Complex Simple Lie Algebra Research Articles

Norm formulae for the Bethe Ansatz on root systems of small rank

The norms of the Bethe Ansatz eigenfunctions for the Lieb–Liniger quantum system of n Bosonic particles on a ring with pairwise repulsive delta potential interactions are given by a beautiful determinantal formula, first conjectured by Gaudin in the early seventies and then proven by Korepin about a decade later. Recently, E Emsiz formulated a similar conjecture generalizing the Gaudin–Korepin norm formula in terms of the root systems of complex simple Lie algebras. Here we confirm the validity of the conjecture in question for small root systems up to rank 3 (thus including the important test case of the exceptional root system G2).

Finite-dimensional representations of hyper loop algebras

We study finite-dimensional representations of hyper loop algebras, i.e., the hyperalgebras over an algebraically closed field of positive characteristic associated to the loop algebra over a complex finite-dimensional simple Lie algebra. The main results are the classification of the irreducible modules, a version of Steinberg's Tensor Product Theorem, and the construction of positive characteristic analogues of the Weyl modules as defined by Chari and Pressley in the characteristic zero setting. Furthermore, we start the study of reduction modulo p and prove that every irreducible module of a hyper loop algebra can be constructed as a quotient of a module obtained by a certain reduction modulo p process applied to a suitable characteristic zero module. We conjecture that the Weyl modules are also obtained by reduction modulo p. The conjecture implies a tensor product decomposition for the Weyl modules which we use to describe the blocks of the underlying abelian category.

Categories of modules over an affine Kac–Moody algebra and finiteness of the Kazhdan–Lusztig tensor product

To each category C of modules of finite length over a complex simple Lie algebra g , closed under tensoring with finite dimensional modules, we associate and study a category AFF ( C ) κ of smooth modules (in the sense of Kazhdan and Lusztig [D. Kazhdan, G. Lusztig, Tensor structures arising from affine Lie algebras, I, J. Amer. Math. Soc. 6 (1993) 905–947]) of finite length over the corresponding affine Kac–Moody algebra in the case of central charge less than the critical level. Equivalent characterizations of these categories are obtained in the spirit of the works of Kazhdan and Lusztig [D. Kazhdan, G. Lusztig, Tensor structures arising from affine Lie algebras, I, J. Amer. Math. Soc. 6 (1993) 905–947] and Lian and Zuckerman [B.H. Lian, G.J. Zuckerman, BRST cohomology and noncompact coset models, in: Proceedings of the XXth International Conference on Differential Geometric methods in Theoretical Physics, New York, 1991, 1992, pp. 849–865; B.H. Lian, G.J. Zuckerman, An application of infinite dimensional Lie theory to semisimple Lie groups, in: Representation Theory of Groups and Algebras, in: Contemp. Math., vol. 145, 1993, pp. 249–257]. In the main part of this paper we establish a finiteness result for the Kazhdan–Lusztig tensor product which can be considered as an affine version of a theorem of Kostant [B. Kostant, On the tensor product of a finite and an infinite dimensional representation, J. Funct. Anal. 20 (1975) 257–285]. It contains as special cases the finiteness results of Kazhdan, Lusztig [D. Kazhdan, G. Lusztig, Tensor structures arising from affine Lie algebras, I, J. Amer. Math. Soc. 6 (1993) 905–947] and Finkelberg [M. Finkelberg, PhD thesis, Harvard University, 1993], and states that for any subalgebra f of g which is reductive in g the “affinization” of the category of finite length admissible ( g , f ) modules is stable under Kazhdan–Lusztig's tensoring with the “affinization” of the category of finite dimensional g modules (which is O κ in the notation of [D. Kazhdan, G. Lusztig, Tensor structures arising from affine Lie algebras, I, J. Amer. Math. Soc. 6 (1993) 905–947; D. Kazhdan, G. Lusztig, Tensor structures arising from affine Lie algebras, II, J. Amer. Math. Soc. 6 (1994) 949–1011; D. Kazhdan, G. Lusztig, Tensor structures arising from affine Lie algebras, IV, J. Amer. Math. Soc. 7 (1994) 383–453]).

Bernstein-Szegö polynomials associated with root systems

We introduce multivariate generalizations of the Bernstein-Szego polynomials, which are associated to the root systems of the complex simple Lie algebras. The multivariate polynomials in question generalize Macdonald's Hall-Littlewood polynomials associated with root systems. For the root system of type A1 (corresponding to the Lie algebra SL (2;C)) the classic Bernstein-Szego polynomials are recovered.

AUTOMORPHISMS OF CERTAIN NILPOTENT LIE ALGEBRAS OVER COMMUTATIVE RINGS

Let L be a finite-dimensional complex simple Lie algebra, Lℤ be the ℤ-span of a Chevalley basis of L and LR = R⊗ℤLℤ be a Chevalley algebra of type L over a commutative ring R. Let [Formula: see text] be the nilpotent subalgebra of LR spanned by the root vectors associated with positive roots. The aim of this paper is to determine the automorphism group of [Formula: see text].

