Related papers: The BGG Category for Generalised Reductive Lie Alg…

We show that, for many Lie superalgebras admitting a compatible $\mathbb{Z}$-grading, Kac induction functor gives rise to a bijection between simple supermodules over a Lie superalgebra and simple supermodules over the even part of this Lie…

For any finite-dimensional simple Lie algebra $\mathfrak{g}$ and commutative associative algebra $S$ of finite type, we give a complete classification of the simple weight modules of $\mathfrak{g}\otimes S$ with bounded weight…

A parabolic subalgebra $\mathfrak{p}$ of a complex semisimple Lie algebra $\mathfrak{g}$ is called a parabolic subalgebra of abelian type if its nilpotent radical is abelian. In this paper, we provide a complete characterization of the…

We determine the Ringel duals for all blocks in the parabolic versions of the BGG category O associated to a reductive finite dimensional Lie algebra. In particular we find that, contrary to the original category O and the specific…

We show that for any singular dominant integral weight $\lambda$ of a complex semisimple Lie algebra $\mathfrak{g}$, the endomorphism algebra $B$ of any projective-injective module of the parabolic BGG category…

Let $\mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus \mathfrak{g}_{\bar{1}}$ be a classical Lie superalgebra and $\mathcal{F}$ be the category of finite dimensional $\mathfrak{g}$-supermodules which are completely reducible over the reductive Lie…

We describe a systematic method to construct arbitrary highest-weight modules, including arbitrary finite-dimensional representations, for any finite dimensional simple Lie algebra $\mathfrak{g}$. The Lie algebra generators are represented…

We determine the Verma multiplicities and the characters of projective modules for atypical blocks in the BGG Category O for the general linear Lie superalgebras $\frak{gl}(2|2)$ and $\frak{gl}(3|1)$. We then explicitly determine the…

For semisimple Lie algebras, the BGG resolution is often viewed as a categorification of the Weyl character formula. For general linear Lie superalgebras, Brundan--Stroppel constructed an infinite resolution of the so-called Kostant simple…

We describe a class (called regular) of invariant generalized complex structures on a real semisimple Lie group G. The problem reduces to the description of admissible pairs (\gk, \omega), where \gk is an appropriate regular subalgebra of…

Let g=g_{0} \oplus g_{1} be a classical Lie superalgebra and F be the category of finite dimensional g-supermodules which are semisimple over g_{0}. In this paper we investigate the homological properties of the category F. In particular we…

In this paper, we introduce a class of infinite Lie conformal algebras $\mathfrak{B}(\alpha,\beta,p)$, which are the semi-direct sums of Block type Lie conformal algebra $\mathfrak{B}(p)$ and its non-trivial conformal modules of $\Z$-graded…

Given a complex semisimple Lie algebra ${\mathfrak g}$ and a commutative ${\mathbb C}$-algebra $A$, let ${\mathfrak g}[A] = {\mathfrak g} \otimes A$ be the corresponding generalized current algebra. In this paper we explore questions…

Let $\mathfrak{g}$ be a classial Lie algebra and $\mathfrak{p}$ be a maximal parabolic subalgebra. Let $M$ be a generalized Verma module induced from a one dimensional representation of $\mathfrak{p}$. Such $M$ is called a scalar type…

We study the algebra of regular functions on the big cell of the Gauss decomposition of a simple complex Lie group G. We prove that it is spanned by the matrix elements of big projective modules in the BGG category O, and admits a…

We study post-Lie algebra structures on pairs of Lie algebras (g,n), and prove existence results for the case that one of the Lie algebras is semisimple. For semisimple g and solvable n we show that there exist no post-Lie algebra…

In this paper, we study the BGG category $\mathcal{O}$ for the quantum Schr{\"o}dinger algebra $U_q(\mathfrak{s})$, where $q$ is a nonzero complex number which is not a root of unity. If the central charge $\dot z\neq 0$, using the module…

Let $G$ be a reductive algebraic group with Lie algebra $\mathfrak{g}$ and $V$ a finite-dimensional representation of $G$. Costello-Gaiotto studied a graded Lie algebra $\mathfrak{d}_{\mathfrak{g}, V}$ and the associated affine Kac-Moody…

We compute projective dimension of translated simple modules in the regular block of the BGG category $\mathcal{O}$ in terms of Kazhdan-Lusztig combinatorics. This allows us to determine which projectives can appear at the last step of a…

We study gradings by noncommutative groups on finite dimensional Lie algebras over an algebraically closed field of characteristic zero. It is shown that if $L$ is gradeg by a non-abelian finite group $G$ then the solvable radical $R$ of…