Finite-dimensional Weight Research Articles

Twisted super-Yangians and their representations

Starting with the super-Yangian Y(M|N) based on gl(M|N), we define twisted super-Yangians Y(M|N)±. Only Y+(M|2n) and Y−(2m|N) can be defined, and appear to be isomorphic one with each other. We study their finite-dimensional irreducible highest weight representations.

Characters of simple bounded modules over an untwisted affine lie algebra

After V. Chari and A. Pressley, a simple integrable module with finite-dimensional weight spaces over an affine Lie algebra is either a standard module (highest or lowest weight), in which case its formal character is given by the famous Weyl–Kac formula, or a subquotient of a tensor product of loop modules. In this paper we compute formal characters of generic simple integrable modules of the latter type.

Classification of Irreducible Integrable Modules for Multi-loop Algebras with Finite-Dimensional Weight Spaces

Classification of Irreducible Integrable Modules for Multi-loop Algebras with Finite-Dimensional Weight Spaces

Weyl modules for classical and quantum affine algebras

We introduce and study the notion of a Weyl module for the classical affine algebras, these modules are universal finite-dimensional highest weight modules. We conjecture that the modules are the classical limit of a family of irreducible modules of the quantum affine algebra, and prove the conjecture in the case of s l 2 sl_2 . The conjecture implies also that the Weyl modules are the classical limits of the standard modules introduced by Nakajima and further studied by Varagnolo and Vasserot.

Classification of Irreducible Nonzero Level Modules with Finite-Dimensional Weight Spaces for Affine Lie Algebras

We classify the irreducible weight affine Lie algebra modules with finite-dimensional weight spaces on which the central element acts nontrivially. In particular, we show that any such module is a quotient of a generalized Verma module. The classification of such irreducible modules is reduced to the classification of irreducible torsion-free modules for algebras of type An and Cn which is due to O. Mathieu (forthcoming).

Equivalence of certain categories of modules for quantized affine lie algebras

AbstractWe show that a quantum Verma-type module for a quantum group associated to an affine Kac-Moody algebra is characterized by its subspace of finite-dimensional weight spaces. In order to do this we prove an explicit equivalence of categories between a certain category containing the quantum Verma modules and a category of modules for a subalgebra of the quantum group for which the finite part of the Verma module is itself a module.

The Admissibility of Simple Bounded Modules for an Affine Lie Algebra

This paper studies a class of simple integrable modules for an affine Lie algebras which are closely related to the finite-dimensional modules studied by V. Chari and A. Pressley, except that the Euler element is assumed to act. They are infinite-dimensional; but are shown to have finite-dimensional weight spaces. It is conjectured that any simple integrable module with a zero weight space belongs to this class and their classification is given. The main interest in studying such modules is that they may occur in the endomorphism rings of highest weight modules whilst those of Chari and Pressley in general do not. Their character theory is also more complicated.

QUANTIZED SYMMETRIC POWERS

Let g = sln(C) be the Lie algebra of n×n matrices over C of trace zero, and let U be the q-analogue of its universal enveloping algebra defined by Drinfel’d [2] and Jimbo [4]. According to [7, 3.5.6, 6.2.3 & 6.3.4], for each dominant weight μ in the weight lattice of g there is an irreducible, finite-dimensional highest weight U -module V (μ) with highest weight μ. Kashiwara [5] and Lusztig [7, 14.4.12] have independently shown the existence of a certain canonical basis B(μ) for V (μ). In [5], Kashiwara proves the existence of a certain crystal basis associated with V (μ) using certain operators, and defines a graph (the crystal graph) which encodes how these operators act on the crystal basis. Using this, he defines B(μ). In §2 we define a certain subspace of the r-th tensor power of the basic module for U , for each integer r ≥ 1, and show that it is a module for U , with highest weight (r, 0, 0, . . . , 0). (We use the same numbering as [1, Planche I]). In §3 we study the irreducible finite-dimensional highest weight module for U with highest weight (r, 0, 0, . . . , 0), and find its canonical basis in terms of monomials in U acting on its highest weight vector. We then bring all of this together by showing that the module in §2 is isomorphic to the module studied in §3, and as a result we have two nice descriptions of the canonical basis for this module. Using these we determine how the generators of U act on this basis.

A constraint on the existence of simple torsion free Lie modules

For any simple Lie algebra L with Cartan subalgebra H the classification of all simple H-diagonalizable L-modules having a finite-dimensional weight space is known to depend on determining the simple torsion-free L-modules of finite degree. It is further known that the only simple Lie algebras which admit simple torsion-free modules of finite degree are those of types A n {A_n} and C n {C_n} . For the case of A n {A_n} we show that there are no simple torsion-free A n {A_n} -modules of degree k for n ≥ 4 n \geq 4 and 2 ≤ k ≤ n − 2 2 \leq k \leq n - 2 . We conclude with some examples showing that there exist simple torsion-free A n {A_n} -modules of degrees 1 , n − 1 1,n - 1 , and n.

Signature characters forA 2 andB 2

The signatures of the inner product matrices on a Lei algebra's highest weight representation are encoded in the representation's signature character. We show that the signature characters of a finite-dimensional Lie algebra's highest weight representations obey simple difference equations that have a unique solution once appropriate boundary conditions are imposed. We use these results to derive the signature characters of allA 2 andB 2 highest weight representations. Our results extend, and explain, signature patterns analogous to those observed by Friedan, Qiu and Shenker in the Virasoro algebra's representation theory.

Lie algebra modules with finite-dimensional weight spaces. I

Let g \mathfrak {g} denote a reductive Lie algebra over an algebraically closed field of characteristic zero, and let h \mathfrak {h} denote a Cartan subalgebra of g \mathfrak {g} . In this paper we study finitely generated g \mathfrak {g} -modules that decompose into direct sums of finite dimensional h \mathfrak {h} -weight spaces. We show that the classification of irreducible modules in this category can be reduced to the classification of a certain class of irreducible modules, those we call torsion free modules. We also show that if g \mathfrak {g} is a simple Lie algebra that admits a torsion free module, then g \mathfrak {g} is of type A A or C C .

Limits of weight spaces, Lusztig’s 𝑞-analogs, and fiberings of adjoint orbits

Let G be a connected complex semisimple algebraic group, and T a maximal torus inside a Borel subgroup B, with g, t, and b their Lie algebras. Let V be a representation in the category a for g. The t-decomposition V = met* VA of V into a direct sum of finite-dimensional weight spaces is central in the representation theory of g. In this paper, we introduce on weight spaces a new structure, the principal filtration Je (V) , where e is a principal nilpotent in g chosen to be compatible with t; for example, e can be the sum of the simple root vectors relative to (t, b) . This filtration is constructed in a very simple way by taking J' (V') to be the space of vectors annihilated by the (p + 1)th power of e, for p > 0. Our approach is motivated by Kostant's fundamental work [Ki, K2] on actions of the principal TDS (three-dimensional subalgebra) and coordinate rings of regular adjoint orbits. In Theorem 3.4, we give a new description, in terms of the dimension jumps of the principal filtration of the weight space, of Lusztig's [L] q-analog mm (q) of dominant ,-weight multiplicity in a finite-dimensional irreducible g-representation V>. The polynomial m (q) was defined algebraically as an alternating sum over the Weyl group, through a q-analog of Kostant's weight multiplicity formula. We prove that m (q) is equal to the jump polynomial