Related papers: Variations on the $S_n$-module $Lie_n$

Let $V$ be a finite-dimensional $FG$-module, where $F$ is a field of prime characteristic $p$ and $G$ is a group. We show that, when $r$ is not a power of $p$, the Lie power $L^r(V)$ has a direct summand $B^r(V)$ which is a direct summand…

In this paper we study the category of graded modules for the current algebra associated to $\mathfrak{sl}_2$. The category enjoys many nice properties, including a tilting theory which was established in previous work of the authors. We…

Let $G$ be a group, $F$ a field of prime characteristic $p$ and $V$ a finite-dimensional $FG$-module. Let $L(V)$ denote the free Lie algebra on $V$, regarded as an $FG$-module, and, for each positive integer $r$, let $L^r(V)$ be the $r$th…

Let $\mathfrak{g}$ be a finite-dimensional simple complex Lie algebra. A layer sum is introduced as the sum of formal exponentials of the distinct weights appearing in an irreducible $\mathfrak{g}$-module. It is argued that the character of…

The symmetric group S_n possesses a nontrivial central extension, whose irreducible representations, different from the irreducible representations of S_n itself, coincide with the irreducible representations of a certain algebra A_n.…

Quantum spaces with $\frak{su}(2)$ noncommutativity can be modelled by using a family of $SO(3)$-equivariant differential $^*$-representations. The quantization maps are determined from the combination of the Wigner theorem for $SU(2)$ with…

The purpose of this paper is to study representations of simple multiplicative Hom-Lie algebras. First, we provide a new proof using Killing form for characterization theorem of simple Hom-Lie algebras given by Chen and Han, then discuss…

Let $n\ge2$ be an integer, $\mathcal{K}_n$ the Weyl algebra over the Laurent polynomial algebra $A_n=\mathbb{C} [x_1^{\pm1}, x_2^{\pm1}, ..., x_n^{\pm1}]$, and $\mathbb{S}_n$ the Lie algebra of divergence zero vector fields on an…

In this paper, first we introduce the notion of a Reynolds operator on an $n$-Lie algebra and illustrate the relationship between Reynolds operators and derivations on an $n$-Lie algebra. We give the cohomology theory of Reynolds operators…

Springer resolution of the set of nilpotent elements in a semisimple Lie algebra plays a central role in geometric representation theory. A new structure on this variety has arisen in several representation theoretic constructions, such as…

We provide an infinite family of counterexamples to the conjecture of Zassenhaus on the solvability of the outer derivation algebra of a simple modular Lie algebra. In fact, we show that the simple modular Lie algebras $H(2;(1,n))^{(2)}$ of…

We study a dual pair of general linear Lie superalgebras in the sense of R. Howe. We give an explicit multiplicity-free decomposition of a symmetric and skew-symmetric algebra (in the super sense) under the action of the dual pair and…

For modular Lie superalgebras, new notions are introduced: Divided power homology and divided power cohomology. For illustration, we give presentations (in terms of analogs of Chevalley generators) of finite dimensional Lie (super)algebras…

We study the representation theory of the symmetric group $S_n$ in positive characteristic $p$. Using features of the LLT-algorithm we give a conjectural description of the projective cover $P(\lambda)$ of the simple module $D(\lambda)$…

In this study, all rings are commutative with non-zero identity and all modules are considered to be unital. Let $M$ be a left $R$-module. A proper submodule $N$ of $M$ is called an $S$-$weakly$ $prime$ submodule if $0_{M}\neq f(m)\in N$…

Let $p$ be an odd prime and let $n$ be a natural number. In this article we determine the irreducible constituents of the permutation module induced by the action of the symmetric group $\mathfrak{S}_n$ on the cosets of a Sylow $p$-subgroup…

We study Demazure modules which occur in a level $\ell$ irreducible integrable representation of an affine Lie algebra. We also assume that they are stable under the action of the standard maximal parabolic subalgebra of the affine Lie…

We introduce and study certain hyperbolic versions of automorphic Lie algebras related to the modular group. Let $\Gamma$ be a finite index subgroup of $\mathrm{SL}(2,\mathbb{Z})$ with an action on a complex simple Lie algebra $\mathfrak…

A Lie conformal algebra is an algebraic structure that encodes the singular part of the operator product expansion of chiral fields in conformal field theory. A Lie pseudoalgebra is a generalization of this structure, for which the algebra…

We first present a filtration on the ring L of Laurent polynomials such that the direct sum decomposition of its associated graded ring gr L agrees with the direct sum decomposition of gr L, as a module over the complex general linear Lie…