Boujemaa Agrebaoui

Karima Benali
email: karimabenali172@yahoo.fr

Abdenacer Makhlouf
email: abdenacer.makhlouf@uha.fr

REPRESENTATIONS OF SIMPLE HOM-LIE ALGEBRAS

Keywords: Hom-Lie algebra, Simple Hom-Lie algebra, Representation. 1

The purpose of this paper is to study representations of simple multiplicative Hom-Lie algebras. First, we provide a new proof using Killing form for characterization theorem of simple Hom-Lie algebras given by Chen and Han, then discuss the representations structure of simple multiplicative Hom-Lie algebras. Moreover, we study weight modules and root space decompositions of simple multiplicative Hom-Lie algebras, characterize weight modules and provide examples of representations of sl2-type Hom-Lie algebras.

Introduction

Nowadays, one of the most modern trends in mathematics has to do with representations and deformations. These two topics are important tools in most parts of Mathematics and Physics.

Hom-type algebras arised first in examples of q-deformations of algebras of vector fields, like Witt and Virasoro algebras, where the usual derivation is replaced by a σ-derivation. Motivated by these examples, Hartwig, Larsson and Silvestrov developed from the algebraic point of view in [START_REF] Hartwig | Deformations of Lie algebras using σ-derivations[END_REF] the deformation theory using σ-derivations and introduced a new category of algebras called Hom-Lie algebras. A Hom-Lie algebra is a triple (g, [•, •] α , α) in which the bracket satisfies a twisted Jacobi identity along the linear map α. It should be pointed that Lie algebras form a subclass of Hom-Lie algebras, i.e. when α equal to the identity map. Various classical algebraic structures where considered and generalized within similar framework like Hom-Lie superalgebras in [START_REF] Ammar | Hom-Lie superalgebras and Hom-Lie admissible superalgebras[END_REF].

Representations of Hom-Lie algebras were introduced and studied in [START_REF] Sheng | Representation of Hom-Lie algebras[END_REF], see also [START_REF] Benayadi | Hom-Lie algebras with symmetric invariant non-degenerate bilinear form[END_REF]. Based on this, we aim in this paper to discuss simple Hom-Lie algebras representations. Simple Hom-Lie algebras were characterized in [START_REF] Chen | Classification of multiplicative simple Hom-Lie algebras[END_REF], where the authors showed that they are obtained by Yau twist of semisimple Lie algebras. This key observation is used here to built a representation theory of simple Hom-Lie algebras. Moreover examples are provided by a study of Hom-type sl(2)-modules.

The paper is organized as follows. In Section 2, we review basic definitions and relevant constructions about Hom-Lie algebras. Then in Section 3 we recall some fundamental results about structure of simple multiplicative Hom-Lie algebras and provide a new proof of the main theorem using Killing form. Section 4 is dedicated to the construction of multiplicative simple Hom-Lie algebras representations, we show that there is a correspondence between representation of multiplicative simple Hom-Lie algebras and representation of the induced semisimple Lie algebras using invertible twisting maps. In Section 5, we introduce and discuss the notion of simple multiplicative Hom-Lie algebra weight-modules. Finally in Section 6 we study and classify Hom-sl(2)-modules.

Basics

In this section, we provide some preliminaries, basic definitions and relevant constructions about Hom-Lie algebras and related structures. Throughout this paper all algebras and vector spaces are considered over K, an algebraically closed field of characteristic 0. Definition 2.1. A Hom-Lie algebra is a triple (g, [•, •] α , α) consisting of a vector space g, a bilinear map [•, •] α : g × g -→ g and a linear map α : g → g that satisfies :

[x, y] α = -[y, x] α , ∀x, y ∈ g (skewsymmetry)

x,y,z [α(x), [y, z] α ] α = 0, ∀x, y, z ∈ g (Hom-Jacobi identity).

A Hom-Lie algebra (g, [•, •] α , α) is said to be multiplicative if α is an algebra morphism, i.e.

α([x, y]

α ) = [α(x), α(y)] α , ∀x, y ∈ g. It is said regular if α is an algebra automorphism. Definition 2.2. Let (g, [•, •] α , α) be a Hom-Lie algebra. A subspace h of g is called Hom-Lie subal- gebra if [h, h] α ⊆ h and α(h) ⊆ h. In particular, a Hom-Lie subalgebra h is said to be an ideal of g if [h, g] α ⊆ h. The Hom-Lie algebra g is called abelian if as usual [x, y] = 0, ∀x, y ∈ g. Definition 2.3. Let (g 1 , [•, •] α 1 , α 1 ) and (g 2 , [•, •] α 2 , α 2 ) be two Hom-Lie algebras. A linear map ϕ : g 1 -→ g 2 is a Hom-Lie algebra morphism if for all x, y ∈ g 1 ϕ([x, y] α 1 ) = [ϕ(x), ϕ(y)] α 2 and ϕ • α 1 = α 2 • ϕ.
In particular, they are isomorphic if ϕ is a bijective linear map.

A linear map ϕ : g 1 -→ g 2 is said to be a weak Hom-Lie algebra morphism if for all x, y ∈ g 1 , we

have only ϕ([x, y] α 1 ) = [ϕ(x), ϕ(y)] α 2 .
There is a key construction introduced by D. Yau that gives rise to a Hom-Lie algebra starting from a Lie algebra and a Lie algebra homomorphism [START_REF] Yau | Hom-Algebras and Homology[END_REF].

Proposition 2.4 (Yau Twist). Let g = (g, [•, •]) be a Lie algebra and α : g -→ g be a Lie algebra homomorphism. Then,

g α = (g, [•, •] α := α([•, •]), α) is a Hom-Lie algebra.
Proof. It is straightforward to prove that the new bracket [•, •] α satisfies the Hom-Jacobi identity.

Remark 2.5. More generally, let (g, [•, •] α , α) be a Hom-Lie algebra and γ : g → g be a weak Hom-Lie algebra morphism. Then

g γ = (g, [•, •] γ := γ([•, •]), γ • α) is a new Hom-Lie algebra. Definition 2.6. Let (g, [•, •] α , α) be a Hom-Lie algebra. If there exists a Lie algebra (g, [•, •]) such that [x, y] α = α([x, y]) = [α(x), α(y)], for all x, y ∈ g, then (g, [•, •] α , α) is said to be of Lie-type and (g, [•, •]) is called the induced Lie algebra of (g, [•, •] α , α). Lemma 2.7. Let (g, [•, •] α , α) be a regular Hom-Lie algebra. Then (g, [•, •] α , α) is of Lie-type with the induced Lie algebra (g, [•, •]), where [x, y] = α -1 ([x, y] α ), ∀x, y ∈ g. Proof. Let [x, y] = α -1 ([x, y] α ) for any x, y ∈ g. Since (g, [•, •] α , α) is multiplicative then, α([x, y] α ) = [α(x), α(y)] α . It implies α 2 ([x, y]) = α([α(x), α(y)]) and thus α([x, y]) = [α(x), α(y)]. In the follow- ing we shall prove that (g, [•, •]) is a Lie algebra. The skewsymmetry of [•, •] is obvious. Now, we
prove that it satisfies the Jacobi identity. Indeed, let's x, y, z ∈ g, we have

x,y,z [x, [y, z]] = x,y,z α -1 [x, α -1 [y, z] α ] α = x,y,z α -2 [α(x), [y, z] α ] α = 0. It follows that (g, [•, •]) is a Lie algebra.
The concept of representation of a Hom-Lie algebra was introduced in [START_REF] Sheng | Representation of Hom-Lie algebras[END_REF], see also [START_REF] Benayadi | Hom-Lie algebras with symmetric invariant non-degenerate bilinear form[END_REF]. Definition 2.8. A representation of a multiplicative Hom-Lie algebra (g, [•, •] α , α) on the vector space V with respect to β ∈ End(V ) is a linear map ρ β : g -→ End(V ), such that for any x, y ∈ g, the following conditions are satisfied:

ρ β (α(x)) • β = β • ρ β (x), (2.1) ρ β ([x, y] α ) • β = ρ β (α(x))ρ β (y) -ρ β (α(y))ρ β (x). (2.2) Hence (V, ρ β , β) is called a g-module via the action x.v = ρ β (x)v, ∀x ∈ g, v ∈ V .
We have the following property. Proposition 2.9. Let (V, ρ β , β) be a representation of a simple multiplicative Hom-Lie algebra (g, [•, •], α) with β invertible. Then, ∀n ∈ N we have, [START_REF] Ammar | Hom-Lie superalgebras and Hom-Lie admissible superalgebras[END_REF]

ρ β (α n (x)) = β n ρ β (x)β -n . (2) ρ β (α n [x, y])β = ρ β (α n+1 (x))ρ β (α n (y)) -ρ β (α n+1 (y))ρ β (α n (x)).
Proof. (1) is straightforward by induction using (2.1) and (2) is proved by induction using (2.2). Definition 2.10. For a g-module

(V, ρ β , β), if a subspace V 1 ⊆ V is invariant under ρ β (x), ∀x ∈ g and under β then (V 1 , ρ β , β| V 1 ) is called a g-submodule of (V, ρ β , β).
A g-module (V, ρ β , β) is called irreducible, if it has precisely two g-submodules (itself and 0) and it

is called completely reducible if V = V 1 ⊕ ... ⊕ V s , where (V i , ρ β , β| V i ) are irreducible g-submodules.
Theorem 2.11 ([15]). Let (g, [•, •] α , α) be a Lie-type Hom-Lie algebra with (g, [•, •]) the induced Lie algebra.

(1) Let (V, ρ β , β) be a representation of the Hom-Lie algebra where

β is invertible. Then (V, ρ) = (V, β -1 • ρ β ) is a representation of the Lie algebra (g, [•, •]).
(2) Suppose that (V, ρ) is a representation of the Lie algebra (g, [•, •]). If there exists β ∈ End(V ) such that

(2.3) β • ρ(x) = ρ(α(x)) • β, ∀x ∈ g, ∀v ∈ V, then (V, ρ β = β • ρ, β) is a representation of the Hom-Lie algebra (g, [•, •] α , α). Definition 2.12. Let (g, [•, •] α , α) be a Lie-type Hom-Lie algebra with (g, [•, •]) the induced Lie algebra. A representation (V, ρ β , β) of (g, [•, •] α , α) is called of Lie-type if ρ β = β • ρ
(V, ρ := β -1 • ρ β ) of the induced Lie algebra (g, [•, •]) is irreducible.
The previous theorem provides a relationship between representations of Lie-type Hom-Lie algebras and those of their induced Lie algebras.

