The index and spectrum of Lie poset algebras of type B, C, and D

Abstract

In this paper, 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. Interestingly, such type-B, C, and D Lie poset algebras can be related to Reiner's notion of a parset. Our primary concern is the index and spectral theories of 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.

Classification of Frobenius, two-step solvable Lie poset algebras

“…The original motivation for this article was to understand if the type-A cohomological result of equation ( 1) could be succinctly extended to the other classical types -this owing to the recent introduction of the definitions of Lie poset algebras in types B, C, and D (see [3] and Definition 4 below).…”

“…Following the suggestion of [1], Coll and Mayers [3] extend the notion of Lie poset algebra to the other classical families: B n = so(2n + 1), C n = sp(2n), and D n = so(2n), by providing the definitions of posets of types B, C, and D which encode the standard matrix forms of Lie poset algebras in types B, C, and D, respectively. There are no conditions on the "generating" posets in either gl(n) or in sl(n).…”

The evolution of the spectrum of a Frobenius Lie algebra under deformation

“…However, we have no examples of deformable Frobenius Lie poset algebras. It is also worth noting that the unbroken spectrum property seems to be a property of Frobenius Lie poset algebras, although the spectrum is "binary", consisting of only 0's and 1's (see [8], [9], [10], and [11]).…”