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Starsailor (band)

Starsailor are an English post-Britpop band, formed in 2000. Since their formation, the band have included guitarist and vocalist James Walsh, drummer Ben Byrne, bassist James Stelfox, and keyboardist Barry Westhead. They are best known for their 2003 single "Silence Is Easy", which reached number 9 in the UK, and "Four to the Floor", which reached number one in France as well as the top five in Australia.

The band have released five studio albums and scored ten Top 40 hit singles in the UK. Their first album Love Is Here was released in 2001, followed by Silence Is Easy (2003) and On the Outside (2005). After the release of its fourth album All the Plans (2009), the band entered into extended hiatus until 2014, during which its members were involved in individual projects.

Starsailor announced their reunion on 23 May 2014 and began performing worldwide. Good Souls: The Greatest Hits was released in September 2015 and featured two new songs. A fifth studio album All This Life was released in 2017.

Formation, signed by EMI (1999–2000)
Bassist James Stelfox and drummer Ben Byrne had been playing together in Warrington, Cheshire for a number of years. The band members met whilst studying a Music course at Wigan and Leigh College. When their regular singer fell ill, they recruited young Chorley singer and songwriter James Walsh from a school choir.

The band, then named Waterface, had tried a number of guitarists before they asked long-time friend Barry Westhead to join the band in 2000 on keyboards. He had been teaching judo and playing organ for a church near his home town. His arrival has been heralded as the most significant event in the band's formation. Walsh also took up the guitar, following frustration over not finding a musician right for the group. The band started to build up a reputation, and their name changed to Starsailor after the 1970 album Starsailor by Tim Buckley.

A journalist from NME saw a gig in 2000 and gave the band a glowing review. "One live encounter was enough to convince many sceptics that here was a band who were genuinely special, blessed with a singer whose voice thrummed like an emotional telegraph wire, that swerved the pitfalls of indie melancholia and were clearly in love with rock 'n' roll and all its possibilities." Their performance at the Glastonbury Festival added to the band's reputation and led to a bidding war amongst UK record companies.

A relative of one band member worked for EMI, and the label then signed them in 2000. In 2008, in a re-organisation, Starsailor were moved to Virgin Records, a division of EMI.

Love Is Here, critical acclaim (2000–2002)
"Fever", the band's first single, was released in early 2001. The song, and its two B-Sides "Love Is Here" and "Coming Down" were cut from a demo recording session in mid-2000. All three tracks went on to feature on Starsailor's first album Love Is Here, but were re-recorded.

In March 2001 the band sold out its first UK Tour, which consisted of eleven dates in England. Their second single, "Good Souls", was released in April and featured a cover of Van Morrison's "The Way Young Lovers Do" as a B-Side. During this period, the band were recording their debut album in Rockfield Studios, Wales. By this time, the band were already previewing many of the album's tracks in their concerts, most notably "Poor Misguided Fool", "Lullaby" and "Way To Fall", which would appear as the ending theme of the video game Metal Gear Solid 3: Snake Eater in 2004. An acoustic version of "Alcoholic" appeared on a promotional CD for NME magazine earlier in 2001. The original release date for the album was August 2001.

Further live dates came supporting the Manic Street Preachers in Glasgow, as well as Doves on their North American tour. The band also marked their first festival appearances, playing at events such as V Festival and Germany's Rock im Park. "Alcoholic" appeared as the third single to be taken off their debut album, and its number ten chart position remains the band's second highest placing to date. The single version was actually an extended version of the album track.

Their album Love Is Here reached number 2 in the UK album charts in October 2001, after receiving great critical acclaim. Combining a mix of acoustic guitar and Walsh's vocals, critics said the songs leave the "hairs on the back of your neck standing."

The year ended with the band winning the "Brightest New Hope" award at the NME Awards, and the release of their fourth single, "Lullaby", which reached number thirty-six. Some fans complained about the difficulty in obtaining copies of the single.

Peter Kay joined the band on stage at their 2002 Christmas Concert held in Warrington's Parr Hall.

Silence Is Easy (2002–2005)
For their second album, Silence Is Easy, which was recorded in Los Angeles, Starsailor teamed up with Phil Spector (in what ended up being his final production work before his conviction of murder in 2009 and his death in 2021). The collaboration came about following Spector's daughter Nicole attending one of the band's American concerts in the winter of 2002. Spector was reported to have been fascinated by "Lullaby", the band's fourth single. After meeting the producer, the band agreed to work with him on their second album. However, the collaboration was short-lived; sessions at London's Abbey Road proved difficult. Spector is said to have proved difficult to work with. Only two tracks made the band's second album: the title track, "Silence Is Easy", and "White Dove".

The band co-produced seven of the other tracks with Danton Supple and former The Stone Roses and Radiohead producer John Leckie was brought in to oversee the recording of "Shark Food".

The first single was "Silence Is Easy", which made the Top Ten (No. 9, the band's highest placing). The album spawned just three singles; the second of which, "Born Again" had evolved from a B-side to "Poor Misguided Fool", released in early 2002. The song was re-recorded for the album, and cut down for a radio edit. "Four to the Floor", which was remixed by Thin White Duke, became a popular club hit. Walsh wanted the track to become the band's "I Am the Resurrection", something to be played in "indie discos everywhere".

The album charted well, but sales were sluggish in comparison to the band's first album. The release dropped out of the Top 40 soon thereafter.