Structure of Simple Multiplicative Hom-Lie algebras

In [START_REF] Chen | Classification of multiplicative simple Hom-Lie algebras[END_REF], the authors have proved that multiplicative simple Hom-Lie algebras are of Lie-type and their induced Lie algebras are semisimple. Moreover they discussed the dimension problem and showed that there is an n-dimensional simple Hom-Lie algebra for any integer n larger than 2. We should mention also the following relevant references [START_REF] Benayadi | Hom-Lie algebras with symmetric invariant non-degenerate bilinear form[END_REF], [START_REF] Li | Representations of 3-dimensional simple multiplicative Hom-Lie Algebras[END_REF] and [START_REF] Yau | Module Hom-algebras[END_REF]. We have the following two propositions. Proof. Let (g, [•, •] α , α) be a simple Hom-Lie algebra, then [g, g] α = g. Suppose that ker(α) = 0.

Then, α(ker(α)) = 0 and α([ker(α), g] α ) = 0. So ker(α) is a non trivial ideal of (g, [•, •] α , α). This contradicts the simplicity of (g, [•, •] α , α), except when ker(α) = 0. Hence, α is an automorphism.

The following theorem summarizes results given in [START_REF] Chen | Classification of multiplicative simple Hom-Lie algebras[END_REF] about a characterization of simple multiplicative Hom-Lie algebras. We provide a new and different proof based on Killing form.

Theorem 3.4. ([5]) Let (g, [•, •] α , α) be a simple multiplicative Hom-Lie algebra. Then the induced Lie algebra (g, [•, •] = α -1 ([•, •] α )
) is semisimple and its n simple ideals are isomorphic mutually besides α acts simply transitively on simple ideals of g. Furthermore g can be generated by a simple ideal g 1 of the Lie algebra (g, [•, •]) and α ∈ Aut(g). Taking α, γ ∈ Aut(g) such that α n and γ n leaves each simple ideal invariant and α n (g 1 ) = g 1 (or γ n (g 1 ) = g 1 ). Then we have

(1) g = g 1 ⊕ α(g 1 ) ⊕ α 2 (g 1 ) ⊕ ... ⊕ α n-1 (g 1 ) or g = g 1 ⊕ γ(g 1 ) ⊕ γ 2 (g 1 ) ⊕ ... ⊕ γ n-1 (g 1 ) .
(2) α and γ are conjugate on g ⇔ α n and γ n are conjugate on g 1 .

Proof. The Killing form

K : g × g → C of (g, [•, •]) is non-degenerate. In fact, let k = {x ∈ g/K(x, y) = 0, ∀y ∈ g} its kernel. It is clear that k is an ideal of g, since K([x, y], z) = K(x, [y, z]) = 0, ∀x ∈ k, y, z ∈ g. Since α is an automorphism and K(α(x), α(y)) = K(x, y). Then K(α(x), y) = K(x, α -1 (y)) = 0, ∀x ∈ k, y ∈ g and α(k) ⊂ k.
Then k is an ideal of the multiplicative simple Hom-Lie algebra g and then k = 0 and K is non-degenerate. We deduce that (g, [•, •]) is a semisimple

Lie algebra and then a direct sum of its simple ideals. Let g 1 be a minimal proper ideal of the induced Lie algebra (g, [•, •]). In particular g 1 is a simple ideal. Let n be the minimal integer such

that α n-1 (g 1 ) = g 1 and α n (g 1 ) = g 1 . The algebra b = g 1 ⊕ α(g 1 ) ⊕ . . . ⊕ α n-1 (g 1 ) is an ideal of the simple multiplicative Hom-Lie algebra (g, [., .] α , α). Then g = b = g 1 ⊕ α(g 1 ) ⊕ . . . ⊕ α n-1 (g 1 ).
Definition 3.5. A multiplicative Hom-Lie algebra is called semisimple if its induced Lie algebra is semisimple.

Proposition 3.6. A multiplicative semisimple Hom-Lie algebra is a direct sum of multiplicative simple Hom-Lie algebras.

Proof. For g semisimple Hom-Lie algebra, the Killing form of the induced Lie algebra will be nondegenerate and from the previous definition the induced Lie algebra is semisimple. We take a minimal ideal g 1 which will be a simple ideal. Let n be the minimal integer such that α n-1 (g 1 ) = g 1 and α n (g 1 ) = g 1 . The algebra b = g 1 ⊕ α(g 1 ) ⊕ . . . ⊕ α n-1 (g 1 ) is a simple ideal of the semisimple multiplicative Lie algebra. Let b ′ be the subspace of g orthogonal (with respect to K) to b.

As K is invariant, b ′ is an ideal of g. In fact, let x ∈ b, y ∈ b ′ , z ∈ g, by invariance of K, we have K(x, [z, y]) = K([x, z], y) = 0 since [x, z] ∈ b. Moreover, since α(b) = b, K(x, α(y)) = K(α -1 (x), y) = 0, then b ′
is an ideal of the Hom-Lie algebra (g, [., .] α , α).

By minimality of b, the intersection b ∩ b ′ can only be (0) or b. We can prove that the second case cannot occur. If not K(x, y) = 0, ∀x, y ∈ b and

x = k i=1 [x i , y i ] since [b, b] = b.
Then for all z ∈ g, using the invariance of K and b an ideal, we have

K(x, z) = K( k i=1 [x i , y i ], z) = k i=1 K([x i , y i ], z) = k i=1 K(x i , [y i , z]) = 0 which contradicts the fact that K is non-degenerate. Hence g = b ⊕ b ′ . The restriction of K to b ′ × b ′ is invariant non-degenerate bilinear form.
The proof is completed by induction on the dimension of g.

Representations of simple multiplicative Hom-Lie algebras

We aim in this section to characterize representations of simple multiplicative Hom-Lie algebras and provide the relationship with those of the induced semisimple Lie algebra.

Proof. Let v ∈ Ker(β), ρ β (α(x)) • β(v) = 0 = β(ρ β (x)(v)), ∀x ∈ g. Then, ρ β (α(x))(v) ∈ Ker(β). So Ker(β) is a submodule of V . Now let v ∈ Im(β), there exists w ∈ V such that v = β(w). Since α is an automorphism ∀x ∈ g, ρ β (x)(v) = ρ β (α(α -1 (x)))β(w) = β(ρ β (α -1 (x)w). So ρ β (x)(v) ∈ Im(β), ∀x ∈ g and therefore Im(β) is a submodule of V. Corollary 4.2. If (V, ρ β , β) is an irreducible representation of a simple multiplicative Hom-Lie algebra (g, [•, •] α , α). Then β is invertible. Proposition 4.3. Let (g, [•, •] α , α) be a simple multiplicative Hom-Lie algebra and (V, ρ β , β) a rep- resentation with β invertible. If (V, ρ := β -1 • ρ β ) is irreducible representation of the induced Lie algebra. Then (V, ρ β , β) is irreducible representation of the multiplicative simple Hom-Lie algebra. Proof. Assume that (V, ρ β , β) is reducible. Then, there exists W = {0 V } a subspace of V such that (V, ρ β , β | W ) is a submodule of (g, [•, •] α , α). That is β(W ) ⊂ W and ρ β (x)W ⊂ W, ∀x ∈ g. Hence, β • ρ(x)W ⊂ W, ∀x ∈ g and then ρ(x)W ⊂ β -1 (W ) ⊂ W, ∀x ∈ g and so W is a submodule for (V, ρ)
which is a contradiction. Proposition 4.4. Let g be a simple Lie algebra and (W, ρ) be a representation of g. Then, for

α ∈ Aut(g), (W, ρ := ρ • α -k ) is a representation of α k (g). Moreover, there exists β ∈ GL(W ) such that ρ(α k (x)) = β k • ρ k (x) • β -k , ∀x ∈ g.
Proof. Let (W, ρ) be a representation of g on W such that the following diagram is commutative (4.1)

g ρ -→ End(W ) α k     T α k (g) ρ -→ End(W )
That is there exists T such that ρ • α k = T • ρ. Using Skolem-Noether Theorem [START_REF] Bourbaki | Eléments de Mathématiques, Livre II, Algèbre, chapitre 8, «Modules et anneaux semi-simples[END_REF], there exists

S ∈ GL(W ) such that ∀x ∈ g, T • ρ(x) = S • ρ(x) • S -1 . Then, we have ρ • α k (x) = S • ρ(x) • S -1 .
By basic linear algebra theory, there exists β ∈ GL(W ) such that S = β k . Then the commutativity

becomes ρ • α k (x) = β k • ρ(x) • β -k , ∀x ∈ g. Proposition 4.5. ([4]) Let g = g 1 ⊕.
..⊕g n be a semisimple Lie algebra and let (V i , ρ i ) be a g i -module,

∀ 0 ≤ i ≤ n.
Then, an irreducible representation of g is given by (V, ρ) where

V = V 1 ⊗ ... ⊗ V n and ρ is given for x = (x 1 , ..., x n ) by ρ : g -→ End(V 1 ⊗ ... ⊗ V n ) x -→ ρ(x) = n i=1 Id V 1 ⊗ ... ⊗ ρ i (x i ) ⊗ ... ⊗ Id Vn .
Using the classification of simple multiplicative Hom-Lie algebras by Chen and Han in Theorem Theorem 4.6. Let (g, [•, •] α , α) be a simple multiplicative Hom-Lie algebra and g 1 be a simple ideal generating the induced semisimple Lie algebra such that g = g 1 ⊕ α(g 1 ) ⊕ ... ⊕ α n-1 (g 1 ).

Let (V 0 , ρ 0 ), (V 1 , ρ 1 ), . . . , (V n-1 , ρ n-1 ) be n representations of g 1 . Then, (1)
There exist

β 0 ∈ GL(V 0 ), β 1 ∈ GL(V 1 ), . . . , β n-1 ∈ GL(V n-1 )
depending on α such that ρ : g → End(V ), where V = V 0 ⊗...⊗V n-1 , defined for all X = (x 0 , α(x 1 ), . . . , α n-1 (x n-1 )) ∈ g by

(4.2) ρ(X) = n-1 k=0 Id V 0 ⊗ . . . ⊗ β k k • ρ k (x k ) • β -k k ⊗ . . . ⊗ Id V n-1 ,
is an irreducible representation of the induced Lie algebra (g, [•, •]) on V .

