The Index and Spectrum of Lie Poset Algebras of Types B, C, and D

Abstract

We define posets of types B, C, and D. These posets encode the matrix forms of certain Lie algebras which lie between the algebras of upper-triangular and diagonal matrices. Our primary concern is the index and spectral theories of such type-B, C, and D Lie poset algebras. For an important restricted class, we develop combinatorial index formulas and, in particular, characterize posets corresponding to Frobenius Lie algebras. In this latter case we show that the spectrum is binary; that is, consists of an equal number of 0's and 1's. Interestingly, type-B, C, and D Lie poset algebras can be related to Reiner's notion of a parset.