PRIMITIVE IDEALS AND AUTOMORPHISM GROUP OF $U_{q}^{+}(B_{2})$

Let 𝔤 be a complex simple Lie algebra of type B2 and q be a nonzero complex number which is not a root of unity. In the classical case, a theorem of Dixmier asserts that the simple factor algebras of Gelfand–Kirillov dimension 2 of the positive part U+(𝔤) of the enveloping algebra of 𝔤 are isomorphic to the first Weyl algebra. In order to obtain some new quantized analogues of the first Weyl algebra, we explicitly describe the prime and primitive spectra of the positive part [Formula: see text] of the quantized enveloping algebra of 𝔤 and then we study the simple factor algebras of Gelfand–Kirillov dimension 2 of [Formula: see text]. In particular, we show that the centers of such simple factor algebras are reduced to the ground field ℂ and we compute their group of invertible elements. These computations allow us to prove that the automorphism group of [Formula: see text] is isomorphic to the torus (ℂ*)2, as conjectured by Andruskiewitsch and Dumas.

RECURRENCE RELATIONS, DYNKIN DIAGRAMS AND PLÜCKER FORMULAE

Abstract.We prove a generalisation of an observation of N. Iwahori concerning the coefficients of the extended Dynkin diagram of a complex simple Lie algebra. We relate the combinatorics of these coefficients to the orders of finite groups that act discontinuously on the Riemann sphere and to the Plücker formulae.

A Note on Exponents vs Root Heights for Complex Simple Lie Algebras

We give an elementary combinatorial proof of a special case of a result due to Bazlov and Ion concerning the Fourier coefficients of the Cherednik kernel. This can be used to give yet another proof of the classical fact that for a complex simple Lie algebra $\mathfrak{ g}$, the partition formed by the exponents of $\mathfrak{ g}$ is dual to that formed by the numbers of positive roots at each height.

Rational solutions of CYBE for simple compact real Lie algebras

In [A.A. Stolin, On rational solutions of Yang–Baxter equation for sl ( n ) , Math. Scand. 69 (1991) 57–80; A.A. Stolin, On rational solutions of Yang–Baxter equation. Maximal orders in loop algebra, Comm. Math. Phys. 141 (1991) 533–548; A. Stolin, A geometrical approach to rational solutions of the classical Yang–Baxter equation. Part I, in: Walter de Gruyter & Co. (Ed.), Symposia Gaussiana, Conf. Alg., Berlin, New York, 1995, pp. 347–357] a theory of rational solutions of the classical Yang–Baxter equation for a simple complex Lie algebra g was presented. We discuss this theory for simple compact real Lie algebras g . We prove that up to gauge equivalence all rational solutions have the form X ( u , v ) = Ω u − v + t 1 ∧ t 2 + ⋯ + t 2 n − 1 ∧ t 2 n , where Ω denotes the quadratic Casimir element of g and { t i } are linearly independent elements in a maximal torus t of g . The quantization of these solutions is also emphasized.

On $$\mathbb{Z}$$ -Gradations of Twisted Loop Lie Algebras of Complex Simple Lie Algebras

We define the twisted loop Lie algebra of a finite dimensional Lie algebra $\mathfrak g$ as the Fr\'echet space of all twisted periodic smooth mappings from $\mathbb R$ to $\mathfrak g$. Here the Lie algebra operation is continuous. We call such Lie algebras Fr\'echet Lie algebras. We introduce the notion of an integrable $\mathbb Z$-gradation of a Fr\'echet Lie algebra, and find all inequivalent integrable $\mathbb Z$-gradations with finite dimensional grading subspaces of twisted loop Lie algebras of complex simple Lie algebras.

Tensor Product Structure of Affine Demazure Modules and Limit Constructions

AbstractLet g be a simple complex Lie algebra, we denote by ĝ the affine Kac-Moody algebra associated to the extended Dynkin diagram of g. Let Λ0 be the fundamental weight of ĝ corresponding to the additional node of the extended Dynkin diagram. For a dominant integral g-coweight λ∨, the Demazure submodule V_λ∨ (mΛ0) is a g-module. We provide a description of the g-module structure as a tensor product of “smaller” Demazure modules. More precisely, for any partition of λ∨ = λ∑j as a sum of dominant integral g-coweights, the Demazure module is (as g-module) isomorphic to ⊗jV_ (mΛ0). For the “smallest” case, λ∨ = ω∨ a fundamental coweight, we provide for g of classical type a decomposition of V_ω∨(mΛ0) into irreducible g-modules, so this can be viewed as a natural generalization of the decomposition formulas in [13] and [16]. A comparison with the Uq(g)-characters of certain finite dimensional -modules (Kirillov-Reshetikhin-modules) suggests furthermore that all quantized Demazure modules V_λ∨,q(mΛ0) can be naturally endowed with the structure of a -module. We prove, in the classical case (and for a lot of non-classical cases), a conjecture by Kashiwara [10], that the “smallest” Demazure modules are, when viewed as g-modules, isomorphic to some KR-modules. For an integral dominant ĝ-weight Λ let V(Λ) be the corresponding irreducible ĝ-representation. Using the tensor product decomposition for Demazure modules, we give a description of the g-module structure of V(Λ) as a semi-infinite tensor product of finite dimensional g-modules. The case of twisted affine Kac-Moody algebras can be treated in the same way, some details are worked out in the last section.