(2) Let β := β 0 ⊗ ... ⊗ β n-1 ∈ GL(V ) and let ρ β : g -→ End(V ) defined by

(4.3) ρ β (X) = β • ρ(X) = n-1 k=0 β 0 ⊗ ... ⊗ β k+1 k • ρ k (x k ) • β -k k ⊗ ... ⊗ β n-1 .
Then the triple (V, ρ β , β) is a regular representation of the simple multiplicative Hom-Lie

algebra (g, [•, •] α , α).
Proof. Let ρ : g 1 -→ End(W ) be a representation of g 1 . By Proposition 4.4, for all k ∈ N, there

exists β k ∈ GL(W ) and a representation ρ k of α k (g 1 ) on W given by ρ k (α k (x)) = β k k • ρ k (x) • β -k k . Now for (V 0 , ρ 0 ), (V 1 , ρ 1 ), . . . , (V n-1 , ρ n-1 ) n representations of g 1 , where g = g 1 ⊕ α(g 1 ) ⊕ ... ⊕ α n-1 (g 1 ) and V = V 0 ⊗ ... ⊗ V n-1 .
Then, according to Proposition 4.4 and Proposition 4.5, there

exists β 0 ∈ GL(V 0 ), ..., β n-1 ∈ GL(V n-1 ) such that ρ : g -→ End(V ) defined by ρ(x 0 , α(x 1 ), . . . , α n-1 (x n-1 )) = ρ 0 (x 0 ) ⊗ Id V 1 ⊗ . . . ⊗ Id V n-1 + Id V 0 ⊗ β 1 • ρ 1 (x 1 ) • β -1 1 ⊗ . . . ⊗ Id V n-1 + . . . + Id V 0 ⊗ Id V 1 ⊗ . . . ⊗ β n-1 n-1 • ρ n-1 (x n-1 ) • β -n+1 n-1 = n-1 k=0 Id V 0 ⊗ . . . ⊗ β k k • ρ k (x k ) • β -k k ⊗ . . . ⊗ Id V n-1
is an irreducible representation of the induced Lie algebra (g, [•, •]) on V , which completes the proof of (1).

For (2), set β := β 0 ⊗ ... ⊗ β n-1 . Condition (2.3) is satisfied, that is for (v 0 ⊗ ... ⊗ v n-1 ) ∈ V , we have β • ρ(X)(v 0 ⊗ ... ⊗ v n-1 ) = β • n-1 k=0 Id(v 0 ) ⊗ ... ⊗ β k k • ρ k (x k ) • β -k k (v k ) ⊗ ... ⊗ Id(v n-1 ) = n-1 k=0 β 0 (v 0 ) ⊗ ... ⊗ β k+1 k • ρ k (x k ) • β -k k (v k ) ⊗ ... ⊗ β n-1 (v n-1 ) = n-1 k=0 β 0 (v 0 ) ⊗ ... ⊗ β k+1 k • ρ k (x k ) • β -k k • β k • β -1 k (v k ) ⊗ ... ⊗ β n-1 (v n-1 ) = n-1 k=0 β 0 (v 0 ) ⊗ ... ⊗ β k+1 k • ρ k (x k ) • β -(k+1) k • β k (v k ) ⊗ ... ⊗ β n-1 (v n-1 ) = n-1 k=0 β 0 (v 0 ) ⊗ ... ⊗ ρ k (α k+1 (x k )) • β k ⊗ ... ⊗ β n-1 (v n-1 ) = ρ(α(X)) • β(v 0 ⊗ ... ⊗ v n-1 ). So (V, ρ β = β • ρ, β) is a representation of the simple multiplicative Hom-Lie algebra (g, [•, •] α , α),
which is regular if the representation of the induced Lie algebra is irreducible.

(g, [•, •] ′ α ) and (g, [•, •] ′ γ ), ϕ is still an isomorphism of semisimple Lie algebras that is, it satisfies ϕ([x, y] ′ α ) = [ϕ(x), ϕ(y)] ′ γ . Let (V, ρ β , β) and (V, ρ δ , δ) be two representations on the vector space V of respectively (g, [•, •] α , α) and (g, [•, •] γ , γ). The corresponding induced representations (V, ρ ′ α ) and (V, ρ ′ γ ) respectively of (g, [•, •] ′ α ) and (g, [•, •] ′ γ ) are equivalent. So there exists a linear automorphism T of End(V ) such that ρ ′ γ • ϕ = T • ρ ′ α
. By Skolem-Noether Theorem [START_REF] Bourbaki | Eléments de Mathématiques, Livre II, Algèbre, chapitre 8, «Modules et anneaux semi-simples[END_REF], there exists an S ∈ GL(V ) such that T = Ad S . Therefore,

(4.4) ∀x ∈ g, ρ ′ γ (ϕ(x)) = S • ρ ′ α (x) • S -1 .
Using Theorem 2.11, Theorem 3.4 and Proposition 2.9,

δ 2 • ρ ′ γ (ϕ(x)) • δ -1 = ρ δ (γ(ϕ(x))) = ρ δ (ϕ(α(x))) = δ • ρ ′ γ (ϕ(α(x))) = δ • S • ρ ′ α (α(x)) • S -1 = δ • S • β • ρ ′ α (x) • β -1 • S -1 .
So we get (4.5)

δ 2 • ρ ′ γ (ϕ(x)) = δ • S • β • ρ ′ α (x) • β -1 • S -1 .
Moreover, using (4.4) we get

(4.6) δ 2 • ρ ′ γ (ϕ(x)) = δ 2 • S • ρ ′ α (x) • S -1 • δ -1 .
Comparing (4.5) and (4.6), we will obtain

S • β = δ • S. Therefore δ = S • β • S -1 .

Weight modules of Simple multiplicative Hom-Lie algebras

We introduce and discuss in the following the root space decomposition of simple multiplicative Hom-Lie algebras, weight modules and Verma modules. be a simple multiplicative Hom-Lie algebra and let g 1 be a simple ideal generating the induced Lie algebra (g, [•, •]) such that g = g 1 ⊕ α(g 1 ) . . . ⊕ α n-1 (g 1 ).

Let h 1 be a Cartan subalgebra of g 1 (we write CSA for abbreviation). The decomposition of g 1 into root spaces relatively to h 1 is given by g

1 = h 1 ⊕ η∈∆ 1 (g 1 ) η
, where ∆ 1 is the set of roots. Since α is an automorphism of g. For all η ∈ (g 1 ) η , x ∈ g 1 and h ∈ h 1 , we have

α k ([h, x]) = η(h)α k (x), ∀0 ≤ k ≤ n -1. Namely, [α k (h), α k (x)] = η(h)α k (x) and so [α k (h), α k (x)] = η • α -k (α k (h))α k (x).
Then, α k ((g 1 ) η ) is a root space of α k (g 1 ) and we have α k ((g

1 ) η ) = (α k (g 1 )) η•α -k . The set of roots of α k (g 1 ) is given by ∆ k+1 = ∆ 1 • α -k = {η • α -k , η ∈ ∆ 1 }. Remark 5.1. If α n is an outer automorphism, then α n (h 1 ) = h 1 , otherwise, α n (h 1 ) = h 1 .
A consequence of the discussion above is the following proposition.

Proposition 5.2. Let g = g 1 ⊕ α(g 1 ) . . . ⊕ α n-1 (g 1 ) the induced Lie algebra of a simple multiplicative Hom-Lie algebra

(g, [•, •] α , α). Then (1) h = h 1 ⊕ α(h 1 ) ⊕ ... ⊕ α n-1 (h 1 ) is a CSA of g and ∆ = n-1 k=0
∆ k+1 is the set of roots with respect to h and the root space decomposition is given by g = h ⊕ n-1 k=0 η∈∆ k+1 α k (g 1 ) η .

(2) Let γ be another automorphism such that g = g 1 ⊕ γ(g 1 ) . . . ⊕ γ n-1 (g 1 ), γ n (g 1 ) = g 1 . Then,

there exists ϕ ∈ Aut(g) such that ϕ • α = γ • ϕ, (γ = ϕ • α • ϕ -1 ) and γ k (g 1 ) = γ k (h 1 ) ⊕ η∈∆ 1 •γ -k (γ k (g 1 )) η . Set ∆ ′ k+1 = ∆ 1 • γ -k = ∆ 1 • ϕ • α -k • ϕ -1 and h ′ = h 1 ⊕ γ(h 1 ) ⊕ ... ⊕ γ n-1 (h 1 ). Hence g = h ⊕ ( n-1 k=0 η∈∆ ′ k+1 (γ k (g 1 )) η )
is the root space decomposition of g with respect to h ′ .

5.2.

Weight modules of simple multiplicative Hom-Lie algebras. Let g = g 1 ⊕ ... ⊕ g n be a semisimple Lie algebra. Let h = h 1 ⊕ h 2 ⊕ ... ⊕ h n be a CSA of g and g = n + ⊕ h ⊕ n -its triangular decomposition. Let ∆ be the set of roots with respect to h. Let ρ : g -→ End(V ) be a representation

of g. Let λ ∈ h * and V λ = {v ∈ V : ρ(h)v = λ(h)v, ∀h ∈ h}. If V λ = {0}, λ is called a weight of
V and V λ is called a weight subspace of V of weight λ. Denote the set of all weights by P (V ).

If V = λ∈P (V )
V λ is a direct sum of its weight subspaces then, we say that V is a weight module. Every weight module of g is a sum of modules of the form V 1 ⊗ ... ⊗ V n where V i is a module of the simple factor g i of g. Let V = V 1 ⊗ ... ⊗ V n , then the set of weights of V is given by:

P (V ) = P (V 0 ) ⊕ ... ⊕ P (V n-1 ) = { n k=1 λ i /λ i ∈ P (V i )}
and we have

V = ⊕ (λ 1 ,...,λn)∈P (V ) V λ 1 ⊗ ... ⊗ V λn .
Let (W, ρ) be a representation of g 1 . By Proposition 4.4, we associate a representation (W, ρ) of

α k (g 1 ) such that there exists β ∈ GL(W ) satisfying ρ(α k (x)) = β k • ρ k • β -k . Proposition 5.3. Let (g, [•, •] α , α
) be a simple multiplicative Hom-Lie algebra and g = g 1 ⊕ α(g 1 ) ⊕ α 2 (g ) ⊕ ... ⊕ α n-1 (g 1 ) be its induced Lie algebra, (α n (g 1 ) = g 1 ).

