Related papers: Blocks of the category $\mathcal{O}^\mathfrak{p}$ …

A simpler definition for a class of two-parameter quantum groups associated to semisimple Lie algebras is given in terms of Euler form. Their positive parts turn out to be 2-cocycle deformations of each other under some conditions. An…

In this paper, we introduce two new families of infinite-dimensional simple Lie algebras and a new family of infinite-dimensional simple Lie superalgebras. These algebras can be viewed as generalizations of the Block algebras.

Let E be an operator algebra on a Hilbert space with finite-dimensional generated C*-algebra. A classification is given of the locally finite algebras and the operator algebras obtained as limits of direct sums of matrix algebras over E…

We establish foundational properties of fractional operators on Lie groups of homogeneous type. We prove embedding theorems for the associated Sobolev-type spaces.

In this paper we consider some properties of semisimple Hopf algebras of dimension pq where p and q are distinct primes. These properties are useful for classification of such Hopf algebras. In particular, we show that for such a Hopf…

In this paper, we explore a canonical connection between the algebra of $q$-difference operators $\widetilde{V}_{q}$, affine Lie algebra and affine vertex algebras associated to certain subalgebra $\mathcal{A}$ of the Lie algebra…

It is known that the category of Lie algebras over a ring admits algebraic exponents. The aim of this paper is to show that the same is true for the category of internal Lie algebras in an additive, cocomplete, symmetric, closed, monoidal…

We study the algebraic structure of the Poisson algebra P(O) of polynomials on a coadjoint orbit O of a semisimple Lie algebra. We prove that P(O) splits into a direct sum of its center and its derived ideal. We also show that P(O) is…

Let L be a finite-dimensional simple Lie algebra over an algebraically closed field of F characteristic p>3. We prove that if the p-envelope of L in the derivation algebra of L contains nonstandard tori of maximal dimension, then p=5 and L…

We show that the blocks of category O for the Lie superalgebra q_n associated to half-integral weights carry the structure of a tensor product categorification for the infinite rank Kac-Moody algebra of type C. This allows us to prove two…

The rank $n$ symplectic oscillator Lie algebra $\mathfrak{g}_n$ is the semidirect product of the symplectic Lie algebra $\mathfrak{sp}_{2n}$ and the Heisenberg Lie algebra $H_n$. In this paper, we study weight modules with finite…

This series of papers is a contribution to the program of classifying $p$-blocks of finite groups up to source algebra equivalence, starting with the case of cyclic blocks. To any $p$-block $\mathbf{B}$ of a finite group with cyclic defect…

We determine the dimensions of $\mathrm{Ext}$-groups between simple modules and dual generalized Verma modules in singular blocks of parabolic versions of category $\mathcal{O}$ for complex semisimple Lie algebras and affine Kac-Moody…

We determine the structure of category $\cO$ for the rational Cherednik algebra of $G(m,1,n)$ in the case where the $\KZ$ functor satisfies a condition called \emph{separating simples}. As a consequence, we show that the property of having…

We compute the number of orbit types for simply connected simple algebraic groups over algebraically closed fields as well as for compact simply connected simple Lie groups. We also compute the number of orbit types for the adjoint action…

The symplectic blob algebra is a physically motivated quotient of the Hecke algebra $H(\tilde{C}_n)$ with a diagram calculus. We find the blocks for the symplectic blob algebra for all specialisations of its parameters over the complex…

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.…

We single out a large class of semisimple singularities with the property that all roots of the Poincar\'e polynomial of the Lie algebra of derivations of the corresponding suitably (not necessarily quasihomogeneously) graded moduli algebra…

We investigate models of algebraic theories in the category of cocommutative coalgebras over a field. We establish some of their categorical properties, similar to those of algebraic varieties. We introduce a class of categories of…

We study some elementary properties of the quantum enveloping algebra associated to a parabolic subalgebra $\mathfrak{p}$ of a semisimple Lie algebra $\mathfrak{g}$. In particular we prove an explicit formula for the degree of this algebra,…