On Lie Algebra Crossed Modules

The goal of this article is to construct a crossed module representing the cocycle ⟨[,],⟩ generating H 3(𝔤;ℂ) for a simple complex Lie algebra 𝔤.

SEXTONIONS AND THE MAGIC SQUARE

Associated to any complex simple Lie algebra is a non-reductive complex Lie algebra which we call the intermediate Lie algebra. We propose that these algebras can be included in both the magic square and the magic triangle to give an additional row and column. The extra row and column in the magic square corresponds to the sextonions. This is a six dimensional subalgebra of the split octonions which contains the split quaternions.

Symplectic resolutions for nilpotent orbits (III)

We prove that two symplectic resolutions of a nilpotent orbit closures in a simple complex Lie algebra of classical type are related by Mukai flops in codimension 2. To cite this article: B. Fu, C. R. Acad. Sci. Paris, Ser. I 342 (2006).

ON LIE INDUCTION AND THE EXCEPTIONAL SERIES

Lie bialgebras occur as the principal objects in the infinitesimalization of the theory of quantum groups — the semi-classical theory. Their relationship with the quantum theory has made available some new tools that we can apply to classical questions. In this paper, we study the simple complex Lie algebras using the double-bosonization construction of Majid. This construction expresses algebraically the induction process given by adding and removing nodes in Dynkin diagrams, which we call Lie induction. We first analyze the deletion of nodes, corresponding to the restriction of adjoint representations to subalgebras. This uses a natural grading associated to each node. We give explicit calculations of the module and algebra structures in the case of the deletion of a single node from the Dynkin diagram for a simple Lie (bi-)algebra. We next consider the inverse process, namely that of adding nodes, and give some necessary conditions for the simplicity of the induced algebra. Finally, we apply these to the exceptional series of simple Lie algebras, in the context of finding obstructions to the existence of finite-dimensional simple complex algebras of types E9, F5 and G3. In particular, our methods give a new point of view on why there cannot exist such an algebra of type E9.

A universal dimension formula for complex simple Lie algebras

We present a universal formula for the dimension of the Cartan powers of the adjoint representation of a complex simple Lie algebra (i.e., a universal formula for the Hilbert functions of homogeneous complex contact manifolds), as well as several other universal formulas. These formulas generalize formulas of Vogel and Deligne and are given in terms of rational functions where both the numerator and denominator decompose into products of linear factors with integer coefficients. We discuss consequences of the formulas including a relation with Scorza varieties.

Fixed point subalgebras of extended affine Lie algebras

It is a well-known result that the fixed point subalgebra of a finite dimensional complex simple Lie algebra under a finite order automorphism is a reductive Lie algebra so it is a direct sum of finite dimensional simple Lie subalgebras and an abelian subalgebra. We consider this for the class of extended affine Lie algebras and are able to show that the fixed point subalgebra of an extended affine Lie algebra under a finite order automorphism (which satisfies certain natural properties) is a sum of extended affine Lie algebras (up to existence of some isolated root spaces), a subspace of the center and a subspace which is contained in the centralizer of the core. Moreover, we show that the core of the fixed point subalgebra modulo its center is isomorphic to the direct sum of the cores modulo centers of the involved summands.

Spherical orbits and representations of Uε(g)

Let Ue(g) be the simply connected quantized enveloping algebra at roots of one associated to a finite dimensional complex simple Lie algebra g. The De Concini-Kac-Procesi conjecture on the dimension of the irreducible representations of Ue(g) is proved for the representations corresponding to the spherical conjugacy classes of the simply connected algebraic group G with Lie algebra g. We achieve this result by means of a new characterization of the spherical conjugacy classes of G in terms of elements of the Weyl group.

On the Classical Double of Parabolic Subalgebras#

Given a complex simple finite-dimensional Lie algebra g with fixed root system, there exists a so-called classical Drinfeld–Jimbo r-matrix, r. Consider any parabolic subalgebra P S ⊆ g defined by a subset S of the set of simple roots. We prove that the Lie bialgebra structure on g defined by r can be restricted to P S . Moreover, it turns out that the corresponding classical double D(P S ) is isomorphic to g ⊕ Red(P S ), where Red(P S ) denotes the reductive part of P S .

Maximal Subalgebras of Kac–Moody Algebras#

We use the classification of finite order automorphisms by Kac to characterize all maximal subalgebras, regular, semisimple, reductive or not of a simple complex Lie algebra (up to conjugacy) that we can determine from its Dynkin diagram. Using Barnea et al. [Barnea, Y., Shalev, A., Zelmanov, E. I. (1998). Graded subalgebras of affine Kac–Moody algebras. Israel J. Math. 104:321–334] we extend our results to the case of affine Kac–Moody algebras. We also point out some inaccuracies in the Dynkin paper [Dynkin, E. B. (1957a). Semisimple subalgebras of semisimple Lie algebras. Amer. Math. Soc. Transl t. 6:111–244].