A full UK tour began in autumn 2003 shortly after the release of the album, culminating at London's Brixton Academy. The show featured the only performance before their American tour in 2006 of "Restless Heart", the closing track on the Silence Is Easy album. Mark Collins, from The Charlatans joined Starsailor for all dates between August 2003 to December 2004, playing additional and lead guitar.

On the Outside, return to form (2005–2007)
EMI allowed the band plenty of time to record their follow-up album; having targeted producer Rob Schnapf to produce it, the group relocated to Los Angeles to record. Five possible titles appeared; ("Faith, Hope, Love"/"Here I Go"/"Ashes" or "In the Crossfire"/"I Do Not Know"/"Counterfeit Life"), but the band settled for On the Outside, a statement of their position in the music industry. The sound was different from the previous two releases; it was far heavier than its predecessors. Starsailor had always received praise for their live sound, but the releases seemed "puny" in comparison, admitted Stelfox. The recording was all done to tape, and the band did not use editing software such as Pro Tools to "perfect" the recording. Byrne later remarked that his drum track on "White Light" took a long time to nail; highlighting the band's desire to make the record they were truly happy with.

Starsailor released their third album, On the Outside, in the UK on 17 October 2005 from which the first single from the album, "In the Crossfire" was taken. It was released by Artists Addiction Records in the USA on 22 August. Critics raved about the release, with many citing it as a return to form. Despite this, promotional appearances including a short live set and signing at London's HMV Oxford Street did not aid sales – the album charted at number 13.

Since September 2005, Richard Warren joined the band when playing live, aiding Walsh with additional guitar, vocals and also harmonica parts. Again, the band's UK tour ended at Brixton Academy. Following that, the band toured Europe, playing several dates in France, where "Four to the Floor" was a number one hit in 2004. Following the Paris concert on 3 April, Walsh and Warren were the DJ's at the after-show party.

The album has spawned three singles; "This Time" and "Keep Us Together" followed "In The Crossfire". Despite promoting the single with appearances in London and Leeds, "Keep Us Together" was the first single not to reach the UK Top 40. The promotional video for "This Time" appears to be a cultural reference to the film Run Lola Run, but may not be intentional.

Starsailor played numerous festivals in the Summer of 2006; most notably "Hyde Park Calling" on 1 July 2006, alongside Roger Waters, and the V Festival, which is held in Stafford and Chelmsford. They also supported the Rolling Stones on their "Bigger Bang" Tour in Munich, Hamburg in 2006 and again in Frankfurt, Paris and Lyon in 2007.

In the autumn of 2006, Starsailor toured North America, playing both headlining shows and supporting James Blunt, in January 2007 they undertook their first trip to Russia, playing in the B2 Club of Moscow. According to the band's official website Barry Westhead's partner Kelly gave birth to a baby boy on 12 April 2007. The baby was named Joseph. Bass player James Stelfox became a father again in 2008, when his partner gave birth to a girl, named Ella, in November while the rest of the band were doing their first trip to Korea.

Starsailor was named the replacement for Amy Winehouse as the support act for The Rolling Stones on the German leg of their tour. They were opening for the rock legends for the first time in Hamburg. Front man James Walsh told BBC 6 Music: "We heard on Sunday evening, our agent gave us a frantic call, and asked what we were doing this week, fortunately we had a week off!"

All the Plans, beginning of hiatus (2007–2009)
In 2007, Starsailor began recording demos for the follow-up to 2005's On the Outside.

Starsailor started to play in various gigs during the months recording their fourth album. James Walsh also played at several events in 2007, like the SXSW festival in Texas in March 2007, and an acoustic show in London on 12 March. The show is notable for not having a set list; rarities such as the hardly played "Restless Heart" was played, due to a request from the front row. In addition to this he played the song "Tell Me It's Not Over", the track from the work-in-progress fourth album (early arrivals saw Walsh sound check half the song).

Walsh and Stelfox played an unplugged set on 13 April 2007 in Switzerland with Mark Collins of The Charlatans.

Further support in Europe for The Rolling Stones took place in mid-June; these were late additions, and fitted in with the band's festival schedule. The previous concert before the Stones' support was in Den Haag, the Netherlands; the band debuted the first full-band version of "Tell Me It's Not Over". Starsailor had embarked on a tour of the UK during October 2007 in support of Fairtrade. They have debuted a number of new songs during the tour as "Boy in Waiting" and "All the Plans". The lyrics for "Tell Me It's Not Over" had changed considerably since the first time they had sung it. Their first travel to Latin America was confirmed after a long period of speculation. Starsailor supported The Killers in November 2007 on their dates in Argentina and Chile. Starsailor played on June at the Isle of Wight Festival on the last day, with The Police headlining on that date. They also performed at Hard Rock Calling in Hyde Park the same month with The Police as headliners. They also confirmed that they would play at "Rockin" Park' in the Netherlands on Saturday 28 June, on the third day of the 2008 Cactus Festival at Minnewaterpark, Brugge, Belgium on Saturday 12 July, and the Open Air Gampel Festival in Switzerland on Saturday 16 August.

On 20 March 2008, front man James posted on the official Starsailor forum announcing the album had been completed.

On 16 October 2008 it was officially confirmed that the new album would be called All the Plans and was to be released in March 2009. On 3 December, the Boy in Waiting EP was released as a free digital download for those who pre-order the deluxe edition of All the Plans. It contained three tracks that did not make the album.

The album debuted at Number 26 in the UK Album Charts on 14 March 2009, while "Tell Me It's Not Over" spent one week in the UK Singles Chart, reaching No. 73, and peaking at No. 5 in the Belgian Charts, becoming Starsailor's most successful single in that country.

