Derivations of the Lie algebra of strictly block upper triangular matrices

Growth of differential identities

In this paper we study the growth of the differential identities of some algebras with derivations, i.e., associative algebras where a Lie algebra $L$ (and its universal enveloping algebra $U(L)$) acts on them by derivations. In particular, we study in detail the differential identities and the cocharacter sequences of some algebras whose sequence of differential codimensions has polynomial growth. Moreover, we shall give a complete description of the differential identities of the algebra $UT_2$ of $2\times 2$ upper triangular matrices endowed with all possible action of a Lie algebra by deri…

Description of B-orbit closures of order 2 in upper-triangular matrices

Let n_n(C) be the algebra of strictly upper-triangular n x n matrices over the field of complex numbers and X_2 the subset of matrices of nilpotent order 2. Let B_n(C) be the group of invertible upper-triangular matrices acting on n_n(C) by conjugation. Let B(u) be the orbit of u in X_2 with respect to this action. Let S_n^2 be the subset of involutions in the symmetric group S_n. We define a new partial order on S_n^2 which gives the combinatorial description of the closure of B(u). We also construct an ideal I(B(u)) in symmetric algebra S(n_n(C)^* whose variety V(I(B(u))) equals the closure …

Jordan types of triangular matrices over a finite field

Let $λ$ be a partition of an integer $n$ and ${\mathbb F}_q$ be a finite field of order $q$. Let $P_λ(q)$ be the number of strictly upper triangular $n\times n$ matrices of the Jordan type $λ$. It is known that the polynomial $P_λ$ has a tendency to be divisible by high powers of $q$ and $Q=q-1$, and we put $P_λ(q)=q^{d(λ)}Q^{e(λ)}R_λ(q)$, where $R_λ(0)\neq0$ and $R_λ(1)\neq0$. In this article, we study the polynomials $P_λ(q)$ and $R_λ(q)$. Our main results: an explicit formula for $d(λ)$ (an explicit formula for $e(λ)$ is known, see Proposition 3.3 below), a recursive formula for $R_λ(q)$ (a…

Coadjoint orbits for $A_{n-1}^+, B_{n}^{+}$, and $D_{n}^{+}$

A complete description of the coadjoint orbits for A_{n-1}^{+}, the nilpotent Lie algebra of n-by-n strictly upper triangular matrices, has not yet been obtained, though there has been steady progress on it ever since the orbit method was devised. We apply methods developed by Andre' to find defining equations for the elementary coadjoint orbits for the maximal nilpotent Lie subalgebras of the orthogonal Lie algebras, and we also determine all the possible dimensions of coadjoint orbits in the case of A_{n-1}^{+}.

A transitivity result for ad-nilpotent ideals in type A

The paper considers subspaces of the strictly upper triangular matrices, which are stable under Lie bracket with any upper triangular matrix. These subspaces are called ad-nilpotent ideals and there are Catalan number of such subspaces. Each ad-nilpotent ideal meets a unique largest nilpotent orbit in the Lie algebra of all matrices. The main result of the paper is that under an equivalence relation on ad-nilpotent ideals studied by Mizuno and others, the equivalence classes are the ad-nilpotent ideals with the same largest nilpotent orbit. We include two applications of the result, one to the…

Prehomogeneous spaces for Borel subgroups of general linear groups

Let $k$ be an algebraically closed field. Let $B$ be the Borel subgroup of $\mGL_n(k)$ consisting of nonsingular upper triangular matrices. Let $\frb = \mLie B$ be the Lie algebra of upper triangular $n \times n$ matrices and $\fru$ the Lie subalgebra of $\frb$ consisting of strictly upper triangular matrices. We classify all Lie ideals $\frn$ of $\frb$, satisfying $\fru' \subseteq \frn \subseteq \fru$, such that $B$ acts (by conjugation) on $\frn$ with a dense orbit. Further, in case $B$ does not act with a dense orbit, we give the minimal codimension of a $B$--orbit in $\frn$. This can be vi…

李代数表示的 Jordan-Kronecker 不变量：示例与计算

Jordan-Kronecker invariants of Lie algebra representations: examples and computations

本文系统计算了多类李代数表示及 GL(n)、SL(n) 合同作用微分的 Jordan-Kronecker 不变量，补充了相关表示论计算实例。

Blocks and modules for Whittaker pairs

Inspired by recent activities on Whittaker modules over various (Lie) algebras we describe some general framework for the study of Lie algebra modules locally finite over a subalgebra. As a special case we obtain a very general setup for the study of Whittaker modules, which includes, in particular, Lie algebras with triangular decomposition and simple Lie algebras of Cartan type. We describe some basic properties of Whittaker modules, including a block decomposition of the category of Whittaker modules and certain properties of simple Whittaker modules under some rather mild assumptions. We e…