论文标题
半膜谎言代数和泊松托管的还原性亚代代代数
Reductive subalgebras of semisimple Lie algebras and Poisson commutativity
论文作者
论文摘要
令$ \ mathfrak g $为半imple lie代数，$ \ mathfrak h \ subset \ mathfrak g $一个还原的子algebra，以至于$ \ mathfrak h^\ perp $是互补的$ \ mathfrak h $ h $ -submodule of $ \ mathfrak g $。 1983年，Bogoyavlenski声称，一个人获得了由Bi-Homogeene os Mathers产生的subalgebra $ {\ Mathcal Z} $，获得了对称代数$ {\ Mathcal s} $的对称代数$ {\ Mathcal s} $的托儿所的副代数的托管亚级数g）^{\ Mathfrak G} $。但这是错误的，我们提出反例。我们还提供了此类subergebras $ {\ Mathcal Z} $的泊松通勤的标准。作为副产品，我们证明$ {\ MATHCAL Z} $如果$ \ Mathfrak h $是Abelian，则在特殊情况下描述$ {\ Mathcal Z} $，当$ \ \ \ Mathfrak H $是Cartan subergebra时。在这种情况下，$ {\ Mathcal z} $似乎是多项式的，并且具有最大的超越度$（\ Mathrm {dim} \，\ Mathfrak G+\ Mathrm {rk} \，\ Mathfrak G）/2 $。
Let $\mathfrak g$ be a semisimple Lie algebra, $\mathfrak h\subset\mathfrak g$ a reductive subalgebra such that $\mathfrak h^\perp$ is a complementary $\mathfrak h$-submodule of $\mathfrak g$. In 1983, Bogoyavlenski claimed that one obtains a Poisson commutative subalgebra of the symmetric algebra ${\mathcal S}(\mathfrak g)$ by taking the subalgebra ${\mathcal Z}$ generated by the bi-homogeneous components of all $H\in{\mathcal S}(\mathfrak g)^{\mathfrak g}$. But this is false, and we present a counterexample. We also provide a criterion for the Poisson commutativity of such subalgebras ${\mathcal Z}$. As a by-product, we prove that ${\mathcal Z}$ is Poisson commutative if $\mathfrak h$ is abelian and describe ${\mathcal Z}$ in the special case when $\mathfrak h$ is a Cartan subalgebra. In this case, ${\mathcal Z}$ appears to be polynomial and has the maximal transcendence degree $(\mathrm{dim}\,\mathfrak g+\mathrm{rk}\,\mathfrak g)/2$.