(1) Consider n weight modules (V 0 , ρ 0 ), (V 1 , ρ 1 ), ..., (V n-1 , ρ n-1 ) of g 1 . Then, for all integer

k = 0, • • • , n -1, there exists a representation (V k , ρ k ) of α k (g 1 ) on V k and β k ∈ GL(V k ) satisfying ρ k (α k (x)) = β k k • ρ k (x) • β -k k for all x ∈ g 1 . Then V k is a weight module of α k (g 1 ) and β k k ((V k ) λ ) is a weight subspace of V k of weight λ • α -k . (2) Let P (V i ) be the set of weights of V i with respect to h 1 . The set of weights of V = ⊗ n-1 i=0 V i with respect to h = h 1 ⊕ α(h 1 ) ⊕ ... ⊕ α n-1 (h 1 ) is given by P h (V ) = n-1 k=0 λ i • α -k /λ i ∈ P (V i ) .
The weight decomposition of V is given by

V = ⊕ λ∈P (V ) (V 0 ) λ 0 ⊗ (V 1 ) λ 1 •α -1 ⊗ ... ⊗ (V n-1 ) λ n-1 •α 1-n . Proof. Let (V k , ρ k ) be a weight module of g 1 . Then V k = λ∈P (V k ) (V k ) λ , where (V k ) λ = {v ∈ V k /ρ k (h)v λ = λ(h)v, h ∈ h 1 }. By Proposition 4.4, there exists ρ k and β k ∈ GL(V k ) such that for h ∈ h 1 , v ∈ (V k ) λ , w have ρ l (α k (h))β k k (v) = β k k ρ(h)v = β k k (λ(h)v) = λ(h)β k k (v) = λ • α -k (α k (h))(β k k (v)). So β k k ((V k ) λ ) = (V k ) λ is a weight subspace of β k k (V k ) = V k with respect to the CSA α k (h 1 ) of α k (g 1 ) of weight λ • α -k . Moreover we have, β k k ((V k ) λ ) = (β k k (V k )) λ•α -k and V k = β k k (V k ) = λ∈P (V k ) β k k ((V k ) λ ) = λ∈P (V k ) (β k k (V k )) λ•α -k = λ∈P (V k ) (V k ) λ•α -k .
Thus, a weight decomposition of V is given by

V = λ∈P (V ) (V 0 ) λ 0 ⊗ (V 1 ) λ 1 •α -1 ⊗ ... ⊗ (V n-1 ) λ n-1 •α 1-n . Remark 5.4. Since α n is an automorphism of g 1 , if α n is an outer automorphism then α n (h 1 ) = h 1 and β n (V λ k ) = V λ k , 0 ≤ k ≤ n -1. If α n is an inner automorphism then α n (h 1 ) = h 1 and β n (V λ k ) = V λ k •α -n .
Definitions 5.5.

(1) We call a module of a simple multiplicative Hom-Lie algebra a weak weight module if it is a weight module of the induced semisimple Lie algebra. We call a weak weight subspace of a module of a simple multiplicative Hom-Lie algebra, every weight subspace of the weight module of the induced Lie algebra.

(2) A strong weight module (V, ρ β , β) of (g, [•, •] α , α) is a weight module of (g, [•, •]) such that β transforms weight subspaces to weight subspaces.

Let g be a simple Lie algebra. Then, every irreducible finite dimensional module of g is a highest weight module and its highest weight is a dominant weight. Let P + be the set of dominant weights of g and λ ∈ P + . Denote such irreducible module by V (λ) and v λ its highest weight vector.

Proposition 5.6. Keeping the same hypothesis as in Proposition 5.3 and let (V, ρ β , β) be a finite dimensional irreducible module of (g, [•, •] α , α). Then (V, ρ β , β) is a weak weight module.

Moreover, there exists (λ 0 , λ 1 , ..., λ n-1 ) dominant weights for g 1 such that

V = V (λ 0 ) ⊗ V (λ 1 • α -1 ) ⊗ ... ⊗ V (λ n-1 • α -(n-1)
)

is an irreducible g-weight module of highest weight (λ 0 , λ 1 • α -1 , ..., λ n-1 • α 1-n ).

Furthermore, there exists β 0 ∈ GL(V (λ 0 )), β 1 ∈ GL(V (λ 1 )), ..., β n-1 ∈ GL(V (λ n-1 )) such that

β k k (V (λ k )) = V (λ k • α -k ).
Proof. Let us assume that V (λ k ) is a highest weight module of highest weight λ k ∈ h * 1 of g 1 and highest weight vector

v λ k , (v λ k = 0). Since α k (g 1 ) = α k (n + 1 ) ⊕ α k (h 1 ) ⊕ α k (n - 1 )
, by Proposition 4.4 we have for all

x ∈ n + , ρ k (α k (x)) • β k k (v λ k ) = β k k • ρ k (x)(v λ k ) = β k k ( -→ 0 ) = -→ 0 . So β k k (v λ k ) is a highest weight vector of highest weight λ k • α -k of β k k (V (λ k )) = V (λ k • α -k ) as α k (g 1 )-module and we have (v λ 0 ⊗ β 1 (v λ 1 ) ⊗ ... ⊗ β n-1 n-1 (v λ n-1 )
) is a highest weight vector of V considered as g-module.

= M (λ 0 ) ⊗ M (λ 1 • α -1 ) ⊗ ... ⊗ M (λ n-1 • α -(n-1) ) is a Verma module of g = g 1 ⊕ α(g 1 ) ⊕ ... ⊕ α n-1 (g 1 ) of highest weight λ = (λ 0 , λ 1 • α -1 , ..., λ n-1 • α -(n-1) ).
Let (M (λ), ρ) the corresponding representation of g. For β ∈ GL(M (λ)), we have (M (λ), ρ β , β) is a weak weight module of (g, [•, •] α , α) where ρ β = β • ρ.

Applications

The simple Lie algebra sl(2) is the smallest simple Lie algebra which plays a distinguish role in Lie theory. In this section, we provide examples and study representations of simple Hom-Lie algebras of sl(2)-type. 2), the Hom-Lie algebras obtained by applying the Yau twist to sl(2). The twisted algebras are obtained along algebra morphisms which are automorphisms in the case of sl(2). They are determined in [START_REF] Jacobson | Lie algebras[END_REF], see also [START_REF] Elchinger | Formalité liées aux algèbres enveloppantes et études des algèbres Hom-(co)Poisson, General Mathematics[END_REF]. In this section we will study in details representations of sl(2)-type Hom-Lie algebras. The Hom-sl(2)-modules will be constructed using Theorem 2.11. One needs first to consider the set of all morphisms on sl(2) and then seek for twistings of sl(2)-modules. Lemma 6.1. Every diagonal Twist of sl( 2) is given by a morphism α defined with respect to the basis {e, f, h} by α(e) = λe, α(f

) = λ -1 f, α(h) = h, where λ is a nonzero parameter in K. Let (V, ρ β , β) be a representation of Hom-sl(2) = (sl(2), [•, •] α = α([•, •]), α) where V is an (n + 1)- dimensional vector space with basis {v 0 , . . . , v n }. Then β(v i ) = λ -i b 0 v i , 0 ≤ i ≤ n, and b 0 ∈ K.
Proof. Straightforward calculations show that the sl(2)-type Hom-Lie algebras with diagonal twist are given by morphisms α of the form α(e) = λe, α(f ) = λ -1 f, α(h) = h, where λ is a parameter different from 0 in K. Therefore Hom-sl(2) is equipped with the following bracket,

[h, e] α = 2λe, [h, f ] α = -2λ -1 f, [e, f ] α = h, where λ = 0, 1.
Let ρ be a representation of sl(2, C) on an (n + 1)-dimensional vector space V generated by {v 0 , ..., v n }. It is defined as follows

         ρ(e)v i = (n -i + 1)v i-1 , ∀ i = 1, ..., n, ρ(e)v 0 = 0, ρ(f )v i = (i + 1)v i+1 , ∀ i = 0, ..., n -1, ρ(f )v n = 0, ρ(h)v i = (n -2i)v i , ∀ i = 0, ..., n.
In the following, we twist the previous representation with respect to Hom-sl(2).

Let β ∈ End(V ) and set β(v i ) = n j=0 a ij v j . We construct maps β that satisfy Condition (2.3).

First, we apply ρ to e, then in the LHS we get ρ(α(e))β(

v i ) = λ n j=1 (n -j + 1)a ij v j-1 and on the RHS we get β(ρ(e)v i ) = (n -i + 1) n j=0 a i-1,j v j .
For j = n, the LHS vanishes; (n -i + 1)a i-1,n = 0 and so we get (6.1) a 0,n = .... = a n-1,n = 0.

Considering the equality, one has (n -i + 1)

n-1 j=0 a i-1,j v j = λ n j=1 (n -j + 1)a ij v j-1 .
Rewriting the equality we get

(n -i + 1)(a i-1,0 v 0 + a i-1,1 v 1 + ... + a i-1,i-1 v i-1 + ... + a i-1,n-1 v n-1 ) = λ(na i1 v 0 + (n -1)a i1 v 1 + ... + (n -(i -1))a ii v i-1 + ... + a in v n-1 ).
More precisely, we have (n -i + 1)a i-1,0 v 0 + (n -i + 1)a i-1,1 v 1 + ... + (n -i + 1)a i-1,i-1 v i-1 + ... + (n -i + 1)a i-1,n-1 v n-1

= λna i1 v 0 + λ(n -1)a i1 v 1 + ... + λ(n -(i -1))a ii v i-1 + ... + λa in v n-1 .

By identification we get the following system of n equations

                                      
(n -i + 1)a i-1,0 = λna i1

(n -i + 1)a i-1,1 = λ(n -1)a i2

(n -i + 1)a i-1,2 = λ(n -2)a i3 . . .

(n -i + 1)a i-1,i-1 = λ(n -i + 1)a ii (n -i + 1)a ii = λ(n -i)a i+1,i+1 . . .

(n -i + 1)a i-1,n-1 = λa in .

Setting i = 1, we get a 00 = λa 11 and a 12 = ... = a 1n and from (6.1) we get (n -2)a 2,n-1 = λa 3n Similarly, (6.3) leads to a 34 = ... = a 3n = 0. One may show that a 30 and a 31 also vanish, and so one.

The case i = n leads to a n-1,0 = λna n1 , a n-1,1 = λ(n -1)a n2 , • • • , a n-1,n-1 = λa nn . To check that coefficients a n1 = a n2 = ... = a n,n-1 = 0, we use the identity (2.3) with generator f . We have ρ(α(f ))β(v i ) = λ -1 n-1 j=0 a ij (j + 1)v j+1 and β(ρ(f )v i ) = (i + 1) n j=0 a i+1,j v j . Writing the equality, we obtain λ -1 a i0 v 1 + 2λ -1 a i1 v 2 + 3λ -1 a i2 v 3 + ... + nλ -1 a i,n-1 v n = (i + 1)a i+1,0 v 0 + ... + (i + 1)a i+1,n v n .

