# 数学学科2020系列学术报告之四、五、六

Abstract: We give a brief introduction to the study of pre-Lie algebras with some recent progress, emphasizing the relationships with other topics in mathematics and mathematical physics.

Absrtract：Let $d\ge1$ be an integer, $W_d$ be the Witt  algebra。 For any admissible $W_d$-module $P$ and any $\gl_d$-module $V$, one can form a $\W_d$-module $F(P,V)$, which as a vector space is $P\ot V$。  Since $W_d$ has a natural subalgebra isomorphic to $\sl_{d+1}$, we can view $F(P,V)$ as an $\sl_{d+1}$-module。 Taking $P=\Omega(\bf{\lambda})$, the rank-$1$ $U(\mathfrak{h})$-free $W_d$-module and $V=V(\bf{a},b)$, the irreducible cuspidal module over $\gl_d$, we get the special $\sl_{d+1}$-module $\F(\bf{\lambda}, \bf{a},b)=F(\Omega(\bf{\lambda}),V(\bf{a},b))$。 We determine the necessary and sufficient conditions for the $\sl_{d+1}$-module $F(\bf{\lambda};\bf{a},b)$ to be irreducible。 And for the reducible case, we constructed their proper submodules explicitly。