Reductive subalgebras of semisimple Lie algebras and Poisson commutativity

Publication date: 7 December 2020

Abstract: Let

m a t h f r a k g

be a semisimple Lie algebra,

m a t h f r a k h s u b s e t m a t h f r a k g

a reductive subalgebra such that

m a t h f r a k h^{p} e r p

is a complementary

m a t h f r a k h

-submodule of

m a t h f r a k g

. In 1983, Bogoyavlenski claimed that one obtains a Poisson commutative subalgebra of the symmetric algebra

m a t h c a l S (m a t h f r a k g)

by taking the subalgebra

m a t h c a l Z

generated by the bi-homogeneous components of all

H i n m a t h c a l S (m a t h f r a k g)^{m a t h f r a k g}

. But this is false, and we present a counterexample. We also provide a criterion for the Poisson commutativity of such subalgebras

m a t h c a l Z

. As a by-product, we prove that

m a t h c a l Z

is Poisson commutative if

m a t h f r a k h

is abelian and describe

m a t h c a l Z

in the special case when

m a t h f r a k h

is a Cartan subalgebra. In this case,

m a t h c a l Z

appears to be polynomial and has the maximal transcendence degree

(m a t h r m d i m, m a t h f r a k g + m a t h r m r k, m a t h f r a k g) / 2

.

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