Then we get the following system of n equations                a i+1,0 = 0, a i+1,1 = 0 2λ -1 a i1 v 2 = (i + 1)a i+1,2 v 2 . . . nλ -1 a i,n-1 v n = (i + 1)a i+1,n-1 v n

Solving the system, we obtain a diagonal matrix where λ -1 a ii = a i+1,i+1 , ∀1 ≤ i ≤ n.

Setting a 00 = b 0 , we have a ii = λ -i b 0 for i = 1, . . . , n.

We proved that β is given by a diagonal matrix that for all v i ∈ V , β(v i ) = λ -i b 0 v i , where b 0 is a nonzero scalar. In the following, we characterize the action with respect to β, see also [START_REF] Li | Representations of 3-dimensional simple multiplicative Hom-Lie Algebras[END_REF]. = λ -2i µ 1 γ 0 + λ -2i η 0 iν 0 -2i 2 η 0 + 2i i -1 2 η 0 = λ -2i µ 1 γ 0 + iη 0 ν 0 -(i + 1)η 0 .

Therefore, we get the following connections characterizing the parameters such that ρ β is a Homsl(2)-module on a (n + 1)-dimensional vector space V :

η i = (λ -1 ) i η 0 , i = 0 and γ i µ i+1 = λ -2i µ 1 γ 0 + iη 0 (ν 0 -(i + 1)η 0 ) .

Setting η 0 = b 0 , µ 1 = 1, γ 0 = n leads to the same result obtained in Theorem 6.2.

  where ρ is the representation of the induced Lie algebra (g, [•, •]). It is called regular if the representation

Proposition 3 . 2 (Proposition 3 . 3 .

 3233 [START_REF] Benayadi | Hom-Lie algebras with symmetric invariant non-degenerate bilinear form[END_REF]). Let (g, [•, •]) be a simple Lie algebra and let α ∈ Aut(g).Then, g α = (g, α([•, •]), α) is a simple Hom-Lie algebra. Simple multiplicative Hom-Lie algebras are regular Hom-Lie algebras.

Proposition 4 . 1 .

 41 Let (V, ρ β , β) be a representation of a simple multiplicative Hom-Lie algebra (g, [•, •] α , α). Then, Im(β) and Ker(β) are submodules of V for (g, [•, •] α , α). Moreover, we have an isomorphism of (g, [•, •] α , α)-modules β : V /Ker(β) -→ Im(β).

3. 4 ,

 4 Li Theorem 2.11 and Proposition 4.4, we provide a construction of representations of simple multiplicative Hom-Lie algebras.

Proposition 4 . 7 .

 47 Let (g, [•, •] α , α) and (g, [•, •] γ , γ) be two isomorphic simple multiplicative Hom-Lie algebras generated by a simple ideal g1 . Let (g, [•, •] ′ α ) and (g, [•, •] ′ γ ) their induced Lie algebras. If (V, ρ β , β) (resp. (V, ρ δ , δ)) is an irreducible representation of (g, [•, •] α , α) (resp. (g, [•, •] γ , γ)) onthe same vector space V . Then δ and β are conjugated by S ∈ GL(V ).Proof. Since (g, [•, •] α , α) and (g, [•, •] γ , γ) are isomorphic, then there exists a linear automorphism ϕ of g such that ϕ • α = γ • ϕ and ϕ([x, y] α ) = [ϕ(x), ϕ(y)] γ . On the induced Lie algebras

5. 1 .

 1 Root space decomposition of simple multiplicative Hom-Lie algebras. Let (g, [•, •] α , α)

Remark 5 . 7 .

 57 Let β = β 0 ⊗...⊗β n-1 and set ρ β := β •ρ, where ρ is the representation of the induced semisimple Lie algebra. Then, (V, ρ β , β) is a representation of the simple multiplicative Hom-Lie algebra (g, [•, •] α , α). It turns out that β k do not necessary transforms weight subspaces to weight subspaces. If it is the case then the weight module of the induced Lie algebra becomes weight module for the multiplicative Hom-Lie algebra (g, [•, •] α , α). The same holds for highest weight modules.Let M (λ) be a Verma module of g of highest weight λ ∈ P . If there exists α ∈ ∆ + (the set of positive roots) such that λ(α) ∈ N, then M (λ) is reducible and there exists a maximal submodule M (λ) of M (λ) such that V (λ) := M (λ)/M (λ) is irreducible. Otherwise, λ(α) ∈ R\N and M (λ) is irreducible. Let M (λ 0 ), ..., M (λ n-1 ) be n Verma modules of g 1 . Then M (λ) :

6. 1 .

 1 Representations of the sl(2)-type Hom-Lie algebras. We consider the usual Lie algebra sl(2) generated by {e, f, h} and defined by the brackets [h, e] = 2e, [h, f ] = -2f, [e, f ] = h. We call sl(2)-type Hom-Lie algebras or Hom-sl(

(6. 2 )

 2 a 12 = ... = a 1n = 0 For i = 2 we have                     (n -1)a 10 = λna 21 (n -1)a 11 = λ(n -1)a 22 (n -1)a 12 = λ(n -2)a 23 . . . (n -1)a 1,n-1 = λa 2nUsing (6.1) and (6.2) we obtain (6.3)a 21 = a 23 = ... = a 2,n-1 = 0It remains to verify that a 20 = 0.For i = 3 we have                      (n-2)a 20 = λna 31 (n -2)a 21 = λ(n -1)a 32 a 22 = λ.a 33 . . .

= λ -2i µ 1 γ 0 + η 0 i- 1 k=0λ

 1 -2i ν 0 -2iη 0 + 2kη 0 = λ -2i µ 1 γ 0 + λ -2i η 0

Theorem 6.2. The representations (V, ρ β , β) of sl(2)-type Hom-Lie algebras on a vector space V equipped with a basis {v 0 , ..., v n } are given by β such that β(v i ) = λ -i b 0 v i , b 0 = 0 and the actions defined by ρ β (e)v i = (n -i + 1)λ -i+1 b 0 v i-1 , i = 1, ..., n.

Proof. It is straightforward using Theorem 2.11.

From the foregoing theorem we can classify the irreducible sl(2)-weight modules as follows.

Theorem 6.3. There are precisely four types of irreducible Hom-sl(2)-modules. The actions are described in the following.

(1) The finite-dimensional irreducible modules with basis {v 0 , v 1 , ..., v n } and where

(2) The irreducible infinite-dimensional lowest weight Hom-sl(2)-modules, with a basis of heigenvectors {v 0 , v 1 , ...} and nonnegative integer τ , such that

(3) The irreducible infinite-dimensional highest weight Hom-sl(2)-modules, with a basis of h-

6.2. General method to compute Hom-sl(2)-modules. Let ρ β : sl(2) → End(V ) be the representation of the Hom-sl(2) and consider the action of the generators {e, f, h} as follows:

The next step is to calculate the parameters ν i , γ i , µ i , and η i so that ρ β be a Hom-sl(2)-module on a (n + 1)-dimensional vector space V . Straightforward computation using Definition 2.8 gives for each i = 0, ..., n the following 5 equations

Conditions (4) and (5) lead to η i = (λ -1 ) i η 0 , i = 0. Then (1) ⇒ aγ i µ i+1 -λ -1 µ i γ i-1 = ν i (λ -1 ) i η 0 .

(

. Now for µ i and γ i we get

) i-λ η 0 + ν i (λ -1 ) i η 0 = λ -2×3 µ i-2 γ i-3 + λ -2×2 ν i-2 (λ -1 ) i-2 η 0 + (λ -2 )ν i-1 (λ -1 ) i-1 η 0 + ν i (λ -1 ) i η 0 = λ -2×3 µ i-2 γ i-3 + ν i-2 (λ -1 ) i+2 η 0 + ν i-1 (λ -1 ) i+1 η 0 + ν i (λ -1 ) i η 0


Louis Azan , "The romantic conception of the entrepreneur in

The aim of this paper is to demonstrate that the figure of the Schumpeterian entrepreneur is deeply marked by a Romantic imagination, critical of Utilitarianism. Schumpeter constructs the entrepreneur as a creative and dynamic agent, who succeeds in creating something radically new by the force of his will and his freedom of spirit, thus destroying the existing equilibrium. He is not a rational economic agent, a homo oeconomicus, but a romantic man who uses imagination and intuition in his actions. Like the Romantic authors, the Austrian economist puts forward the idea that economic life is marked by an incessant flow of innovations, destroying the old so that the new may emerge. Moreover, the decline of the entrepreneurial function is interpreted by Schumpeter from a romantic perspective, with the idea that capitalist modernity is a force for the rationalization of the world and the routinization of human existence, which no longer allows entrepreneurs to deploy their creative energy.

INTRODUCTION

The links between Romanticism and Economics have rarely been properly investigated, and for a good reason; Economic Science was born out of the 18th century, the Enlightenment and Utilitarianism, whereas Romanticism is precisely an anti-Enlightenment intellectual movement, critical of the claim that reason alone can think and explain the world. While Romanticism is generally regarded as one of the PhD Student in History of Economic Thought (third year), in LEFMI (Université Picardie Jules 1 Verne) and PHARE (Paris 1 Panthéon-Sorbonne).

greatest turning points in Western thought (see, for example, Isaiah Berlin (1999)), it seems to have left no trace on modern economic thought. The aim of this paper is therefore to show that the thought of one of the most influential economists of the twentieth century, Joseph Aloïs Schumpeter, is deeply marked by Romanticism, particularly in his way of representing the figure of the entrepreneur.

By Romanticism, we do not mean here simply an aesthetic movement born in Germany and England at the end of the 18th century. This artistic movement is characterized by a revolt against classicism and the Enlightenment, with a praise of the feeling, of the individuality, considered as the source of the creation. Among the Romantic authors, we find a valorization of the creative freedom, in rupture with the codes and the classical conventions. Moreover, on the political level, Romanticism constitutes a reaction to the French universalism resulting from the Enlightenment, with a counter-revolutionary dimension.

However, in this article, we will understand romanticism in a broader sense, namely a truly Weltanschauung, an alternative conception of the world to that of the Enlightenment. As Schumpeter mentioned in his famous History of Economic Analysis (1954), Romanticism is a "cultural antipode" to Utilitarianism. For Berlin in The Roots of Romanticism (1999), this Weltanschauung is notably characterized by (i) an emphasis on individual creativity and the role of human imagination; (ii) a critique of the reduction of men to calculating machines.

