are some odds and ends that I find useful from time to time:

 * Policy Library
 * Utilities
 * Cite your sources
 * Verifiability
 * Wikiquette
 * Civility
 * Conflict resolution
 * Brilliant prose
 * Pages needing attention
 * Peer review
 * Bad jokes and other deleted nonsense
 * Village pump
 * Boilerplate text

Feel free to ask me anything the links and talk pages don't answer. You can most easily reach me by posting on my talk page.

You can sign your name on any page by typing 4 tildes, likes this: ~~~~.

Best of luck, and have fun!

ClockworkTroll 04:17, 21 Oct 2004 (UTC)

Cuspidal representation
Hi, I've noticed that you have removed the tags from the above mentioned page, calling for cleanup of the article. Please note that the cleanup tags are in no way reflecting your knowledge of the subject or editing ability, they simply suggest that the article be made more user friendly. Myself, I have no knowledge in the subject, so I am unable to preform the edits to the page, but the page does need to be accessable to a broader audience. It currently is a description of a subject, which by itself has no broad explaination of its applications or usages. Also, you did not cite any sources throughout this article. Please find a verifyable book, website, or other published source and cite it as the source for the information on the page. If you need any further editing help, or clarifications on Wikipedia style, feel free to contact me through my talk page. Thanks. --KPWM_Spotter 18:26, 23 July 2006 (UTC)

Rename request
Hi Trace ! I also use the Trace login - on the french wiki and more - and try to have a single unified login, thus I could sign my interwiki edits.

Would you be kind enough to rename your account, which would allow me to use the Trace login on the english wikipedia ?

I would be very grateful if you could enlighten me with your opinion and help. Thank you.

Trace 82.241.15.104 (talk) 20:21, 11 October 2008 (UTC)

See also : Unified login on meta wiki
 * Logged in as Footmark, I confirm I request this renaming. Footmark (talk) 07:59, 13 October 2008 (UTC)

title: User talk:Trace (usurped)