Related papers: Components of $V(m\rho) \otimes V(n\rho)$

Let g be an affine Kac-Moody Lie algebra and let $\lambda, \mu$ be two dominant integral weights for g. We prove that under some mild restriction, for any positive root $\beta$, $V(\lambda)\otimes V(\mu)$ contains $V(\lambda+\mu-\beta)$ as…

In this paper, using crystal theory we prove the existence of a new family of irreducible components appearing in the tensor product of two irreducible integrable highest weight modules over symmetrizable Kac-Moody algebras motivated by the…

Kostant asked the following question: Let $\mathfrak{g}$ be a simple Lie algebra over the complex numbers. Let $\lambda$ be a dominant integral weight. Then, $V(\lambda)$ is a component of $V(\rho)\otimes V(\rho)$ if and only if $\lambda…

In this paper we discuss the structure of the tensor product V'_{\alpha,\beta}\otimes L(c,h) of irreducible module from intermediate series and irreducible highest weight module over the Virasoro algebra. We generalize Zhang's…

We consider the problem of decomposing tensor powers of the fundamental level 1 highest weight representation $V$ of the affine Kac-Moody algebra $\g(E_9)$. We describe an elementary algorithm for determining the decomposition of the…

In this paper we construct a class of new irreducible modules over untwisted affine Kac-Moody algebras $\widetilde{\mathfrak{g}}$, generalizing and including both highest weight modules and Whittaker modules. These modules allow us to…

In this paper, the tensor product of highest weight modules with intermediate series modules over the Neveu-Schwarz algebra is studied. The weight spaces of such tensor products are all infinitely dimensional if the highest weight module is…

For a complex simple Lie algebra $\mathfrak{g}$ or rank $r$, let $\rho$ be the half sum of positive roots and $P(2\rho)\subset \mathbb{R}^r$ be the convex hull of all dominant weights $\lambda$ of the form $\lambda=2\rho-\sum_{i=1}^r…

In this paper, we are interested in the decomposition of the tensor product of two representations of a symmetrizable Kac-Moody Lie algebra $\mathfrak g$. Let $P\_+$ be the set of dominant integral weights. For $\lambda\in P\_+$ ,…

Let $\mathfrak{g}_m=\mathfrak{sl}(2)\ltimes V(m)$, $m\ge 1$, where $V(m)$ is the irreducible $\mathfrak{sl}(2)$-module of dimension $m+1$ viewed as an abelian Lie algebra. It is known that the isomorphism classes of uniserial…

We study structure of the semigroup Tens(G) consisting of triples of dominant weights (\lambda,\mu,\nu) of a complex reductive Lie group G such that the triple tensor product of the corresponding irreducible representations of G has a…

Kostant asked the following question: Let $\mathfrak{g}$ be a simple Lie algebra over the complex numbers. Let $\lambda$ be a dominant integral weight. Then, $V(\lambda)$ is a component of $ V(\rho)\otimes V(\rho)$ if and only if $\lambda…

Let $G$ be a simple algebraic group in defining characteristic $p>0$, and let $V$ be an irreducible $G$-module which is the tensor product of exactly two non-trivial modules. We obtain a criterion for $V$ to have the zero weight. In…

We consider a large class of series of symmetrizable Kac-Moody algebras (generically denoted X_n). This includes the classical series A_n as well as others like E_n whose members are of Indefinite type. The focus is to analyze the behavior…

We consider the semigroup S of highest weights appearing in tensor powers V^k of a finite dimensional representation V of a connected reductive group. We describe the cone generated by S as the cone over the weight polytope of V intersected…

We categorify the highest weight integrable representations and their tensor products of a symmetric quantum Kac-Moody algebra. As byproducts, we obtain a geometric realization of Lusztig's canonical bases of these representations as well…

We consider integrable, category O-modules of indecomposable symmetrizable Kac-Moody algebras. We prove that unique factorization of tensor products of irreducible modules holds in this category, upto twisting by one dimensional modules.…

In this article, we give geometric constructions of tensor products in various categories using quiver varieties. More precisely, we introduce a lagrangian subvariety $\Zl$ in a quiver variety, and show the following results: (1) The…

We consider integrable category $\mathcal{O}$ representations of Borcherds--Kac--Moody algebras whose Cartan matrix is finite dimensional, and determine the necessary and sufficient conditions for which the tensor product of irreducible…

Let $\mathfrak{sl}(2)\ltimes \mathfrak{h}_n$, $n\ge 1$, be the Galilean Lie algebra over a field of characteristic zero, here $\mathfrak{h}_{n}$ is the Heisenberg Lie algebra of dimension $2n+1$, and $\mathfrak{sl}(2)$ acts on…