According to Schumpeter, Romanticism appears to be a "revolt against convention, particularly against rationalized convention: feeling rose against cold reason; spontaneous impulse against utilitarian logic; intuition against analysis; the 'soul' against the intellect; the romance of national history against the artefacts of the Enlightenment" (Schumpeter, 1954, p.396). He adds: "A romanticist or any writer influenced by the romantic attitude would, of course, look upon industrial life and its problems in a nonbourgeois spirit and take views quite different from the Benthamite ones. More generally, he would feel a healthy disgust at the utilitarian tendency to reduce the colorful variety of social patterns and processes to a few bald generalizations about thoroughly rationalized hedonic interests. And he would build where utilitarianism leaves a void-or else provides a dump for what is simply nonsense from its standpoint-a shrine for the historically unique and for the values of the extrarational" (Schumpeter, 1954, p.387).

Schumpeter thus characterizes the romantic attitude as an alternative representation of the world to Utilitarianism, stemming from the philosophy of the Enlightenment.

The goal of this paper is to demonstrate that Schumpeter's conception of the entrepreneur, his role and his place in the capitalist system, reflects a romantic attitude and conception of the economic and social world. In a negative way, Schumpeter criticized Utilitarianism, the figure of Homo Oeconomicus (O, Boyle, 2016), as well as the phenomenon of rationalization and bureaucratization of the world, undermining the creative possibilities of the innovative entrepreneur. In a positive way, he emphasized the role of intuition and imagination in the economic decision of the entrepreneur, he transposed the chivalric figure to the business world, and he insists on the dynamism of social life.

It seems that few commentators have insisted on this romantic dimension of Schumpeter's thinking. [START_REF] Shionoya | Schumpeter and evolution : a philosophical interpretation[END_REF] underlined this romantic dimension in an article on Schumpeter's conception of economic evolution. And Bronk, in his book The Romantic Economist (2009) , has characterized 2 Schumpeter as one of the economists whose view of the economic world was marked by the romantic imagination. Among the lessons of romanticism identified by Richard Bronk, which he believes can be used to enrich economic analysis, it seems to us that one of them corresponds to Schumpeter's approach: « the need for a fuller psychology of human motivation than is allowed for by some versions of rationalism -one that recognises, in particular, the role of imagination and sentiment as well as reason » (Bronk, p.87). But the existing literature mostly focuses on the influences of German-speaking authors of the late nineteenth and early twentieth centuries: notably Nietzsche (Santarelli, [START_REF] Sayre R | The Emergence of a Vision : The Development of Schumpeter's Theory of Entrepreneurship[END_REF][START_REF] Reinert | Creative Destruction in Economics : Nietzsche, Sombart, Schumpeter[END_REF][START_REF] Lapied | L'entrepreneur schumpéterien est-il surhumain ?[END_REF], Sombart [START_REF] Gislain | Les origines de l'entrepreneur schumpéterien[END_REF] or Weber [START_REF] Gislain | Les origines de l'entrepreneur schumpéterien[END_REF], Love, 2017), who are authors who seem to have influenced Schumpeter at that time.

Through a work of history of economic thought, the aim of this article is then to establish the romantic influence of Schumpeter's conception of the entrepreneur, in order to highlight the way in which a Weltanschauung is always underlying any economic theory. Moreover, it seems to us that this romantic dimension of Schumpeter's work is what constitutes its profound originality, which is very often evaded by economists claiming to be part of his legacy. Indeed, endogenous growth models are called neo-Schumpeterian models, insofar as they insist on the central role of innovation in economic growth, taking up the Schumpeterian concept of creative destruction . In our view, however, these models tend to betray 3 the specificity and depth of Schumpeter's thought, and in particular his conception of rationality or his vision of the dynamics of capitalism. By trying to fit the Austrian economist's thought into a standard framework, with perfectly rational agents and equilibrium thinking, the originality of his thought is partly denied, and in particular the romantic dimension that we would like to emphasize. As Bronk points out: "Endogenous Growth Theory may build in creative destruction in a stylised manner, but it still models its effect on the 'steady-state growth rate' ; and it still assumes that economic agents are driven by rational probability-calculating expectations and the tendency to maximise expected consumption and profits" (Bronk, 2009, p.80). Conversely, Schumpeter's romantic influence allows us to think of a more original and richer conception of the individual and of rationality, which has an influence on the way we think about capitalist economic dynamics.

We will focus on two romantic dimensions: (i) a macro-social dimension of criticism of the rationalization of the world induced by capitalist modernity, which undermines the creative possibilities of First, a few brief reminders about Schumpeter's entrepreneur, who occupies a central place in his theoretical edifice . According to Schumpeter, the entrepreneur is the economic agent who implements 4 new productive combinations, who innovates, which is the basis of the dynamics of capitalism. He is therefore a creative agent, who tends to upset the productive order by introducing new methods of production, or by setting up radically new forms of work organization . The entrepreneur is a leader , and Policy, 1990, n°4. The affinity between this conception of the entrepreneur and Nietzsche's thought has been the subject of several studies (notably [START_REF] Reinert | Creative Destruction in Economics : Nietzsche, Sombart, Schumpeter[END_REF][START_REF] Lapied | L'entrepreneur schumpéterien est-il surhumain ?[END_REF].

Indeed, on this point, Schumpeter is opposed to other Austrian economists who are more often 7 opposed to financing credit investments on the basis of credit, because of the inflation that this generates. While the Austrians defended the financing of innovations on the basis of prior savings, Schumpeter emphasized the importance of credit in capitalism. It is credit that allows money to be diverted from a declining sector to an innovative sector. For a more detailed presentation of the role of money and credit for Schumpeter, see Lakomski-Laguerre, The moneraty institutions of capitalism, The economic thought of J.A. Schumpeter, 2002. In their article entitled "Figures of anti-capitalist Romanticism", Robert Sayre and Michel Löwy 8

insist on the idea that Romanticism appears as a reaction to capitalist modernity, and notably its rationalizing force. In particular, they take up the Weberian thesis of capitalism as a force of disenchantment of the world : "an important aspect of romanticism (...) is the "re-enchantment" of the 9 world by the imagination" (Sayre, Löwi, p.106). Indeed, for the romantic sensibility, the capitalist modernity led to the loss of a "set of qualitative values -ethical, social, cultural -as opposed to the rational commercial calculation and the exchange value" (Sayre, Löwi, p.109). One of the great values lost by modernity, or at least that can't fully developed in modern times, is "individual subjectivity".

Indeed, according to them, if modernity tends to produce independent individualities within the framework of the division of labor, when "these individuals turn into subjective individualities, and begin to explore the inner world of their particular feelings, they enter into contradiction with a system based on quantitative calculation and standardization. And when they claim the free play of their imaginative fantasy, they come up against the extreme mechanization and flatness of the world created by capitalist relations. Romanticism represents the revolt of the affectivity repressed, channeled and deformed under capitalism, and of the "magic" of the imagination banished from the capitalist world" (Sayre, Löwi, p.110).

According to us, Schumpeter also develops a "critique" of capitalist modernity as a force of 10 rationalization of the world. Indeed, in Capitalism, Socialism and Democracy (1942) , the decline of the 11 entrepreneurial function is interpreted by the Austrian economist in a romantic perspective, with the idea of capitalism as a force of rationalization, bureaucratization, and routinization. Schumpeter writes: "The romance of earlier commercial adventure is rapidly wearing away, because so many more things can be strictly calculated that had of old to be visualized in a flash of genius" (CSD, p.132). We find here the opposition between calculating rationality and genial intuition; modernity would value calculation, The disenchantment of the world is the central thesis that runs through Weber's entire work. It refers 9 to the idea that modernity is a force of rationalization of all social activities. The modern world is also marked by a form of "demagification", in the sense that the world is no longer interpreted by religious or mythical categories, but by a purely instrumental and scientific rationality.

Our aim is obviously not to make of Schumpeter a critical thinker of capitalism, but to identify 10 within his work a critique of capitalist modernity, as a force of rationalization.

CSD now.

11 nevertheless essential to the perpetuation of the dynamics of capitalism. Basically, capitalism undermines the foundations on which it rests by the bureaucratization and routinization of human existence that it engenders .

12 Furthermore, another aspect of a romantic representation of the world is a nostalgia for a precapitalist past, in which certain values were able to unfold. In Capitalism, Socialism and Democracy (1942) again, we also find a form of nostalgia for a medieval past (partly fantasized and idealized), in 13 which singular individualities deployed their energy on the battlefield. Thus, in the military field, we find this modern break with an individualistic conception of war:

« Of old, roughly up to and including the Napoleonic Wars, generalship meant leadership and success meant the personal success of the man in command who earned corresponding "profits" in terms of social prestige. The technique of warfare and the structure of armies being what they were, the individual decision and driving power of the leading man-even his actual presence on a showy horse-were essential elements in the strategical and tactical situations. Napoleon's presence was, and had to be, actually felt on his battlefields » (CSD, p.133).

Conversely, modernity is characterized by a routinization of war, which is no longer the prerogative of singular individualities:

"Rationalized and specialized office work will eventually blot out personality, the calculable result, the "vision". The leading man no longer has the opportunity to fling himself into the fray. He is becoming just another office worker-and one who is not always difficult to replace" (CSD, p.133).

For Schumpeter, we find such a process in the economic field, with the routinization of innovation induced by capitalism, which is therefore no longer implemented by entrepreneurs: « Now a similar social process -in the last analysis the same social process-undermines the role and, along with the role, the social position of the capitalist entrepreneur » (CSD, p.134).

We find in Schumpeter two romantic themes at a macro-social level : (i) the idea of capitalist modernity as a force for rationalizing the world, undermining individual subjectivity and thus the possibility for creative individualities to implement innovations, which are nevertheless at the foundation of the capitalist dynamic ; (ii) a nostalgia for a pre-capitalist past (notably medieval), in which these individualities were not stifled by this phenomenon of bureaucratization of human activities, and notably of the entrepreneurial activity.

According to Schumpeter, this bureaucratization would lead to a socialist society, not because of a 12 proletarian revolution, but because the entrepreneurial function would decline. This would lead to a bureaucratized innovation, implemented by offices and commissions, rather than by individuals with extraordinary qualities (which he identifies in the 1940s as an ongoing process).

The Romantic movement is characterized precisely by an idealization of a bygone past, notably the 13 Antiquity and the Middle Ages.

