Centers and Azumaya loci of finite WW-algebras

In this paper, we study the center $Z$ of the finite $W$-algebra $\mathcal{T}(\mathfrak{g},e)$ associated with a semi-simple Lie algebra $\mathfrak{g}$ over an algebraically closed field \mathds{k} of characteristic $p\gg0$, and an arbitrarily given nilpotent element $e\in\mathfrak{g}$. We obtain an analogue of Veldkamp's theorem on the center. For the maximal spectrum $\text{Specm}(Z)$, we show that its Azumaya locus coincides with its smooth locus of smooth points. The former locus reflects irreducible representations of maximal dimension for $\mathcal{T}(\mathfrak{g},e)$.Comment: 31 page

Super Vust theorem and Schur-Sergeev duality for principal finite WW-superalgebras

In this paper, we first formulate a super version of Vust theorem associated with a regular nilpotent element $e\in\mathfrak{gl}(V)$. As an application of this theorem, we then obtain the Schur-Sergeev duality for principal finite $W$-superalgebras which is partially a super version of Brundan-Kleshchev's higher level Schur-Weyl duality.Comment: 35 pages, comments are welcom

Study on the Experiment of Cultivating the Peer Interpersonal Skill of the Children in the Kindergarten

This purpose of research is that cultivating the peer interpersonal skill of the children in the kindergarten of the linguistic& non-linguistic competence, pro-social behavior & social obstacle. Research methods are on-the-spot experimental and observational method. The results are that the difference is significance in the total level of the children’s peer interpersonal skill between the experimental group and the control group (p＜0.05), that the difference is significance in the linguistic & non-linguistic competence and the children’s interaction obstacle between the experimental group and the control group (p＜0.05). But that the difference is no significance in the social initiative and pro-social between the experimental group and the control group (p＞0.05). The conclusion is that it is effective for teachers to intervene in children’s peer interaction during the experiment. It can become the appropriate education measures to improve the peer interpersonal skill of the children in the kindergarten