Kostant theorem for special filtered algebras

Linear algebra meets Lie algebra: the Kostant-Wallach theory

In two languages, Linear Algebra and Lie Algebra, we describe the results of Kostant and Wallach on the fibre of matrices with prescribed eigenvalues of all leading principal submatrices. In addition, we present a brief introduction to basic notions in Algebraic Geometry, Integrable Systems, and Lie Algebra aimed at specialists in Linear Algebra.

The Kostant invariant and special $ε$-orthogonal representations for $ε$-quadratic colour Lie algebras

Let k be a field of characteristic not two or three, let $\mathfrak{g}$ be a finite-dimensional colour Lie algebra and let V be a finite-dimensional representation of $\mathfrak{g}$. In this article we give various ways of constructing a colour Lie algebra $\tilde{\mathfrak{g}}$ whose bracket in some sense extends both the bracket of $\mathfrak{g}$ and the action of $\mathfrak{g}$ on V. Colour Lie algebras, originally introduced by R. Ree ([Ree60]), generalise both Lie algebras and Lie superalgebras, and in those cases our results imply many known results ([Kos99], [Kos01], [CK15], [SS15]). Fo…

Adjoint quotient maps for restricted Lie algebras Wn and Sn

We give an explicit description of adjoint quotient maps for Jacobson-Witt algebra Wn and special algebra Sn. An analogue of Kostant's differential criterion of regularity is given for Wn. Furthermore, we describe the fiber of adjoint quotient map for Sn and construct the analogs of Kostant's transverse slice. In addition, we analysis the nilpotent elements and generalize some Premet's results to Sn.

三变量多项式代数的过滤形变

Filtered deformations of three-variable polynomial algebras

本文分类三变量多项式环的所有过滤形变，并据此证明正特征代数闭域上具有有限维过滤且相伴分次环为三变量多项式环的代数满足多项式恒等式。

Krull 维数为 2 的交换代数的滤过形变

Filtered deformations of commutative algebras of Krull dimension two

本文研究正特征域上有限生成滤过代数，当其相伴分次环为 Krull 维数 2 的整环时，证明该代数满足多项式恒等式，部分肯定回答 Etingof 的问题。

Differential forms on smooth operadic algebras

The classical Hochschild--Kostant--Rosenberg (HKR) theorem computes the Hochschild homology and cohomology of smooth commutative algebras. In this paper, we generalise this result to other kinds of algebraic structures. Our main insight is that producing HKR isomorphisms for other types of algebras is directly related to computing quasi-free resolutions in the category of left modules over an operad; we establish that an HKR-type result follows as soon as this resolution is diagonally pure. As examples we obtain a permutative and a pre-Lie HKR theorem for smooth commutative and smooth brace al…