Finite-dimensional Simple Modules Research Articles

The Second Cohomology Groups of Simple Modules Over S p 4(k)

The second cohomology groups for simple, simply connected algebraic group Sp 4(k) over an algebraically closed field of characteristic p ≥ 7 with coefficients in the simple finite-dimensional modules are described.

Combinatorics of character formulas for the lie superalgebra $ \mathfrak{g}\mathfrak{l}\left( {m,n} \right) $

Let \( \mathfrak{g} \) be the Lie superalgebra \( \mathfrak{g}\mathfrak{l}\left( {m,n} \right) \). Algorithms for computing the composition factors and multiplicities of Kac modules for \( \mathfrak{g} \) were given by the second author, [12] and by J. Brundan [1]. We give a combinatorial proof of the equivalence between the two algorithms. The proof uses weight and cap diagrams introduced by Brundan and C. Stroppel, and cancelations between paths in a graph \( \mathcal{G} \) defined using these diagrams. Each vertex of \( \mathcal{G} \) corresponds to a highest weight of a finite dimensional simple module, and each edge is weighted by a nonnegative integer. If \( \mathcal{E} \) is the subgraph of \( \mathcal{G} \) obtained by deleting all edges of positive weight, then \( \mathcal{E} \) is the graph that describes nonsplit extensions between simple highest weight modules. We also give a procedure for finding the composition factors of any Kac module, without cancelation. This procedure leads to a second proof of the main result.

Cohomology of generalized supergrassmannians and character formulae for basic classical Lie superalgebras

In this paper, we use geometrical methods adapted from the Borel-Weil-Bott theory to compute the character of every finite dimensional simple module over a basic classical Lie superalgebra.

Extensions between finite-dimensional simple modules over a generalized current Lie algebra

We calculate the first extension groups for finite-dimensional simple modules over an arbitrary generalized current Lie algebra, which includes the case of loop Lie algebras and their multivariable analogs.

Representations of multiloop algebras

We describe the finite-dimensional simple modules of all the (twisted and untwisted) multiloop algebras and classify them up to isomorphism.

A family of Koszul algebras arising from finite-dimensional representations of simple Lie algebras

Let g be a finite-dimensional simple Lie algebra and let S g be the locally finite part of the algebra of invariants ( End C V ⊗ S ( g ) ) g where V is the direct sum of all simple finite-dimensional modules for g and S ( g ) is the symmetric algebra of g . Given an integral weight ξ, let Ψ = Ψ ( ξ ) be the subset of roots which have maximal scalar product with ξ. Given a dominant integral weight λ and ξ such that Ψ is a subset of the positive roots we construct a finite-dimensional subalgebra S Ψ g ( ⩽ Ψ λ ) of S g and prove that the algebra is Koszul of global dimension at most the cardinality of Ψ. Using this we construct naturally an infinite-dimensional non-commutative Koszul algebra of global dimension equal to the cardinality of Ψ. The results and the methods are motivated by the study of the category of finite-dimensional representations of the affine and quantum affine algebras.

Finite-Dimensional Simple Poisson Modules

We prove a result that can be applied to determine the finite-dimensional simple Poisson modules over a Poisson algebra and apply it to numerous examples. In the discussion of the examples, the emphasis is on the correspondence with the finite-dimensional simple modules over deformations and on the behaviour of finite-dimensional simple Poisson modules on the passage from a Poisson algebra to the Poisson subalgebra of invariants for the action of a finite group of Poisson automorphisms.

Hopf structures on ambiskew polynomial rings

We derive necessary and sufficient conditions for an ambiskew polynomial ring to have a Hopf algebra structure of a certain type. This construction generalizes many known Hopf algebras, for example U ( s l 2 ) , U q ( s l 2 ) and the enveloping algebra of the three-dimensional Heisenberg Lie algebra. In a torsion-free case we describe the finite-dimensional simple modules, in particular their dimensions, and prove a Clebsch–Gordan decomposition theorem for the tensor product of two simple modules. We construct a Casimir type operator and prove that any finite-dimensional weight module is semisimple.

Tensor Products and Whittaker Vectors for Quantum Groups

We describe the action of the center of the quantum group Uq (𝔤) on the tensor product V ⊗ L(λ) of an infinite-dimensional representation V having an infinitesimal character χτ and an irreducible finite-dimensional Uq (𝔤) representation L(λ) of highest weight λ. We apply this result in order to describe the tensor product of a Whittaker module and a finite-dimensional simple module for the algebra Uq(𝔰l2).

PRESENTATION OF QUANTUM GENERALIZED SCHUR ALGEBRAS

We obtain the explicit generators of the annihilator ideal for the tensor product of any finite dimensional simple module over quantum group [Formula: see text], by using the weight property of ideals in [Formula: see text] when q is not a root of unity. As an application, we give a presentation of quantum generalized Schur algebra.

Representation theory of two families of quantum projective 3-spaces

Stephenson and Vancliff recently introduced two families of quantum projective 3-spaces (quadratic and Artin–Schelter regular algebras of global dimension 4) which have the property that the associated automorphism of the scheme of point modules is finite order, and yet the algebra is not finite over its center. This is in stark contrast to theorems of Artin, Tate, and Van den Bergh in global dimension 3. We analyze the representation theory of these algebras. We classify all of the finite-dimensional simple modules and describe some zero-dimensional elements of Proj, i.e., so called fat point modules. In particular, we observe that the shift functor on zero-dimensional elements of Proj, which is closely related to the above automorphism, actually has infinite order.