On 26 April 2009, the band performed at The Tavern pub in Wigan, as part of Channel 4 television's "Grassroot Gigs" series. They had never played a concert in the town before, and have played there only once since then.

Lead singer, James Walsh began to work on a solo project at the end of 2009. On 13 November 2009 it was officially announced by The Lancashire Evening Post that Starsailor was on hold and that James Walsh was concentrating on his solo career.

Reunion, touring and Greatest Hits album (2014–2015)
Starsailor have reformed, playing support slots at Summer in the City festival in 10 and 11 July at Castlefield Bowl, headlined by Pixies and James. Starsailor performed on Rock Zottegem festival in Belgium on 14 July 2014. On 3 August 2014 the band performed at the Pentaport Rock Festival held in Incheon, South Korea. Their support tour continued in November 2014, supporting them in England at venues like Parr Hall, National Indoor Arena and Royal Albert Hall. They returned to the stage in June 2015, when they co-headlined several concerts with Embrace in the USA.

Starsailor's first greatest hits album, called Good Souls: The Greatest Hits, was released on 18 September 2015. The album contained 19 tracks, including all of band's singles to date and two brand new songs "Give Up the Ghost" and "Hold on", recorded with producer Harry Rutherford.

Starsailor played a second gig in Wigan in 2016, supported by a local band who shared the same management agency.

All This Life (2017–present)
In 2017, the band released "Listen To Your Heart", their first release in almost two years. Their fifth album, All This Life, was released on 1 September 2017 through the band's new record label, Cooking Vinyl. Flick of the Finger gave the album an overwhelmingly positive 5 star review, with the publication praising the song "Sunday Best", saying that 'the album is best summarised by this one melancholic and emotional track.' The album entered the UK album chart at No. 23.

In December 2021, the band were to release an expanded 20th anniversary edition of their album Love Is Here, as well as going out on tour to support it. The release was delayed until January 2022 due to pressing delays of the vinyl edition. Due to a partial cancelation due to COVID-19, the band were eventually able to complete their delayed tour in September 2022, and gave a live debut to a new song recorded earlier in the year entitled Heavyweight.

October 2023 saw the band announce details of their sixth studio album, entitled Where The Wild Things Grow, with a release date of March 2024. A video for the title track was also released at the same time. In November 2023, Starsailor released a 20th anniversary of their album Silence Is Easy alongside a tour to promote the re-issue.

Charity
The band played the London Astoria on 4 and 8 February 2002. The first date was a concert in aid of Warchild, and also included Travis and Ryan Adams on the bill. The 8th saw the debut of "Born Again", which would eventually be released as a single. The concert is notable since the band were joined on stage by two backing singers and a cellist. James Walsh told NME.COM after the show that he thought the gig was "amazing, something really special" while bassist James Stelfox said it was "one of the greatest ever." This flirtation with these additional voices and a cello only lasted a brief period of time.

On 19 September 2008 Starsailor performed in the second edition of the "Stars of Europe" concert at Brussels in support of UNICEF, where they covered U2's single "All I Want Is You"

On 20 May 2009, the band was named Hard Rock's "Philanthropic Artist of the Year" in recognition of the work they'd done in support of the Caron Keating Foundation, the AECC of Barcelona and Nordoff-Robbins Music Therapy, as well as having contributed an exclusive cover of Graham Nash's "Military Madness" to the World Hunger Year Serve3 charity album.

Band members
Current members
 * James Walsh – guitar and lead vocals
 * James Stelfox – bass
 * Barry Westhead – keyboards
 * Ben Byrne – drums

Session and touring members
 * Mark Collins – guitar and vocals (2003–2004, 2006, 2014)
 * Richard "Echoboy" Warren – guitar, vocals and harmonica (2005–2006)
 * Pete Greenwood – guitar (2008–2009)

Discography
Studio albums
 * Love Is Here (2001)
 * Silence Is Easy (2003)
 * On the Outside (2005)
 * All the Plans (2009)
 * All This Life (2017)
 * Where the Wild Things Grow (2024)

title: Starsailor (band)


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F136

F136 may refer to:

 * Ferrari F136 engine
 * General Electric/Rolls-Royce F136

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title: F136


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E8 (mathematics)

In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8. The designation E8 comes from the Cartan–Killing classification of the complex simple Lie algebras, which fall into four infinite series labeled An, Bn, Cn, Dn, and five exceptional cases labeled G2, F4, E6, E7, and E8. The E8 algebra is the largest and most complicated of these exceptional cases.

Basic description
The Lie group E8 has dimension 248. Its rank, which is the dimension of its maximal torus, is eight.

Therefore, the vectors of the root system are in eight-dimensional Euclidean space: they are described explicitly later in this article. The Weyl group of E8, which is the group of symmetries of the maximal torus that are induced by conjugations in the whole group, has order 2$14$3$5$5$2$7 = 696 729  600.

The compact group E8 is unique among simple compact Lie groups in that its non-trivial representation of smallest dimension is the adjoint representation (of dimension 248) acting on the Lie algebra E8 itself; it is also the unique one that has the following four properties: trivial center, compact, simply connected, and simply laced (all roots have the same length).

There is a Lie algebra Ek for every integer k ≥ 3. The largest value of k for which Ek is finite-dimensional is k = 8, that is, Ek is infinite-dimensional for any k > 8.

