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.

Classification of contact seaweeds

“…If ϕ(h) = 0, construct a Cartan subalgebra h ⊂ s that contains h, and let B h be a Cartan-Weyl basis for s generated from h (see Coll et al [9], Lemma 55). Since, for all x ∈ s, dim(ker(B ϕ )) is upper semicontinuous with respect to ϕ(x), there exists ε > 0 and a one-form ψ ∈ s * such that if ψ(x) = ϕ(x) for all x ∈ B h \ {h} and |ψ(h)| < 2ε, then ind g = 1 ≤ dim ker(B ψ ) ≤ dim ker(B ϕ ) = 1;…”

“…Therefore, if x ϕ is a semisimple element of P, then ϕ is a form of reductive type. Such an approach is reliant on a bevy of contact Lie proset algebras; however, such examples are difficult to come by, mainly due to lack of a comprehensive computational index theory which specializes to the (rather complete) index theory of seaweeds ( [5], [12], [19]) and the nascent (though substantive) index theory of Lie poset algebras ( [7], [9]).…”