Basic properties of generalized down–up algebras

We introduce a large class of infinite dimensional associative algebras which generalize down–up algebras. Let K be a field and fix f∈ K[ x] and r, s, γ∈ K. Define L= L( f, r, s, γ) to be the algebra generated by d, u and h with defining relations: [d,h] r+γd=0, [h,u] r+γu=0, [d,u] s+f(h)=0. Included in this family are Smith's class of algebras similar to U( sl 2), Le Bruyn's conformal sl 2 enveloping algebras and the algebras studied by Rueda. The algebras L have Gelfand–Kirillov dimension 3 and are Noetherian domains if and only if rs≠0. We calculate the global dimension of L and, for rs≠0, classify the simple weight modules for L, including all finite dimensional simple modules. Simple weight modules need not be classical highest weight modules.

Rank ideals and Capelli identities

We define and study a family of completely prime rank ideals in the universal enveloping algebra U( gl n) . A rank ideal is a noncommutative analogue of a determinantal ideal, the defining ideal for the closure of the set of n× n matrices of fixed rank. We introduce a notion of rank for gl n -modules and determine the rank of simple highest weight modules and of simple finite-dimensional modules. The main tools are Capelli-type identities and filtered algebra.

Quantum loop modules

We classify the simple infinite-dimensional integrable modules with finite-dimensional weight spaces over the quantized enveloping algebra of an untwisted affine algebra. We prove that these are either highest (lowest) weight integrable modules or simple submodules of a loop module of a finite-dimensional simple integrable module and describe the latter class. Their characters and crystal basis theory are discussed in a special case.

Some Algebras Similar to the Enveloping Algebra of s1(2)

Consider the algebra R = ℂ[A,B,H] subject to relations where and . We study these algebras for different f and ç. We study highest weight modules and finite dimensional simple modules. We introduce the categories p with p ≥ 1, with an interesting decomposition in categories p I and p II . We also consider the BGG-category and develop new techniques towards the study of .

Tensor product multiplicities, canonical bases and totally positive varieties

We obtain a family of explicit polyhedral combinatorial expressions for multiplicities in the tensor product of two simple finite-dimensional modules over a complex semisimple Lie algebra. Here polyhedral means that the multiplicity in question is expressed as the number of lattice points in some convex polytope. Our answers use a new combinatorial concept of $\ii$-trails which resemble Littelmann's paths but seem to be more tractable. We also study combinatorial structure of Lusztig's canonical bases or, equivalently of Kashiwara's global bases. Although Lusztig's and Kashiwara's approaches were shown by Lusztig to be equivalent to each other, they lead to different combinatorial parametrizations of the canonical bases. One of our main results is an explicit description of the relationship between these parametrizations. Our approach to the above problems is based on a remarkable observation by G. Lusztig that combinatorics of the canonical basis is closely related to geometry of the totally positive varieties. We formulate this relationship in terms of two mutually inverse transformations: tropicalization and geometric lifting.

Down–Up Algebras and Their Representation Theory

A class of algebras called down–up algebras was introduced by G. Benkart and T. Roby (1998, J. Algebra209, 305–344). We classify the finite dimensional simple modules over Noetherian down–up algebras and show that in some cases every finite dimensional module is semisimple. We also study the question of when two down–up algebras are isomorphic.

Down–Up Algebras and Ambiskew Polynomial Rings

We show that the down–up algebras of G. Benkart (1998, in “Recent Progress in Algebra,” Contemporary Mathematics Vol. 224, Am. Math. Soc., Providence) and G. Benkart and T. Roby (1998, J. Algebra209, 305–344) lie in a certain class of iterated skew polynomial rings, called ambiskew polynomial rings, in two indeterminates x and y over a commutative ring B. In such rings, commutation of the indeterminates with elements of B involve the same endomorphism σ of B, but from different sides, that is, yb=σ(b)y and bx=xσ(b), and, for some scalar p, yx−pxy∈ B. In previous studies of ambiskew polynomial rings, σ was required to be an automorphism but, in order to cover all down–up algebras, this requirement must be dropped. The Noetherian down–up algebras are those where σ is an automorphism and, in this case, we apply existing results on ambiskew polynomial rings to determine the finite-dimensional simple modules and the prime ideals. We adapt the methods underlying these results so as to apply to the non-Noetherian down–up algebras for which they reveal a surprisingly rich structure.

The homogenization of the three dimensional skew polynomial algebras of type I

This paper studies a special case of graded central extensions of three dimen-sional Artin-Schelter regular algebras, see [9, §3]. The algebras are homoge-nizations of two classes of three dimensional skew polynomial algebras. We refer to these algebras as Type I and Type II algebras. We describe the non-commutative projective geometry and compute the finite dimensional simple modules for the homogenization of Type I algebras in the case that α is not a primitive root of unity. In this case, all finite dimensional simple modules are quotients of line modules that are homogenizations of Verma modules.

Finite-dimensional simple modules over quantised Weyl algebras

We classify finite-dimensional simple modules over quantised n-th Weyl algebras. over an algebraically closed field under a certain condition on the parameters.