Related papers: Blocks and modules for Whittaker pairs
This book is mainly an exposition of the author's works and his joint works with his former students on explicit representations of finite-dimensional simple Lie algebras, related partial differential equations, linear orthogonal algebraic…
In this short note we announce three formulas for the set of weights of various classes of highest weight modules $\V$ with highest weight \lambda, over a complex semisimple Lie algebra $\lie{g}$ with Cartan subalgebra $\lie{h}$. These…
In this paper, we define and study Whittaker modules for the super-Viraoro algebras, including the Neveu-Schwarz algebra and the Ramond algebra. We classify the simple Whittaker modules and obtain necessary and sufficient conditions for…
If a Lie algebra structure g on a vector space is the sum of a family of mutually compatible Lie algebra structures g_i's, we say that g is simply assembled from the g_i's. Repeating this procedure with a number of Lie algebras, themselves…
We study the structure of weight modules $V$ with restrictions neither on the dimension nor on the base field, over split Lie algebras $L$. We show that if $L$ is perfect and $V$ satisfies $LV=V$ and ${\mathcal Z}(V)=0$, then $$\hbox{$L…
In this paper we analyze Whittaker modules for two families of Wittaker pairs related to the subalgebras of the Virasoro algebra generated by L_r,..., L_{2r} and L_1,L_n. The structure theorems for the corresponding universal Whittaker…
Given an algebraically closed field $\Bbbk$ of characteristic zero, a Lie superalgebra $\mathfrak{g}$ over $\Bbbk$ and an associative, commutative $\Bbbk$-algebra $A$ with unit, a Lie superalgebra of the form $\mathfrak{g} \otimes_\Bbbk A$…
We investigate weight modules for finite and infinite Weyl algebras, classifying all such simple modules. We also study the representation type of the blocks of locally-finite weight module categories and describe indecomposable modules in…
In this paper, a family of infinite dimensional Lie algebras $\tilde{\mathcal{L}}$ is introduced and investigated, called the extended Heisenberg-Virasoro algebra,denoted by $\tilde{\mathcal{L}}$. These Lie algebras are related to the $N=2$…
In this note, we use give some algebraic applications of a previous result by the author which compares the deformations parameterized by the Maurer-Cartan elements of a differential graded Lie algebra, and a differential graded Lie…
The main purpose of this article is to develop an explicit derived deformation theory of algebraic structures at a high level of generality, encompassing in a common framework various kinds of algebras (associative, commutative, Poisson...)…
We classify irreducible finite-dimensional modules of a collection of real Lie superalgebras that includes the simple ones, their classical variants, complex Lie superalgebras after restriction of scalars, and all real Lie algebras. Our…
We study representations of a deformed Heisenberg-Virasoro algebra that does not admit a triangular decomposition. Despite this, its $\mathbb{Z}$-gradation allows the classification of simple restricted modules. We show that all such…
One of the four well-known series of simple Lie algebras of Cartan type is the series of Lie algebras of Special type, which are divergence-free Lie algebras associated with polynomial algebras and the operators of taking partial…
We show that, for an arbitrary quasi-reductive Lie superalgebra with a triangular decomposition and a character $\zeta$ of the nilpotent radical, the associated Backelin functor $\Gamma_\zeta$ sends Verma modules to standard Whittaker…
This is a companion article to my papers on Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebras gl(m|n) (much revised!) and q(n). The goal is to develop the general theory of tilting modules for Lie superalgebras,…
In this work, we consider Lie algebras L containing a subalgebra isomorphic to sl3 and such that L decomposes as a module for that sl3 subalgebra into copies of the adjoint module, the natural 3-dimensional module and its dual, and the…
This paper gives a classification of parabolic subalgebras of simple Lie algebras over $\CC$ that are complexifications of parabolic subalgebras of real forms for which Lynch's vanishing theorem for generalized Whittaker modules is…
This contribution is mainly based on joint papers with Lepowsky and Milas, and some parts of these papers are reproduced here. These papers further extended works by Lepowsky and by Milas. Following our joint papers, I explain the general…
We investigate the structure and representation theory of finite-dimensional $\mathbb{Z}$-graded Lie algebras, including the corresponding root systems and Verma, irreducible, and Harish-Chandra modules. This extends the familiar theory for…