Induced Modules for Affine Lie Algebras

We study induced modules of nonzero central charge with arbitrary multiplicities over affine Lie algebras. For a given pseudo parabolic subalgebra ${\mathcal P}$ of an affine Lie algebra ${\mathfrak G}$, our main result establishes the equivalence between a certain category of ${\mathcal P}$-induced ${\mathfrak G}$-modules and the category of weight ${\mathcal P}$-modules with injective action of the central element of ${\mathfrak G}$. In particular, the induction functor preserves irreducible modules. If ${\mathcal P}$ is a parabolic subalgebra with a finite-dimensional Levi factor then it defines a unique pseudo parabolic subalgebra ${\mathcal P}^{ps}$, ${\mathcal P}\subset {\mathcal P}^{ps}$. The structure of ${\mathcal P}$-induced modules in this case is fully determined by the structure of ${\mathcal P}^{ps}$-induced modules. These results generalize similar reductions in particular cases previously considered by V. Futorny, S. K\"onig, V. Mazorchuk [Forum Math. 13 (2001), 641-661], B. Cox [Pacific J. Math. 165 (1994), 269-294] and I. Dimitrov, V. Futorny, I. Penkov [Comm. Math. Phys. 250 (2004), 47-63]

Faces of polytopes and Koszul algebras

Let \g be a reductive Lie algebra and $V$ a \g-semisimple module. In this article, we study the category \G of graded finite-dimensional representations of \g \ltimes V. We produce a large class of truncated subcategories, which are directed and highest weight. Suppose $V$ is finite-dimensional with weights \wt(V). Let \Psi \subset \wt(V) be the set of weights contained in a face \F of the polytope that is the convex hull of \wt(V). For each such $\Psi$, we produce quasi-hereditary Koszul algebras. We use these Koszul algebras to construct an infinite-dimensional graded subalgebra \spg of the locally finite part of the algebra of invariants (END{\C} (\V) \otimes \Sym V)^{\g}, where \V is the direct sum of all simple finite-dimensional \g-modules. We prove that \spg is Koszul of finite global dimension.Comment: v3: Significant revisions. To appear in the Journal of Pure and Applied Algebra; 20 page

On R-matrix representations of Birman-Murakami-Wenzl algebras

We show that to every local representation of the Birman-Murakami-Wenzl algebra defined by a skew-invertible R-matrix $R\in Aut(V\otimes V)$ one can associate pairings $V\otimes V\to C$ and $V^*\otimes V^*\to C$, where V is the representation space. Further, we investigate conditions under which the corresponding quantum group is of SO or Sp type.Comment: 9 page

Minimal Affinizations of Representations of Quantum Groups: the simply--laced case

We continue our study of minimal affinizations for algebras of type D, E.Comment: 25 page

Quivers with relations arising from Koszul algebras of g\mathfrak g-invariants

Let $\mathfrak g$ be a complex simple Lie algebra and let $\Psi$ be an extremal set of positive roots. One associates with $\Psi$ an infinite dimensional Koszul algebra \bold S_\Psi^{\lie g} which is a graded subalgebra of the locally finite part of ((\bold V)^{op}\tensor S(\lie g))^{\lie g}, where $\bold V$ is the direct sum of all simple finite dimensional \lie g-modules. We describe the structure of the algebra \bold S_\Psi^{\lie g} explicitly in terms of an infinite quiver with relations for \lie g of types $A$ and $C$. We also describe several infinite families of quivers and finite dimensional algebras arising from this construction.Comment: 49 pages, AMSLaTeX+amsref