Algebra & Number Theory
- Algebra Number Theory
- Volume 7, Number 6 (2013), 1505-1534.
Commuting involutions of Lie algebras, commuting varieties, and simple Jordan algebras
Let and be commuting involutions of a connected reductive algebraic group with . Let
be the corresponding -grading. If , then maps into , and the zero fiber of this bracket is called a -commuting variety. The commuting variety of and commuting varieties related to one involution are particular cases of this construction. We develop a general theory of such varieties and point out some cases, when they have especially good properties. If is a Hermitian symmetric space of tube type, then one can find three conjugate pairwise commuting involutions , , and . In this case, any -commuting variety is isomorphic to the commuting variety of the simple Jordan algebra associated with . As an application, we show that if is the Jordan algebra of symmetric matrices, then the product map is equidimensional, while for all other simple Jordan algebras equidimensionality fails.
Algebra Number Theory, Volume 7, Number 6 (2013), 1505-1534.
Received: 19 September 2012
Accepted: 24 January 2013
First available in Project Euclid: 20 December 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 14L30: Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17]
Secondary: 17B08: Coadjoint orbits; nilpotent varieties 17B40: Automorphisms, derivations, other operators 17C20: Simple, semisimple algebras 22E46: Semisimple Lie groups and their representations
Panyushev, Dmitri I. Commuting involutions of Lie algebras, commuting varieties, and simple Jordan algebras. Algebra Number Theory 7 (2013), no. 6, 1505--1534. doi:10.2140/ant.2013.7.1505. https://projecteuclid.org/euclid.ant/1513730036