Real and complex forms
There is a unique complex Lie algebra of type E8, corresponding to a complex group of complex dimension 248. The complex Lie group E8 of complex dimension 248 can be considered as a simple real Lie group of real dimension 496. This is simply connected, has maximal compact subgroup the compact form (see below) of E8, and has an outer automorphism group of order 2 generated by complex conjugation.

As well as the complex Lie group of type E8, there are three real forms of the Lie algebra, three real forms of the group with trivial center (two of which have non-algebraic double covers, giving two further real forms), all of real dimension 248, as follows:

 * The compact form (which is usually the one meant if no other information is given), which is simply connected and has trivial outer automorphism group.
 * The split form, EVIII (or E8(8)), which has maximal compact subgroup Spin(16)/(Z/2Z), fundamental group of order 2 (implying that it has a double cover, which is a simply connected Lie real group but is not algebraic, see below) and has trivial outer automorphism group.
 * EIX (or E8(−24)), which has maximal compact subgroup E7×SU(2)/(−1,−1), fundamental group of order 2 (again implying a double cover, which is not algebraic) and has trivial outer automorphism group.

For a complete list of real forms of simple Lie algebras, see the list of simple Lie groups.

E8 as an algebraic group
By means of a Chevalley basis for the Lie algebra, one can define E8 as a linear algebraic group over the integers and, consequently, over any commutative ring and in particular over any field: this defines the so-called split (sometimes also known as "untwisted") form of E8. Over an algebraically closed field, this is the only form; however, over other fields, there are often many other forms, or "twists" of E8, which are classified in the general framework of Galois cohomology (over a perfect field k) by the set H1(k,Aut(E8)), which, because the Dynkin diagram of E8 (see below) has no automorphisms, coincides with H1(k,E8).

Over R, the real connected component of the identity of these algebraically twisted forms of E8 coincide with the three real Lie groups mentioned above, but with a subtlety concerning the fundamental group: all forms of E8 are simply connected in the sense of algebraic geometry, meaning that they admit no non-trivial algebraic coverings; the non-compact and simply connected real Lie group forms of E8 are therefore not algebraic and admit no faithful finite-dimensional representations.

Over finite fields, the Lang–Steinberg theorem implies that H1(k,E8)=0, meaning that E8 has no twisted forms: see below.

The characters of finite dimensional representations of the real and complex Lie algebras and Lie groups are all given by the Weyl character formula. The dimensions of the smallest irreducible representations are :
 * 1, 248, 3875, 27000, 30380, 147250, 779247, 1763125, 2450240, 4096000, 4881384, 6696000, 26411008, 70680000, 76271625, 79143000, 146325270, 203205000, 281545875, 301694976, 344452500, 820260000, 1094951000, 2172667860, 2275896000, 2642777280, 2903770000, 3929713760, 4076399250, 4825673125, 6899079264, 8634368000 (twice), 12692520960...

The 248-dimensional representation is the adjoint representation. There are two non-isomorphic irreducible representations of dimension 8634368000 (it is not unique; however, the next integer with this property is 175898504162692612600853299200000 ). The fundamental representations are those with dimensions 3875, 6696000, 6899079264, 146325270, 2450240, 30380, 248 and 147250 (corresponding to the eight nodes in the Dynkin diagram in the order chosen for the Cartan matrix below, i.e., the nodes are read in the seven-node chain first, with the last node being connected to the third).

The coefficients of the character formulas for infinite dimensional irreducible representations of E8 depend on some large square matrices consisting of polynomials, the Lusztig–Vogan polynomials, an analogue of Kazhdan–Lusztig polynomials introduced for reductive groups in general by George Lusztig and David Kazhdan (1983). The values at 1 of the Lusztig–Vogan polynomials give the coefficients of the matrices relating the standard representations (whose characters are easy to describe) with the irreducible representations.

These matrices were computed after four years of collaboration by a group of 18 mathematicians and computer scientists, led by Jeffrey Adams, with much of the programming done by Fokko du Cloux. The most difficult case (for exceptional groups) is the split real form of  E8 (see above), where the largest matrix is  of size 453060×453060. The Lusztig–Vogan polynomials for all other exceptional simple groups have been known for some time; the calculation for the split form of E8 is far longer than any other case. The announcement of the result in March 2007 received extraordinary attention from the media (see the external links), to the surprise of the mathematicians working on it.

The representations of the E8 groups over finite fields are given by Deligne–Lusztig theory.

Constructions
One can construct the (compact form of the) E8 group as the automorphism group of the corresponding e8 Lie algebra. This algebra has a 120-dimensional subalgebra so(16) generated by Jij as well as 128 new generators Qa that transform as a Weyl–Majorana spinor of spin(16). These statements determine the commutators
 * $$\left[J_{ij}, J_{k\ell}\right] = \delta_{jk}J_{i\ell} - \delta_{j\ell}J_{ik} - \delta_{ik}J_{j\ell} + \delta_{i\ell}J_{jk}$$

as well as
 * $$\left[J_{ij}, Q_a\right] = \frac 14 \left(\gamma_i\gamma_j-\gamma_j\gamma_i\right)_{ab} Q_b,$$

while the remaining commutators (not anticommutators!) between the spinor generators are defined as
 * $$\left[Q_a, Q_b\right] = \gamma^{[i}_{ac}\gamma^{j]}_{cb} J_{ij}.$$

It is then possible to check that the Jacobi identity is satisfied.

