Faithful representations of SL(2) over truncated Witt vectors

有限截断当前李代数上的简单模

Simple modules over truncated current Lie algebras of the Witt algebra

本文研究了有限截断当前李代数W_ℓ上p-特征为χ且高度不超过1的简单模，并给出其分类；进一步探讨了p-特征高度大于1的简单模家族。

有限截断赋值环上幂零群的忠实维数的多项式性

Polynomiality of the faithful dimension of nilpotent groups over finite truncated valuation rings

本文研究有限截断赋值环上幂零群的忠实维数，证明在有限域且素数充分大时其由有限多项式列表给出，并提出一般情形的多项式性猜想并证明特殊情形，最后精确计算自由亚交换幂零Lie代数的忠实维数。

关于截断移位iYangians上的权模

On weight modules over truncated shifted iYangians

本文研究截断移位iYangians的权模，引入oKLRW代数、KLR iYangians等，建立其表示性质，并证明其权模与区间oKLRW代数幂零模的等价关系。

Representing Lie algebras using approximations with nilpotent ideals

We prove a refinement of Ado's theorem for Lie algebras over an algebraically-closed field of characteristic zero. We first define what it means for a Lie algebra $L$ to be approximated with a nilpotent ideal, and we then use such an approximation to construct a faithful representation of $L$. The better the approximation, the smaller the degree of the representation will be. We obtain, in particular, explicit and combinatorial upper bounds for the minimal degree of a faithful $L$-representation. The proofs use the universal enveloping algebra of Poincaré-Birkhoff-Witt and the almost-algebraic…

Regular derivations of truncated polynomial rings

Let $\Bbbk$ be an algebraically closed field of characteristic $p>2$. Let $\mathcal{O}_n=\Bbbk[X_1,\ldots,X_n]/(X_1^p,\ldots, X_n^p)$, a truncated polynomial ring in $n$ variables, and denote by $\mathcal{L}$ the derivation algebra of $\mathcal{O}_n$. It is known that the ring of all polynomial functions on $\mathcal{L}$ invariant under the action of the group of $\mathrm{Aut}(\mathcal{L})$ is freely generated by $n$ elements. Furthermore, the related quotient morphism is faithfully flat and all its fibres are irreducible complete intersections. An element $x\in\mathcal{L}$ is called ${\it reg…

The Automorphisms group of a Current Lie algebra

Let $\mathfrak{g}$ be a finite dimensional complex Lie algebra and let $A$ be a finite dimensional complex, associative and commutative algebra with unit. We describe the structure of the derivation Lie algebra of the current Lie algebra $\mathfrak{g}_A= \mathfrak{g} \otimes A$, denoted by $\operatorname{Der}(\mathfrak{g}_A)$. Furthermore, we obtain the Levi decomposition of $\operatorname{Der}(\mathfrak{g}_A)$. As a consequence of the last result, if $\mathfrak{h}_m$ is the Heisenberg Lie algebra of dimension $2 m + 1$, we obtain a faithful representation of $\operatorname{Der}(\mathfrak{h}_{…