As Schumpeter puts it:

« Also, capitalist civilisation is rationalistic and "anti-heroic". The two go together of course. Success in industry and commerce requires a lot of stamina, yet industrial and commercial activity is essentially unheroic in the knight's sense -no flourishing of swords about it, not much physical prowess, no chance to gallop the armoured horse into the enemy, preferably a heretic or heathen -and the ideology that glorifies the idea of fighting for fighting's sake and victory for victory's sake understandably withers in the office among all the columns of figures » Further on, the Austrian economist adds that the industrial bourgeoisie hates a military ideology that clashes with its "rational" utilitarianism. Yet, in Schumpeter's analysis of the entrepreneur's motivations, we find this "military ideology" (of fighting for the sake of fighting), as well as a heroic and romantic dimension to the entrepreneur's action .

14

In a disenchanted civilization, the entrepreneur paradoxically appears as a break with bourgeois ideology (of utilitarian inspiration for Schumpeter, marked by calculation and the optimal articulation between means and ends) ; he is this subsistence in the capitalist universe of a force with a romantic mentality (and therefore partly anti-modern), but one that allows for the disruption of the productive order necessary for the capitalist dynamic.

At a micro-social level, it is a matter of clarifying Schumpeter's vision of these creative individualities, i.e. of these entrepreneurs, and in particular the difference between this figure of the entrepreneur and that of the homo oeconomicus.

A romantic micro-social dimension: the entrepreneur as a creative and dynamic economic agent.

According to Berlin, one of the main lessons of Romanticism is the central importance of individual creativity, including the creation of new values, goals and visions (Bronk, 2009). And, precisely, the role of the entrepreneur in Schumpeter's view is the implementation of innovations (i.e new productive combinations), breaking with routine. The Schumpeterian entrepreneur is a dynamic and creative agent, which places him at odds with the traditional figure of the economic man.

In his book, Richard Bronk makes a distinction between two types of men: (1) homo oeconomicus, i.e the economic man maximizing his utility, who deploys an optimizing rationality in his action;

(2) homo romanticus, "the self-creating, sentimental (…) and imaginative social anima", an agent that uses imagination and creativity in his action. It seems to us that this distinction resonates with Schumpeter's

We will insist on this dimension in the second part of this article.

14 opposition between two types of economic agents : the ordinary and simple executor, and the 15 entrepreneur in the strict sense.

a. The ordinary executor as a routine agent of statics.

According to Schumpeter, the ordinary executor corresponds to the economic agent of the neoclassics, i.e the homo oeconomicus. He is a perfectly rational agent, who does not innovate or create anything; he merely adapts to quantitative variations resulting from external disturbances [START_REF] Gislain | Les origines de l'entrepreneur schumpéterien[END_REF] .

16

He can sometimes change his production techniques at the margin, but only by passive adaptation to exogenous changes, and above all "without truly leaving the usual path" (TEE, pp.79-80). In fact, "it is never a question of doing something entirely new, but only of adapting to new circumstances" (TEE, p.41).

In other words, the ordinary executor is a routine agent, who fails to break with habit and routine.

For the Austrian economist, this routine agent is the Walrasian economic agent, adapted to a general equilibrium model, in which "any creative role is absent" (TEE, p.28). According to Schumpeter, the Walrasian theory fails to adequately describe the capitalist economic system, which is marked by evolution: it is only relevant to describe a relatively stable economic world, marked by homogeneous and continuous growth. The ordinary executor is therefore the economic agent of the "circuit", whose mechanical behaviour contributes to maintaining the stability of the equilibrium.

b. The entrepreneur as a creative agent of dynamics.

The figure of the Schumpeterian entrepreneur appears as a theoretical counterpart of this ordinary executor. Indeed, he has a creative spirit, and an excess of willpower, which leads him to introduce something radically new into the productive order; and this creative spirit is necessary to the economic development. The goal of the entrepreneur on the economic level is the accomplishment of something other than what is accomplished by the usual conduct (Gislain, 2012, p.9). Thus, we find an explicit break between the figure of the entrepreneur and the economic man (O'Boyle, 2016): "choosing new methods is not self-evident and is not without more a conceptual element of rational economic activity" (TEE, p.79). Schumpeter adds: "the picture of an individualistic, rational and hedonistic egoism" (TEE, p.88) does not exactly capture the behaviour of the entrepreneurial type he constructs; if "economic man (...) can be reduced to problems of maximum and minimum, then "entrepreneurs are not economic men in the theoretical sense" (Schumpeter, 1946, quoted by Gislain, 2012, p.11).

In other words, the behaviour of the Schumpeterian entrepreneur cannot be understood within the standard framework of an optimizing rationality, articulating means and ends in an optimal way.

Thus, the motivations of the entrepreneur are partly orthogonal to those of the economic man.

Among these motivations, the Austrian economist insists in particular on the will to conquer, and to achieve success for the sake of success itself:

"there is the will to conquer: the impulse to fight, to prove oneself superior to others, to succeed for the sake, not of the fruits of success, but of success itself. From this aspect, economic action becomes akin to sport -there are financial races, or rather boxing-matches. The financial result is a secondary consideration, or, at all events, mainly valued as an index of success and as a symptom of victory (…) And again we are faced with a motivation characteristically different from that of "satisfaction of wants" in the sense defined above, or from, to put the same thing into other words, "hedonistic adaptation." (TED,p.45).

With this motivation, we find the military ideology evoked by Schumpeter in Capitalism, Socialism and Democracy; the motivation that has precisely vanished in the disenchanted civilization of capitalism. This motive for action is completely different from traditional economic reason, since combat and victory are sought for their own sake, and not for the utility and profit that they provide. This conception of the entrepreneur is close to the ancient, feudal meaning of the term entrepreneur described by Vérin in her book Entrepreneur, Enterprise, History of an Idea (1982). In this medieval, chivalric sense, we find the idea of an absence of distance "between an end pursued and an action to be produced to reach it, since it is the action alone which is realization of oneself, without separable finality of the act itself" (Vérin, 1982, p.45).

Thus, the action of the Schumpeterian entrepreneur is not justified solely by a previously established end (Bronk, 2009), which would be the search for profit, with an action that would only aim at an optimal articulation between the means and this pursued end. Such a conception of the entrepreneur breaks with the instrumental rationality traditionally associated with the economic man.

Another motivation of the entrepreneur evoked by Schumpeter is "the joy of creating" a new form,

i.e a joy in acting and innovating, in getting off the beaten track, which presupposes a freedom of imagination, as well as a strong will.

Indeed, the entrepreneur is the economic agent who manages, by the force of his will, to extricate himself from the beaten track followed by the ordinary executors (prisoners of the routine), in order to set up new productive combinations. If he manages to break out of the routine, it is because the entrepreneur is characterized by an "excess of force" that is deployed in the economic field (energy that was previously deployed in the military field) . As Schumpeter puts it in Imperialism and Social Classes (1972):

17 "There is much less excess energy to be vented in war and conquest than in any precapitalist society. What excess energy there is flows largely into industry itself, accounts for its shining figures -the type of the captain of industry -and for the rest is applied to art, science, and the social struggle » (ISC, p.69).

According to Schumpeter, to fight against the natural tendency of man to act in a routine way, "an expense of new will and of another kind becomes necessary (...), it is necessary to conquer of high struggle space and time for the conception and the elaboration of new combinations" (TEE, p.84). Such "freedom of spirit supposes a force which exceeds by far the requirements of the everyday life, it is by nature something specific and rare" (TEE, p.84). For the most banal individual, the ordinary man, that of the crowd, of the masses, to act outside the daily routine is impossible, because that would suppose to reinvent the world in which one lives (Dannequin, 2012, p.7 ). On the other hand, the entrepreneur is capable of 18 producing something radically new through his strength and imagination, using his creative freedom.

Once more, it is precisely the characteristic of Romanticism to insist on the importance of imagination, of creative freedom, which are the reflection of the dynamism of social life. For the Romantics, human energy is the producer of new and original creations, which is opposed to a rational, predictable, immobile world (which characterizes the Enlightenment (Berlin, 1999)).

Finally, because of this joy in creation and innovation, the entrepreneur does not hesitate to act even though he lacks all the data necessary for the decision; in this, he makes "risky attempts" to bring changes in the economy (TEE, p.92). As Schumpeter puts is, in a "given strategic situation, we must act, even if the data that could be obtained for the action are lacking; in the same way, in economic life we must act without having worked out in all its details what is to happen" (TEE, p.83). This supposes to have a "glance", and the capacity to see things in a way that the experience confirms then, even if on the moment we cannot justify it" (Ibid). The entrepreneur is thus characterized by his intuition and his ability to project himself into an uncertain horizon.

Schumpeter's work is marked by the presence of numerous military metaphors, with multiple Schumpeter », Revue Intervention économiques. In this article, Dannequin puts forward the idea that the Schumpeterian conception of the entrepreneur is inspired by eugenic thought, with a form of hierarchization between individuals, between those who are made to obey (the mass), and those who are made to lead (the entrepreneur in the economic field). We will have the opportunity to come back to this idea in the third part of this article.

We can see the difference between the entrepreneur's action and the action of a reasonable man, patiently and prudently deliberating on the different possible outcomes of his decision in order to maximize his chances of success. Conversely, the entrepreneur is imbued with chivalric virtues of romantic inspiration, namely courage, willpower, and the ability to seize an uncertain opportunity (Vérin, 1983, p.23). The action of the entrepreneur is thus carried out by intuition rather than by calculation, by imagination rather than by custom and routine (thus showing a freedom of spirit). That is why he is a romantic man rather than an economic man: he is not the rational and calculating man of statics, but the energetic and creative man, allowing economic evolution to continue.

If the rationality of the Schumpeterian entrepreneur is of a different type than standard rationality, it is because the entrepreneur does not adapt to existing data, but produces a new world. He is capable of projecting himself into an uncertain environment, by forming anticipations about the future, and by shaping the future (Lakomski-Laguerre, 2002, p.177). In this respect, Schumpeter's thinking can be compared to that of the post-Keynesian economist Shackle, who insists on the role of imagination in economic decision-making processes, and in particular in the anticipations formed by agents operating in a world marked by radical uncertainty (Ibid). Besides, in his book, Richard Bronk notes that Shackle's thought is also underpinned by a romantic dimension, insofar as it insists on this central role of the imagination. As Shackle puts it in Epistemics and Economics : « Not everything that economics touches is fit to be turned to certainty and pure reason. By tacitly assuming that the right conduct can always be discovered by taking orderly thought, and that this is how men' s conduct is formed, economics has precluded itself from understanding the vast area of human enterprise where disorder is the essence of the situation, the areas of break-away, of origination, of poetic creation or innovation in elevated contexts or in the mundane one of business, and of conflict and cut-throat struggle » (Shackle, 1991, p.23). We find precisely the same conception of economic dynamics in Schumpeter, with the entrepreneur as an agent capable of upsetting the productive order.

