完全可约SL（2）-同态

完全可约李超代数的零化定理

The Annihilation theorem for the completely reducible Lie superalgebras

Duflo的一个著名定理是复半单李代数的包络代数中Verma模的零化子由它与中心的交生成。对于李超代数，这个结果不成立。例如，在正交辛李超代数osp（1，2）的情况下，Pinczon在[Pi]中给出了一个Verma模的例子，其零化子不是由它与泛包络代数的中心的交集生成的。更一般地说，Musson在[Mu 1]中提出了一个这样的"奇异"韦尔马模族，用于osp（1，2l）情形。本文给出了$\osp(1,2l)$-韦尔马模的零化子由它与中心的交生成的最大权的一个充要条件.这就回答了Musson的一个问题。

The Criterion of Completely Reducibility for Continuous Representations of Group Algebras

It is shown that every nonsingular continuous representation of the group algebra $L^{1}(G)$ in Banach spaces is completely reducible if and only if $G$ is a compact group.

Reducible subgroups of exceptional algebraic groups

Let $G$ be a simple algebraic group over an algebraically closed field. A closed subgroup $H$ of $G$ is called $G$-completely reducible ($G$-cr) if, whenever $H$ is contained in a parabolic subgroup $P$ of $G$, it is contained in a Levi factor of $P$. In this paper we complete the classification of connected $G$-cr subgroups when $G$ has exceptional type, by determining the $L_{0}$-irreducible connected reductive subgroups for each simple classical factor $L_{0}$ of a Levi subgroup of $G$. As an illustration, we determine all reducible, $G$-cr semisimple subgroups when $G$ has type $F_4$ and v…

On complete reducibility of tensor products of simple modules over simple algebraic groups

Let $G$ be a simply connected simple algebraic group over an algebraically closed field $k$ of characteristic $p>0$. The category of rational $G$-modules is not semisimple. We consider the question of when the tensor product of two simple $G$-modules $L(λ)$ and $L(μ)$ is completely reducible. Using some technical results about weakly maximal vectors (i.e. maximal vectors for the action of the Frobenius kernel $G_1$ of $G$) in tensor products, we obtain a reduction to the case where the highest weights $λ$ and $μ$ are $p$-restricted. In this case, we also prove that $L(λ)\otimes L(μ)$ is comple…

$G$-完全可约性与饱和性

$G$-complete reducibility and saturation

本文研究连通约化线性代数群中完全可约性与饱和性的关系，在特征限制下证明 $H$-完全可约子群在 $G$ 中亦完全可约，并推广饱和性结果至一般群及李型有限子群。

Relative complete reducibility and normalised subgroups

We study a relative variant of Serre's notion of $G$-complete reducibility for a reductive algebraic group $G$. We let $K$ be a reductive subgroup of $G$, and consider subgroups of $G$ which normalise the identity component $K^{\circ}$. We show that such a subgroup is relatively $G$-completely reducible with respect to $K$ if and only if its image in the automorphism group of $K^{\circ}$ is completely reducible. This allows us to generalise a number of fundamental results from the absolute to the relative setting. We also derive analogous results for Lie subalgebras of the Lie algebra of $G$, …

The Jacobson--Morozov theorem and complete reducibility of Lie subalgebras

In this paper we determine the precise extent to which the classical sl_2-theory of complex semisimple finite-dimensional Lie algebras due to Jacobson--Morozov and Kostant can be extended to positive characteristic. This builds on work of Pommerening and improves significantly upon previous attempts due to Springer--Steinberg and Carter/Spaltenstein. Our main advance arises by investigating quite fully the extent to which subalgebras of the Lie algebras of semisimple algebraic groups over algebraically closed fields k are G-completely reducible, a notion essentially due to Serre. For example i…

稳定的高阶 Specht 多项式与无限对称群的表示

Stable Higher Specht Polynomials and Representations of Infinite Symmetric Groups

本文引入最终对称函数环 $\ ildeΛ$，在其中构造稳定高阶 Specht 多项式，并刻画无限对称群表示的不可约极限、最大完全可约子表示及显式滤过。

G-complete reducibility and the exceptional algebraic groups

Let $G=G(K)$ be a simple algebraic group defined over an algebraically closed field $K$ of characteristic $p>0$. A subgroup $X$ of $G$ is said to be $G$-completely reducible if, whenever it is contained in a parabolic subgroup of $G$, it is contained in a Levi subgroup of that parabolic. A subgroup $X$ of $G$ is said to be $G$-irreducible if $X$ is in no parabolic subgroup of $G$; and $G$-reducible if it is in some parabolic of $G$. In this thesis, we consider the case that $G$ is of exceptional type. When $G$ is of type $G_2$ we find all conjugacy classes of closed, connected, reductive subgr…

映入量子特征为二时钩号标记的 Specht 模块的同态

Homomorphisms into Specht modules labelled by hooks in quantum characteristic two

本文研究了量子特征为二时 KLR 代数中任意划分到钩号 Specht 模块的同态问题，完整描述了生成元作用，显式构造了一族非零同态及其分次，并推广了平凡模块的 James 结果。

n-李代数的表示

Representations of n-Lie algebras

设$V_n=<e_1,...,e_{n+1}>$是复数域上具有n-李交换子$[e_1,...,\hat{e_i},...,e_{n+1}]=(-1)^ie_i$的向量积n-李代数.任何有限维n-李$V_n$ -模都是完全可约的。对于某个非负整数t，任何有限维不可约n-Lie $V_n$ -模同构于$so_{n+1}$ -模的一个具有最高权$tπ_1$的n-Lie扩张.