Finite dimensional irreducible representations of finite W-algebras associated to even multiplicity nilpotent orbits in classical Lie algebras

We consider finite W-algebras ${U(\mathfrak{g},e)}$ associated to even multiplicity nilpotent elements in classical Lie algebras. We give a classification of finite dimensional irreducible ${U(\mathfrak{g},e)}$ -modules with integral central character in terms of the highest weight theory from Brundan et al. (Int. Math. Res. Notices 15, art. ID rnn051, 2008). As a corollary, we obtain a parametrization of primitive ideals of ${U(\mathfrak{g})}$ with associated variety the closure of the adjoint orbit of e and integral central character.     

Quantum Dynamical coBoundary Equation for finite dimensional simple Lie algebras

For a finite dimensional simple Lie algebra g, the standard universal solution R(x)∈Uq(g)^⊗2 of the Quantum Dynamical Yang-Baxter Equation quantizes the standard trigonometric solution of the Classical Dynamical Yang-Baxter Equation. It can be built from the standard R-matrix and from the solution F(x)∈Uq(g)^⊗2 of the Quantum Dynamical coCycle Equation as . F(x) can be computed explicitly as an infinite product through the use of an auxiliary linear equation, the ABRR equation.Inspired by explicit results in the fundamental representation, it has been conjectured that, in the case where g=sl(n+1)(n?1) only, there could exist an element M(x)∈Uq(sl(n+1)) such that the dynamical gauge transform RJ of R(x) by M(x),

RJ=M₁⁻¹(x)M₂(xqh₁⁾⁻¹R(x)M₁(xqh₂)M₂(x),     

On the classification of finite dimensional irreducible modules for affine BMW algebras

In this paper, we classify the finite dimensional irreducible modules for affine BMW algebra over an algebraically closed field with arbitrary characteristic.     

Explicit quantization of dynamical r-matrices for finite dimensional semisimple Lie algebras

We provide an explicit quantization of dynamical r-matrices for semisimple Lie algebras, classified earlier by the third author, which includes the Belavin-Drinfeld r-matrices. We do so by constructing an appropriate (dynamical) twist in the tensor square of the Drinfeld-Jimbo quantum group , which twists the R-matrix of into the desired quantization. The construction of this twist is based on the method stemming from the work of Jimbo-Konno-Odake-Shiraishi and Arnaudon-Buffenoir-Ragoucy-Roche, i.e. on defining the twist as a unique solution of a suitable difference equation. This yields a simple closed formula for the twist.

This construction allows one to confirm the alternate version of the Gerstenhaber-Giaquinto-Schack conjecture (about quantization of Belavin-Drinfeld r-matrices for in the vector representation), which was stated earlier by the second author on the basis of computer evidence. It also allows one to define new quantum groups associated to semisimple Lie algebras. We expect them to have a rich structure and interesting representation theory.

     

Lie p-algebras of finite p-subalgebra rank

We say that a Lie p-algebra L has finite p-subalgebra rank if the minimal number of generators required to generate every finitely generated p-subalgebra is uniformly bounded by some integer r. This paper is concerned with the following problem: does L being of finite p-subalgebra rank force ad(L) to be finite-dimensional? Although this seems unlikely in general, we show that this is indeed the case for Lie p-algebras in a large class including all locally, residually, and virtually soluble Lie p-algebras.     

Gradings of positive rank on simple Lie algebras

We complete the classification of positive rank gradings on Lie algebras of simple algebraic groups over an algebraically closed field k whose characteristic is zero or not too small, and we determine the little Weyl groups in each case. We also classify the stable gradings and prove Popov’s conjecture on the existence of a Kostant section.     

Generalized Clifford algebras and certain infinite dimensional Lie algebras

The embedding of certain infinite dimensional Lie algebras in generalized Clifford algobras C(N, p) is given. The correspondence between C(N,2) andgl(N, C) asN⇌∞ is pointed out.     

Invertible linear transformations and the Lie algebras

With the help of invertible linear transformations and the known Lie algebras, a way to generate new Lie algebras is given. These Lie algebras obtained have a common feature, i.e. integrable couplings of solitary hierarchies could be obtained by using them, specially, the Hamiltonian structures of them could be worked out. Some ways to construct the loop algebras of the Lie algebras are presented. It follows that some various loop algebras are given. In addition, a few new Lie algebras are explicitly constructed in terms of the classification of Lie algebras proposed by Ma Wen-Xiu, which are bases for obtaining new Lie algebras by using invertible linear transformations. Finally, some solutions of a (2 + 1)-dimensional partial-differential equation hierarchy are obtained, whose Hamiltonian form-expressions are manifested by using the quadratic-form identity.     

Classification of irreducible weight modules over higher rank Virasoro algebras

Let G be a rank n additive subgroup of C and Vir[G] the corresponding Virasoro algebra of rank n. In the present paper, irreducible weight modules with finite dimensional weight spaces over Vir[G] are completely determined. There are two different classes of them. One class consists of simple modules of intermediate series whose weight spaces are all 1-dimensional. The other is constructed by using intermediate series modules over a Virasoro subalgebra of rank n−1. The classification of such modules over the classical Virasoro algebra was obtained by O. Mathieu in 1992 using a completely different approach.     

Graded Lie algebras with finite polydepth

If A is a graded connected algebra then we define a new invariant, polydepthA, which is finite if for some A-module M of at most polynomial growth. Theorem 1: If f:X→Y is a continuous map of finite category, and if the orbits of acting in the homology of the homotopy fibre grow at most polynomially, then has finite polydepth. Theorem 5: If L is a graded Lie algebra and polydepthUL is finite then either L is solvable and UL grows at most polynomially or else for some integer d and all r, ∑_i=k+1^k+ddimL_i?k^r, k? some k(r).