Geometry
The compact real form of E8 is the isometry group of the 128-dimensional exceptional compact Riemannian symmetric space EVIII (in Cartan's classification). It is known informally as the "octooctonionic projective plane" because it can be built using an algebra that is the tensor product of the octonions with themselves, and is also known as a Rosenfeld projective plane, though it does not obey the usual axioms of a projective plane. This can be seen systematically using a construction known as the magic square, due to Hans Freudenthal and Jacques Tits.

E8 root system
[[File:4 21 E8 to 3D H3 symmetry concentric hulls.png|thumb|Shown in 3D projection using the basis vectors [u,v,w] giving H3 symmetry:

The projected 421 polytope vertices are sorted and tallied by their 3D norm generating the increasingly transparent hulls of each set of tallied norms. These show: 1. 4 points at the origin

2. 2 icosahedrons

3. 2 dodecahedrons

4. 4 icosahedrons

5. 1 icosadodecahedron

6. 2 dodecahedrons

7. 2 icosahedrons

8. 1 icosadodecahedron

for 240 vertices. These are two concentric sets of hulls from the H4 symmetry of the 600-cell scaled by the golden ratio. ]]

A root system of rank r is a particular finite configuration of vectors, called roots, which span an r-dimensional Euclidean space and satisfy certain geometrical properties. In particular, the root system must be invariant under reflection through the hyperplane perpendicular to any root.

The E8 root system is a rank 8 root system containing 240 root vectors spanning R8. It is irreducible in the sense that it cannot be built from root systems of smaller rank. All the root vectors in E8 have the same length. It is convenient for a number of purposes to normalize them to have length $\sqrt{2}$. These 240 vectors are the vertices of a semi-regular polytope discovered by Thorold Gosset in 1900, sometimes known as the 421 polytope.

Construction
In the so-called even coordinate system, E8 is given as the set of all vectors in R8 with length squared equal to 2 such that coordinates are either all integers or all half-integers and the sum of the coordinates is even.

Explicitly, there are 112 roots with integer entries obtained from
 * $$\left(\pm 1,\pm 1,0,0,0,0,0,0\right)\,$$

by taking an arbitrary combination of signs and an arbitrary permutation of coordinates, and 128 roots with half-integer entries obtained from
 * $$\left(\pm\tfrac12,\pm\tfrac12,\pm\tfrac12,\pm\tfrac12,\pm\tfrac12,\pm\tfrac12,\pm\tfrac12,\pm\tfrac12\right) \,$$

by taking an even number of minus signs (or, equivalently, requiring that the sum of all the eight coordinates be even). There are 240 roots in all.

The 112 roots with integer entries form a D8 root system. The E8 root system also contains a copy of A8 (which has 72 roots) as well as E6 and E7 (in fact, the latter two are usually defined as subsets of E8).

In the odd coordinate system, E8 is given by taking the roots in the even coordinate system and changing the sign of any one coordinate. The roots with integer entries are the same while those with half-integer entries have an odd number of minus signs rather than an even number.

Dynkin diagram
The Dynkin diagram for E8 is given by.

This diagram gives a concise visual summary of the root structure. Each node of this diagram represents a simple root. A line joining two simple roots indicates that they are at an angle of 120° to each other. Two simple roots that are not joined by a line are orthogonal.

Cartan matrix
The Cartan matrix of a rank $r$ root system is an $r × r$ matrix whose entries are derived from the simple roots. Specifically, the entries of the Cartan matrix are given by
 * $$A_{ij} = 2\frac{\left(\alpha_i, \alpha_j\right)}{\left(\alpha_i, \alpha_i\right)}$$

where $$ is the Euclidean inner product and $α_{i}$ are the simple roots. The entries are independent of the choice of simple roots (up to ordering).

The Cartan matrix for E8 is given by
 * $$\left [

\begin{array}{rr} 2 & -1 & 0 &  0 &  0 &  0 &  0 & 0 \\ -1 &  2 & -1&  0 &  0 &  0 &  0 & 0 \\ 0 & -1 &  2 & -1 &  0 &  0 &  0 & 0 \\ 0 &  0 & -1 &  2 & -1 &  0 &  0 & 0 \\ 0 &  0 &  0 & -1 &  2 & -1 &  0 & -1 \\ 0 &  0 &  0 &  0 & -1 &  2 & -1 & 0 \\ 0 &  0 &  0 &  0 &  0 & -1 &  2 & 0 \\ 0 &  0 & 0 &  0 &  -1 &  0 &  0 & 2 \end{array}\right ].$$ The determinant of this matrix is equal to 1.

Simple roots

A set of simple roots for a root system Φ is a set of roots that form a basis for the Euclidean space spanned by Φ with the special property that each root has components with respect to this basis that are either all nonnegative or all nonpositive.

Given the E8 Cartan matrix (above) and a Dynkin diagram node ordering of:

One choice of simple roots is given by the rows of the following matrix:
 * $$\left [\begin{array}{rr}

1&-1&0&0&0&0&0&0 \\ 0&1&-1&0&0&0&0&0 \\ 0&0&1&-1&0&0&0&0 \\ 0&0&0&1&-1&0&0&0 \\ 0&0&0&0&1&-1&0&0 \\ 0&0&0&0&0&1&1&0 \\ -\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}\\ 0&0&0&0&0&1&-1&0 \\ \end{array}\right ].$$

With this numbering of nodes in the Dynkin diagram, the highest root in the root system has Coxeter labels (2, 3, 4, 5, 6, 4, 2, 3). Using this representation of the simple roots, the lowest root is given by
 * $$\left [\begin{array}{rr}

-1&0&0&0&0&0&0&1 \\ \end{array}\right ].$$ The only simple root that can be added to the lowest root to obtain another root is the one corresponding to node 1 in this labeling of the Dynkin diagram — as is to be expected from the affine Dynkin diagram for $${\tilde{\mathrm{E}}}_{8}$$. The Hasse diagram to the right enumerates the 120 roots of positive height relative to any particular choice of simple roots consistent with this node numbering.

