Related papers: \v{S}apovalov elements for simple Lie algebras and…

We provide upper bounds on the degrees of the coefficients of \v{S}apovalov elements for a simple Lie algebra. If $\fg$ is a contragredient Lie superalgebra and $\gc$ is a positive isotropic root of $\fg,$ we prove the existence and…

If $\mathfrak{g}$ is a contragredient Lie superalgebra and $\gamma$ is a root of $\mathfrak{g},$ we prove the existence and uniqueness of \v{S}apovalov elements for $\gamma$ and give upper bounds on the degrees of their coefficients. Then…

We give explicit expressions for \vSapovalov elements in Type A Lie algebras and superalgebras. Explicit expressions were already given in arXiv:1710.10528 Section 9, using non-commutative determinants, and in fact our first main results,…

For the exceptional finite-dimensional modular Lie superalgebras $\mathfrak{g}(A)$ with indecomposable Cartan matrix $A$, and their simple subquotients, we computed non-isomorphic Lie superalgebras constituting the homologies of the odd…

Let $G$ be a simple algebraic group over an algebraically closed field of characteristic $p>0$ and suppose that $p$ is a very good prime for $G$. We prove that any maximal Lie subalgebra $M$ of $\mathfrak{g} = {\rm Lie}(G)$ with ${\rm…

Shapovalov elements $\theta_{\beta,m}$ are special elements in a Borel subalgebra of a classical or quantum universal enveloping algebra parameterized by a positive root $\beta$ and a positive integer $m$. They relate the canonical…

We study some variants of Verma modules of basic Lie superalgebras obtained via changing Borel subalgebras. These allow us to demonstrate that the principal block of \(\mathfrak{gl}(1|1)\) is realized as (non-Serre) full subcategories of…

Shapovalov elements $\theta _{\beta,m}$ of the classical or quantized universal enveloping algebra of a simple Lie algebra $\mathfrak{g}$ are parameterized by a positive root $\beta$ and a positive integer $m$. They relate the highest…

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…

Axioms for the generalization of root systems were defined and classified (irreducible) by V. Serganova, which precisely correspond to the root systems of basic classical Lie Superalgebras. Here, we present a unified method for constructing…

For a simple Lie algebra, Shapovalov elements give rise to highest weight vectors in Verma modules. The usual construction of these elements uses induction on the length of a certain Weyl group element. If $\mathfrak{g}= \mathfrak{sl}(N+1)$…

Let $G$ be a complex simple Lie group and let $\g = \hbox{\rm Lie}\,G$. Let $S(\g)$ be the $G$-module of polynomial functions on $\g$ and let $\hbox{\rm Sing}\,\g$ be the closed algebraic cone of singular elements in $\g$. Let ${\cal L}\s…

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…

Let $\mathfrak{g}$ be a simple complex Lie algebra and let $\mathfrak{t} \subset \mathfrak{g}$ be a toral subalgebra of $\mathfrak{g}$. As a $\mathfrak{t}$-module $\mathfrak{g}$ decomposes as \[\mathfrak{g} = \mathfrak{s} \oplus…

Let G be a simply connected simple algebraic group over an algebraically closed field K of characteristic p>0 with root system R, and let ${\mathfrak g}={\cal L}(G)$ be its restricted Lie algebra. Let V be a finite dimensional ${\mathfrak…

We study {\em disemisimple} Lie algebras, i.e., Lie algebras which can be written as a vector space sum of two semisimple subalgebras. We show that a Lie algebra $\mathfrak{g}$ is disemisimple if and only if its solvable radical coincides…

For an arbitrary affine Lie algebra we study an analog of the category O for the natural Borel subalgebra and zero central charge. We show that such category is semisimple having the reduced imaginary Verma modules as its simple objects.…

Let $V$ be a simple vertex algebra of countable dimension, $G$ be a finite automorphism group of $V$ and $\sigma$ be a central element of $G$. Assume that ${\cal S}$ is a finite set of inequivalent irreducible $\sigma$-twisted $V$-modules…

Making the first steps towards a classification of simple partial comodules, we give a general construction for partial comodules of a Hopf algebra \(H\) using central idempotents in right coideal subalgebras and show that any…

It is well known that the ring radical theory can be approached via language of modules. In this work, we present some generalizations of classical results from module theory, in the two-sided and graded sense. Let $\mathsf{G}$ be a group,…