Poisson-commutative subalgebras of S(\mathfrak g) associated with involutions

2018/09/02 by Panyushev, Dmitri, Yakimova, Oksana
#14L30 #17B08 #17B20 #17B63 #22E46 #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1809.00350

Abstract

The symmetric algebra S(\mathfrak g) of a reductive Lie algebra \mathfrak g is equipped with the standard Poisson structure, i.e., the Lie-Poisson bracket. Poisson-commutative subalgebras of S(\mathfrak g) attract a great deal of attention, because of their relationship to integrable systems and, more recently, to geometric representation theory. The transcendence degree of a Poisson-commutative subalgebra \mathcal C⊂ S(\mathfrak g) is bounded by the "magic number" \boldsymbolb(\mathfrak g) of \mathfrak g. The "argument shift method" of Mishchenko-Fomenko was basically the only known source of \mathcal C with \rm trdeg \mathcal C=\boldsymbolb(\mathfrak g). We introduce an essentially different construction related to symmetric decompositions \mathfrak g=\mathfrak g₀⊕\mathfrak g₁. Poisson-commutative subalgebras \mathcal Z,\mathcal Z⊂ S(\mathfrak g)^{\mathfrak g₀} of the maximal possible transcendence degree are presented. If the \mathbb Z₂-contraction \mathfrak g₀\ltimes\mathfrak g₁^{\sf ab} has a polynomial ring of symmetric invariants, then \mathcal Z is a polynomial maximal Poisson-commutative subalgebra of S(\mathfrak g)^{\mathfrak g₀}, and its free generators are explicitly described.

Related

• Reductive subalgebras of semisimple Lie algebras and Poisson commutativity
• Shifts of semi-invariants and complete commutative subalgebras in polynomial Poisson algebras
• Periodic automorphisms, compatible Poisson brackets, and Gaudin\n subalgebras
• Generalized argument shift method and complete commutative subalgebras in polynomial Poisson algebras
• The argument shift method and maximal commutative subalgebras of Poisson algebras