Note that the Hasse diagram does not represent the full Lie algebra, or even the full root system. The 120 roots of negative height relative to the same set of simple roots can be adequately represented by a second copy of the Hasse diagram with the arrows reversed; but it is less straightforward to connect these two diagrams via a basis for the eight-dimensional Cartan subalgebra. In the notation of the exposition of Chevalley generators and Serre relations: Insofar as an arrow represents the Lie bracket by the generator $$e_i$$ associated with a simple root, each root in the height -1 layer of the reversed Hasse diagram must correspond to some $$f_i$$ and can have only one upward arrow, connected to a node in the height 0 layer representing the element of the Cartan subalgebra given by $$h_i = [e_i, f_i]$$. But the upward arrows from the height 0 layer must then represent $$[e_i, h_j] = -a_{ji} e_i$$, where $$a_{ji}$$ is (the transpose of) the Cartan matrix. One could draw multiple upward arrows from each $$h_j$$ associated with all $$e_i$$ for which $$[e_i, h_j]$$ is nonzero; but this neither captures the numerical entries in the Cartan matrix nor reflects the fact that each $$e_i$$ only has nonzero Lie bracket with one degree of freedom in the Cartan subalgebra (just not the same degree of freedom as $$h_i$$).

More fundamentally, this organization implies that the span of the generators designated as "the" Cartan subalgebra is somehow inherently special, when in most applications, any mutually commuting set of eight of the 248 Lie algebra generators (of which there are many!) — or any eight linearly independent, mutually commuting Lie derivations on any manifold with E8 structure — would have served just as well. Once a Cartan subalgebra has been selected (or defined a priori, as in the case of a lattice), a basis of "Cartan generators" (the $$h_i$$ among the Chevalley generators) and a root system are a useful way to describe structure relative to this subalgebra. But the root system map is not the Lie algebra (let alone group!) territory. Given a set of Chevalley generators, most degrees of freedom in a Lie algebra and their sparse Lie brackets with $${e_i}$$ can be represented schematically as circles and arrows, but this simply breaks down on the chosen Cartan subalgebra. Such are the hazards of schematic visual representations of mathematical structures.

Weyl group
The Weyl group of E8 is of order 696729600, and can be described as O$+ 8$(2): it is of the form 2.G.2 (that is, a stem extension by the cyclic group of order 2 of an extension of the cyclic group of order 2 by a group G) where G is the unique simple group of order 174182400 (which can be described as PSΩ8+(2)).

E8 root lattice
The integral span of the E8 root system forms a lattice in R8 naturally called the E8 root lattice. This lattice is rather remarkable in that it is the only (nontrivial) even, unimodular lattice with rank less than 16.

Simple subalgebras of E8

The Lie algebra E8 contains as subalgebras all the exceptional Lie algebras as well as many other important Lie algebras in mathematics and physics. The height of the Lie algebra on the diagram approximately corresponds to the rank of the algebra. A line from an algebra down to a lower algebra indicates that the lower algebra is a subalgebra of the higher algebra.

Chevalley groups of type E8
showed that the points of the (split) algebraic group E8 (see above) over a finite field with q elements form a finite Chevalley group, generally written E8(q), which is simple for any q, and constitutes one of the infinite families addressed by the classification of finite simple groups. Its number of elements is given by the formula :
 * $$q^{120}\left(q^{30} - 1\right)\left(q^{24} - 1\right)\left(q^{20} - 1\right)\left(q^{18} - 1\right)\left(q^{14} - 1\right)\left(q^{12} - 1\right)\left(q^8 - 1\right)\left(q^2 - 1\right)$$

The first term in this sequence, the order of E8(2), namely 337 804  753  143  634  806  261  388  190  614  085  595  079  991  692  242  467  651  576  160  959  909  068  800  000 ≈ $3.38$, is already larger than the size of the Monster group. This group E8(2) is the last one described (but without its character table) in the ATLAS of Finite Groups.

The Schur multiplier of E8(q) is trivial, and its outer automorphism group is that of field automorphisms (i.e., cyclic of order f if q = pf, where p is prime).

described the unipotent representations of finite groups of type E8.

Subgroups, subalgebras, and extensions
The smaller exceptional groups E7 and E6 sit inside E8. In the compact group, both E6×SU(3)/(Z/3Z) and E7×SU(2)/(+1,−1) are maximal subgroups of E8.

