论文标题
泊松包裹代数的dixmier定理
A Dixmier theorem for Poisson enveloping algebras
论文作者
论文摘要
我们考虑在多项式代数$ k上进行偏斜的对称$ n $ n $ - y-ary括号[x_1，\ ldots，x_n，x_n，x_ {n+1}] $（$ n \ geq 2 $），由字段$ k $的特征性零定义为$ \ {a_1，\ ldots，a_n \} = j（a_1，\ ldots，a_n，c）$，其中$ c $是$ k [x_1，\ ldots，x_n，x_n，x__ {n+1}] $的固定元素和$ j $是jacobian。如果$ n = 2 $，则此支架是泊松支架，如果$ n \ geq 3 $，那么它是$ k [x_1，\ ldots，x_n，x_n，x_ {n+1}] $ k [x_1，\ ldots on $ k [x_1，\ ldots] $。 We describe the center of the corresponding $n$-Lie-Poisson algebra and show that the quotient algebra $K[x_1,\ldots,x_n,x_{n+1}]/(C-λ)$, where $(C-λ)$ is the ideal generated by $C-λ$, $0\neq λ\in K$, is a simple central $n$-Lie-Poisson代数如果$ c $是均质的多项式，它不是任何非零多项式的适当幂。该结构包括$ p（\ mathrm {sl} _2（k））/（c-λ）$ $ \ mathrm {sl} _2（k）$ in $ p（\ mathrm {sl} _2（k））$。还可以证明，托管封封的代数$ p（\ mathbb {m}）/（c-λ）$的代数$ p（\ mathbb {m}）$ p（\ mathbb {m}）$的七个维度malcev algebra $ \ mathbb {m mathbb {m} $很简单。
We consider a skew-symmetric $n$-ary bracket on the polynomial algebra $K[x_1,\ldots,x_n,x_{n+1}]$ ($n\geq 2$) over a field $K$ of characteristic zero defined by $\{a_1,\ldots,a_n\}=J(a_1,\ldots,a_n,C)$, where $C$ is a fixed element of $K[x_1,\ldots,x_n,x_{n+1}]$ and $J$ is the Jacobian. If $n=2$ then this bracket is a Poisson bracket and if $n\geq 3$ then it is an $n$-Lie-Poisson bracket on $K[x_1,\ldots,x_n,x_{n+1}]$. We describe the center of the corresponding $n$-Lie-Poisson algebra and show that the quotient algebra $K[x_1,\ldots,x_n,x_{n+1}]/(C-λ)$, where $(C-λ)$ is the ideal generated by $C-λ$, $0\neq λ\in K$, is a simple central $n$-Lie-Poisson algebra if $C$ is a homogeneous polynomial that is not a proper power of any nonzero polynomial. This construction includes the quotients $P(\mathrm{sl}_2(K))/(C-λ)$ of the Poisson enveloping algebra $P(\mathrm{sl}_2(K))$ of the simple Lie algebra $\mathrm{sl}_2(K)$, where $C$ is the standard Casimir element of $\mathrm{sl}_2(K)$ in $P(\mathrm{sl}_2(K))$. It is also proven that the quotients $P(\mathbb{M})/(C-λ)$ of the Poisson enveloping algebra $P(\mathbb{M})$ of the exceptional simple seven dimensional Malcev algebra $\mathbb{M}$ are central simple.