To conclude on this part, let's quote Shionoya : "Schumpeter's division of the static-dynamic typology can (…) be compared with Nietzsche's distinction of artistic forms between Dionysian creation and destruction and Apollonian equilibrium and order. Life resides only in the animate and organic entities.

This is a Romanticist legacy to economics in emphasizing the individual creative spirit » (Shionoya, 2008, p.18). This valorization of the figure of the artist is a theme dear to Romanticism: man is not a calculator, but he is a creator. The creative flow, the vital impulse , must be at the heart of social life, rather than 19 being smothered by hedonistic and rational considerations.

c. The economic dynamics.

Some commentators (notably Santarelli, [START_REF] Sayre R | The Emergence of a Vision : The Development of Schumpeter's Theory of Entrepreneurship[END_REF] have noted the influence of Bergson's thought on Schumpeter's thinking, particularly the concept of creative evolution. Bergson's ideas were part of the cultural bath in which intellectuals were immersed at the time.

As we have said, Romanticism insists on the dynamism of social life, and breaks with a static and routine vision of the world. In Schumpeter's, the consequence of this conception of the entrepreneur is a singular vision of economic development as cyclical and discontinuous.

It is notably in chapter IV of the Theory of Economic Evolution that Schumpeter sets out his version of economic development, with an analysis of the "decisive phenomenon of cyclical movement" (TEE, II, p.77). In the first place, Schumpeter clearly affirms his break with the vision of a "continuous and uninterrupted" evolution, i.e. of a "progressive development" (Ibid). In contrast to such a vision, the Austrian economist affirms that "this capital movement of the national economy does not take place in an uninterrupted way and that nothing disturbs it. Counter-movements, backlashes, and events of all kinds appear that impede this evolutionary march" (Ibid). Even more: these contrary movements "put an end to this evolution", so that "the new evolution starts from new premises and in part from new people, many old hopes and old values have been buried forever; new values have been born" (TEE, II, p.78). Here we see the Schumpeterian idea of destruction-creation appearing, where the break of the entrepreneurs with the established order leads to the destruction of the old order.

Such a process leads the economy to be perpetually disrupted in its equilibrium, thus characterized by an "apparently irregular movement, which (...) appears as an effort towards a different equilibrium" (TEE, II, p.92). However, for Schumpeter, this cyclical and bumpy dynamic of economic evolution should not be considered as a problem, and necessarily negative (contrary to the Austrian tradition). As Schumpeter writes:

"But we shall grasp insufficiently the essence and influences of the crisis, if we regard it only as the cessation of the tendency towards prosperity and characterize it by purely negative signs. Rather, it is two positive acts that constitute its essence and that in the national economy are much more characteristic of it than the features we have just retained" (TEE, II, p.96).

In other words, certainly, the innovations of the entrepreneurs upset radically the balance and the productive order, involving economic crises and difficulties for certain sectors of activity, but that is not only harmful. Indeed: "the process of destruction leads (...) to an equilibrium which is different from the previous one" (TEE, II, p.97), eliminating from then on the companies unable to adapt to the new data of the economy, that is to say the least able : "(T)he old enterprises (...) have three options: either to disappear, if for personal or objective reasons they cannot be adapted; or to fold up their sails and try to live in a position that is now modest; or finally, of their own accord or with the help of others, either to change economic branch, or according to the new circumstances, to pass to other technical or commercial arrangements, which in many cases amounts to an extension of the production" (TEE, II, 97).

The economic depression caused by the implementation of new combinations is thus characterized by the elimination of the least robust enterprises, which has a positive dimension for Schumpeter. For the Austrian economist, this "process of depression produces yet another effect (...): it carries out what the boom has promised. This action is lasting, while the phenomena felt unpleasantly are temporary: the stream of goods is enriched, production is partially reorganized, the cost of production is decreased, and what at first appeared to be profit, finally increases the lasting income in kind" (TEE, II, p.99). In other words, boom and bust go hand in hand, in the same way that creation and destruction are inseparable.

Schumpeter returns to this concept of creative destruction in Capitalism, Socialism and Democracy, in which he insists on the evolutionary dimension of capitalism, which cannot be thought of in a static way : "the problem that is usually being visualized is how capitalism administers existing structures, whereas the relevant problem is how it creates and destroys them" (CSD, p.84). He also writes:

"The essential point to grasp is that in dealing with capitalism we are dealing with an evolutionary process.

(…) Capitalism, then, is by nature a form or method of economic change and not only never is but never can be stationary" (CSD, p.82).

Furthermore, Schumpeter adds that the innovations implemented by the entrepreneurs are at the origin of a:

« process of industrial mutation-if I may use that biological term-that incessantly revolutionizes the economic structure from within, incessantly destroying the old one, incessantly creating a new one. This process of Creative Destruction is the essential fact about capitalism. It is what capitalism consists in and what every capitalist concern has got to live in » (CSD, p.83).

Capitalism is thus in perpetual mutation, it is an "evolutionary process", which leads to the destruction of the old so that the new can emerge. Once again, this valorization of the dynamics of creation appears to be largely inspired by Romanticism, of which Berlin tells us that one of the main lessons is « the 'endless self-creativity of the universe', the impossibility of nailing down with laws and formulae this 'unceasing flow', and the unfathomable depth and complexity of the 'process of perpetual forward creation' central to social life » (Berlin, 1999, cited by Bronk, 2009, p.207). What's more, Romanticism aims precisely at marking "a difference between a narrow mechanical model of scientific explanation and rationality (focusing on formal calculus, precise prediction and the constrained optimisation of given factors and preferences) and a more holistic, dynamic and indeterminate explanatory framework (focusing on organic interdependence and the role of creativity)" (Ibid, p.47).

It is interesting to note that we also find such a conception of economic dynamics in Marshall, for whom the conception of a static equilibrium does not make it possible to grasp the specificity of the capitalist dynamic. Indeed, according to Marshall, 'mechanical analogies' have a large place in economics' textbooks only because they are easier to handle mathematically and conceptually than biological or organic analogies. He thought that the term 'equilibrium' 'suggests something of statical analogy', and that the 'fragmentary statical hypotheses' used in equilibrium analysis should be seen as no more than 'temporary auxiliaries to dynamical -or rather biological -conceptions', and insisted that 'the central idea of economics … must be that of living force and movement' (Bronk, p.70).

  entrepreneurs; (ii) a micro-social dimension of valuing a creative and dynamic economic agent, breaking with the figure of Homo Oeconomicus; leading to an original conception of economic dynamics. BRONK R. (2009), The Romantic Economist, Imagination in Economics, Cambridge University 2 Press, Cambridge. AGHION P., HOWITT P. (1992), « A Model of Growth Through Creative Destruction », Econometrica.

  macro-social dimension : the rationalization of the world.

  optimization and the use of reason in decision making, which leaves no room for uncertain action, based on intuition. Schumpeter also highlights the shift from an individual conception of innovation to a routinized and bureaucratized collective conception of innovation, typical of the capitalist modernity: « economic progress tends to become depersonalized and automatized. Bureau and committee work tend to replace individual action » (CSD, p.133). Thus, the problem of capitalism is that it rests on a dynamic of innovation, which presupposes the existence of singular individualities, capable by their genius and their intuition of creating something radically new. But capitalism tends precisely to stifle the existence of such individualities, which are SAYRE R., LÖWY M. (1983), « Figures du romantisme anti-capitaliste », L'homme et la société.

  8

17

  references to the figure of the medieval knight. DANNEQUIN F. (2012), « L'influence de l'eugénisme daltonien dans la pensée de Joseph Aloïs 18

  This conception of the entrepreneur as a leader and an extraordinary individual gives an elitist and

		5	6
	succeeds in imposing his will by relying on the mechanism of credit , which allows him to finance his 7
	innovations. It is the entrepreneur and its singular motivations that is at the basis of the phenomenon of
	creative destruction, which is, according to Schumpeter, constitutive of economic life.
		The entrepreneur is thus at the heart of Schumpeter's thinking, which many commentators have
	been able to emphasize. Moreover, in his book Capitalism, Socialism and Democracy (1942), the Austrian
	economist develops the idea that capitalism is doomed to disappear in the long run, due to the decline of
	the entrepreneurial function, which cannot continue in modern societies.
	4	For a more detailed presentation of Schumpeter's concept of the entrepreneur, see the article by
		Gislain, « The origins of the schumpetarian entrepreneur », Interventions Economiques, 2012 .
	5	In fact, Schumpeter distinguishes five types of innovation : (1) The introduction of a new good -
	that is one with which consumers are not yet familiar -or of a new quality of a good. (2) The
	introduction of a new method of production, that is one not yet tested by experience in the branch of
	manufacture concerned, which need by no means be founded upon a discovery scientifically new, and
	can also exist in a new way of handling a commodity commercially. (3) The opening of a new market,
	that is a market into which the particular branch of manufacture of the country in question has not
	previously entered, whether or not this market has existed before. (4) The conquest of a new source of
	supply of raw materials or half-manufactured goods, again irrespective of whether this source already
	exists or whether it has first to be created. (5) The carrying out of the new organisation of any
	industry, like the creation of a monopoly position (for example through trustification) or the breaking
	up of a monopoly position. (Theory of Economic Development, p.38-39)

6 aristocratic tone to Schumpeter's thinking. On this aspect, see Santarelli, Pesciarelli, « The Emergence of a Vision : The Development of Schumpeter's Theory of Entrepreneurship », in History of Economic

CONCLUDING REMARKS

The romantic representation of the world is an alternative way of thinking about economic phenomena in relation to the standard conception nourished by Utilitarianism and Rationalism, and thus makes it possible to enrich the conception of the economic world. Schumpeter's thought appears to be an example of such an enrichment, by emphasizing the role of intuition and imagination in economic decisions, or by the insistence on the dynamism of social life, and of economic dynamics. Moreover, we could add a methodological dimension of romantic (and historicist) inspiration to Schumpeter's thought, which is characterized by the insistence on the role of institutions, and on the singular historical and national context.

According to Berlin, Romanticism is the greatest turning point in Western consciousness (after the Enlightenment). However, as Schumpeter notes in his History of Economic Analysis, Romanticism did not leave deep marks in economic science. Schumpeter's intellectual and scientific project is therefore partly to revive this neglected part of modern thought within economics (Shionoya, 2008, p.25).

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