Search

In this paper, we consider the tensor product of local Weyl modules for $\mathfrak{sl}_{n+1}[t]$ whose highest weights are multiples of the first and $n^{th}$ fundamental weights. We determine the graded character of these tensor product modules in terms of the graded character of local Weyl modules and prove that these modules admit a filtration whose successive quotients are either truncated Weyl modules or fusion products of Demazure modules. Furthermore, we establish that the truncated Weyl modules appearing as quotients in the filtration of tensor products of local Weyl modules of $\mathfrak{sl}_3[t]$ are indeed isomorphic to fusion products of irreducible $\mathfrak{sl}_3[t]$-modules which establish the independence of a family of fusion product modules of $\mathfrak{sl}_3[t]$ from the set of its evaluation parameters.

We investigate the commuting automorphisms of nilpotent Lie algebras $L$ with coclass $\leq 3$. Our examination exposes the conditions under which the set of commuting automorphisms of $L$ forms a subgroup within its automorphism group.

We investigate a specific class of CV modules for $\mathfrak{sl}_3$ and establish an exact sequence for these modules. Utilizing dimension arguments, we demonstrate that this module is isomorphic to the fusion product of irreducible modules, thereby offering a new proof of the conjecture regarding the independence of fusion products from parameters. By analyzing the filtration of the kernel within the exact sequence, we derive the graded character formula for fusion product modules. Moreover, we leverage the graded character to deduce the algebraic characterization of the Littlewood-Richardson (LR) coefficients and present an alternative proof of the saturation theorem in type $A_2$., Comment: A new section is added, the title is changed; and some typos are corrected

We delineate the image of multilinear Lie polynomial of degree $2$ evaluated on $L$ where $L$ is a finite-dimensional nilpotent Lie algebra over field $k$ with $\dim L' \leq 4$., Comment: 21 pages

Let $\mathfrak g$ be a Borcherds-Kac-Moody Lie superalgebra (BKM superalgebra in short) with the associated graph $G$. Any such $\mathfrak g$ is constructed from a free Lie superalgebra by introducing three different sets of relations on the generators: (1) Chevalley relations, (2) Serre relations, and (3) Commutation relations coming from the graph $G$. By Chevalley relations we get a triangular decomposition $\mathfrak g = \mathfrak{ n_+} \oplus \mathfrak h \oplus \mathfrak n_{-}$ and each roots space $\mathfrak{ g_{\alpha}}$ is either contained in $\mathfrak{n_+}$ or $\mathfrak{ n_{-}}$. In particular, each $\mathfrak{ g_{\alpha}}$ involves only the relations (2) and (3). In this paper, we are interested in the root spaces of $\mathfrak g$ which are independent of the Serre relations. We call these roots free roots$^{1}$ of $\mathfrak g$. Since these root spaces involve only commutation relations coming from the graph $G$ we can study them combinatorially. We use heaps of pieces to study these roots and prove many combinatorial properties. We construct two different bases for these root spaces of $\mathfrak g$: One by extending the Lalonde's Lyndon heap basis of free partially commutative Lie algebras to the case of free partially commutative Lie superalgebras and the other by extending the basis given in \cite{akv17} for the free root spaces of Borcherds algebras to the case of BKM superalgebras. This is done by studying the combinatorial properties of super Lyndon heaps. We also discuss a few other combinatorial properties of free roots.