The 248-dimensional adjoint representation of E8 may be considered in terms of its restricted representation to the first of these subgroups. It transforms under E6×SU(3) as a sum of tensor product representations, which may be labelled as a pair of dimensions as (78,1) + (1,8) + ($\overline{27}$,3) + (27,$\overline{3}$. (Since the maximal subgroup is actually the quotient of this group product by a finite group, these notations may strictly be taken as indicating the infinitesimal (Lie algebra) representations.) Since the adjoint representation can be described by the roots together with the generators in the Cartan subalgebra, we may choose a particular E6 root system within E8 and decompose the sum representation relative to this E6. In this description,
 * (78,1), a copy of $$\mathfrak{e}_6$$, consists of the 72 roots with (−$1/2$,−$1/2$,−$1/2$), (0,0,0), or ($1/2$,$1/2$,$1/2$) in the last three dimensions, together with five Cartan generators corresponding to the first five dimensions and one Cartan generator corresponding to an equally weighted sum of the last three Cartan generators of the E8 system;
 * (1,8), a copy of $$\mathfrak{su}(3)$$, consists of the six roots with permutations of (0,1,−1) in the last three dimensions, together with two Cartan generators corresponding to the two trace-free combinations of the last three Cartan generators of the E8 system;
 * (27 ,3) consists of all roots with permutations of (-1,0,0), (−$1/2$,$1/2$,$1/2$), or (0,1,1) in the last three dimensions; and
 * (27,3 ) consists of all roots with permutations of (1,0,0), ($1/2$,−$1/2$,−$1/2$), or (0,-1,-1) in the last three dimensions.

The 248-dimensional adjoint representation of E8, when similarly restricted to the second maximal subgroup, transforms under E7×SU(2) as: (133,1) + (1,3) + (56,2). We may again see the decomposition by looking at the roots together with the generators in the Cartan subalgebra. In this description,
 * (133,1), a copy of $$\mathfrak{e}_7$$, consists of the 126 roots with (−1,−1), (−$1/2$,−$1/2$), (0,0), ($1/2$,$1/2$), or (1,1) in the last two dimensions, together with six Cartan generators corresponding to the first six dimensions and one Cartan generator corresponding to an equally weighted sum of the last two Cartan generators of the E8 system;
 * (1,3), a copy of $$\mathfrak{su}(2)$$, consists of the two roots (0,0,0,0,0,0,1,−1), (0,0,0,0,0,0,−1,1), together with one Cartan generator corresponding to the trace-free combination of the last two Cartan generators of the E8 system; and
 * (56,2) consists of all roots with permutations of (0,-1), ($1/2$,−$1/2$), or (1,0) in the last two dimensions.

The connection between these two descriptions is given by the graded exceptional Lie algebra constructions of J. Tits and B. N. Allison. Any 27-dimensional representation of E6 can be equipped with a non-associative (but strictly power-associative) Jordan product operation to form an Albert algebra (an important exceptional case in algebraic constructions). The Kantor–Koecher–Tits construction applied to this Albert algebra recovers the 78-dimensional $$\mathfrak{e}_6$$ as the reduced structure algebra of the Albert algebra. This $$\mathfrak{e}_6$$, together with the 27 and 27 representations and the grade operator (the element of the Cartan subalgebra with weight -1 on the 27, +1 on the 27, and 0 on the 78), forms an $$\mathfrak{e}_7$$ 3-graded Lie algebra. A complete exposition of this construction may be found in standard texts on Jordan algebras such as Jacobson 1968 or McCrimmon 2004.

Starting this 3-graded Lie algebra construction with any particular 27-dimensional representation, embedded within $$\mathfrak{e}_8$$, of any particular E6 subgroup of E8 produces the corresponding $$\mathfrak{e}_7$$ subalgebra. The particular $$\mathfrak{e}_7$$ in the E7×SU(2) decomposition given above corresponds to choosing the 27 consisting of all roots with (1,0,0), ($1/2$,-$1/2$,-$1/2$), or (0,-1,-1) in the last three dimensions (in order), with the grade operator having weight (-1,$1/2$,$1/2$) in these dimensions; or equivalently to choosing the "27 " consisting of all roots with (-1,0,0), (−$1/2$,$1/2$,$1/2$), or (0,1,1) in the last three dimensions (in order), with the grade operator having weight (1,-$1/2$,-$1/2$) in these dimensions. Note that there is nothing special about this choice of dimensions — the $$\mathbb R^{8}$$ within which the root system is embedded is not the set of eight independent but non-orthogonal axes corresponding to the simple roots, and any three dimensions will do — and there are also constructions using other equivalent groupings of roots. What matters is that the kernel of the Lie bracket with the generator chosen as the "grade operator" be an $$\mathfrak{e}_6$$ subalgebra (plus a central $$\mathbb{R}^{2}$$ associated with the grade operator itself and the remaining generator of the Cartan subalgebra), not $$\mathfrak{a}_6 \oplus \mathbb{R}^{2}$$, $$\mathfrak{a}_7 \oplus \mathbb{R}$$, $$\mathfrak{e}_7 \oplus \mathbb{R}$$, etc.

(In the simple Lie algebra case, the sign of the grade of the 27 versus the 27 representation is a matter of convention, as is the scale of the grading. However, choosing -1 as the grade of the 27-dimensional "vector" representation is consistent with an extension of the 3-graded algebra to higher positive grades via the exterior algebra over the 27 "covector" representation. The "vector" representation then lies, not in this nonnegative-graded exterior algebra, but in the graded algebra of derivations over the exterior algebra; the 78-dimensional $$\mathfrak{e}_6$$ is the grade 0 subalgebra (a subalgebra of the inner derivations by "vector-valued 1-forms") of this graded algebra of derivations. For details on how this asymmetric structure works starting from a general 3-graded algebra, see references at Frölicher–Nijenhuis bracket. The relevance of this observation to E8 is simply that E7 and E8 are their own clusters of structures, distinguished as exceptional simple Lie groups/algebras, and that any particular reconstruction of them using representations of their subgroups/subalgebras will have extensions beyond the motivating case. Varying conventions of sign, scale, and conjugate relationship in the literature are due not just to inaccuracies but also to the directions in which the authors seek to extend their constructions.)

The distinguished $$\mathfrak{su}(2)$$ in the E7×SU(2) decomposition above is then given by the subalgebra of $$\mathfrak{su}(3)$$ that commutes with the grade operator (which lies in the Cartan subalgebra of this $$\mathfrak{su}(3)$$). Of the four remaining roots in the $$\mathfrak{su}(3)$$, two are of grade $1/2$ and two are of grade -$1/2$. In the convention where the 27 of E6 used to construct the $$\mathfrak{e}_7$$ has grade -1 and the 27 has grade +1, the other two 27's have grade +$1/2$ and the other two 27 's have grade -$1/2$, as is apparent from permuting the values of the last three roots in the description above. Grouping these 4×(1+27)=112 generators to form the grade +$1/2$ and -$1/2$ subspaces of $$\mathfrak{e}_8$$ (relative to the original choice of grade operator within $$\mathfrak{e}_7$$), each subspace may be given a quite particular non-associative (nor even power-associative) product operation, resulting in two copies of Brown's 56-dimensional structurable algebra. Allison's 5-graded Lie algebra construction based on this structurable algebra recovers the original $$\mathfrak{e}_8$$. (Allison's 5-grading differs from the above by a factor -2.) Grouping these generators differently, based on their weights relative to the Cartan generator of the $$\mathfrak{su}(2)$$ orthogonal to $$\mathfrak{e}_7$$, gives two 56-dimensional subspaces that each carry the lowest-dimensional non-trivial irreducible representation of E7. Either of these may be combined with the Cartan generator to form a 57-dimensional Heisenberg algebra, and adjoining this to $$\mathfrak{e}_7$$ produces the (non-simple) Lie algebra E7 1/2 described by Landsberg and Manivel.

From the perspective in which the 27-dimensional grade -1 subspace of $$\mathfrak{e}_8$$ (relative to a choice of grade operator) plays the role of "vector" representation of E6 and the 27 with roots opposite it plays the role of "covector" representation, it is natural to look for "spinor" representations in the grade +$1/2$ and -$1/2$ subspaces, or in some other combination of the (27 ,3) and (27,3 ) representations of E6×SU(3), and to attempt to relate these to geometrical spinors in the Clifford algebra sense as employed in quantum field theory. Variations on this idea are common in the physics literature. See Distler and Garibaldi 2009 for discussion of the mathematical obstacles to constructing a chiral gauge theory based on E8. The structure of $$\mathfrak{e}_8$$ relative to its $$\mathfrak{e}_6$$ subalgebra, together with the conventional scaling of elements of the Cartan subalgebra, invites extensions by geometric analogy but does not necessarily imply a relationship to low-dimensional geometry or low-energy physics. The same may be said of connections to Jordan and Heisenberg algebras, whose historical origins are intertwined with the development of quantum mechanics. Not every visual representation evocative of a tobacco pipe will hold tobacco.

The finite quasisimple groups that can embed in (the compact form of) E8 were found by.

The Dempwolff group is a subgroup of (the compact form of) E8. It is contained in the Thompson sporadic group, which acts on the underlying vector space of the Lie group E8 but does not preserve the Lie bracket. The Thompson group fixes a lattice and does preserve the Lie bracket of this lattice mod 3, giving an embedding of the Thompson group into E8(F3).

The embeddings of the maximal subgroups of E8 up to dimension 248 are shown to the right.

Applications
The E8 Lie group has applications in theoretical physics and especially in string theory and supergravity. E8×E8 is the gauge group of one of the two types of heterotic string and is one of two anomaly-free gauge groups that can be coupled to the N = 1 supergravity in ten dimensions. E8 is the U-duality group of supergravity on an eight-torus (in its split form).

One way to incorporate the standard model of particle physics into heterotic string theory is the symmetry breaking of E8 to its maximal subalgebra SU(3)×E6.

In 1982, Michael Freedman used the E8 lattice to construct an example of a topological 4-manifold, the E8 manifold, which has no smooth structure.

Antony Garrett Lisi's incomplete "An Exceptionally Simple Theory of Everything" attempts to describe all known fundamental interactions in physics as part of the E8 Lie algebra.

reported an experiment where the electron spins of a cobalt-niobium crystal exhibited, under certain conditions, two of the eight peaks related to E8 that were predicted by.

History
discovered the complex Lie algebra E8  during his classification of simple compact Lie algebras, though he did not prove its existence, which was first shown by Élie Cartan. Cartan determined that a complex simple Lie algebra of type E8 admits three real forms. Each of them gives rise to a simple Lie group of dimension 248, exactly one of which (as for any complex simple Lie algebra) is compact. introduced algebraic groups and Lie algebras of type E8 over other fields: for example, in the case of finite fields they lead to an infinite family of finite simple groups of Lie type. E8 continues to be an area of active basic research by Atlas of Lie Groups and Representations, which aims to determine the unitary representations of all the Lie groups.

Lusztig–Vogan polynomial calculation

 * Atlas of Lie groups
 * Kazhdan–Lusztig–Vogan Polynomials for E8
 * Narrative of the Project to compute Kazhdan–Lusztig Polynomials for E8
 * The n-Category Café, a University of Texas blog posting by John Baez on E8.
 * The n-Category Café, a University of Texas blog posting by John Baez on E8.

Other links

 * Graphic representation of E8 root system.
 * The list of dimensions of irreducible representations of the complex form of E8 is sequence A121732 in the OEIS.

title: E8 (mathematics)


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