Homological Properties of Color Lie Superalgebras

Abstract

Let L = L + ⊕ L − be a finite dimensional color Lie superalgebra over a field of characteristic 0 with universal enveloping algebra U (L). We show that gldim(U (L + )) = lFPD(U (L)) = rFPD(U (L)) = injdim U (L) (U (L)) = dim(L + ).We also prove that U (L) is Auslander-Gorenstein and Cohen-Macaulay and thus that it has a QF classical quotient ring.

Generic Lie colour algebras

“…Conditions for U (L) to be semiprime or prime are provided in [13, Thoerem 2.5], which extends the analogous result, [4, Theorem 1.5], for Lie superalgebras. When U (L) has finite global dimension, then it must equal dim L + by [12,Theorem 3.1], which extends the analogous result, [9, Proposition 2.3], for Lie superalgebras.…”

Cohomology of 3-dimensional color Lie algebras

“…The (co)homology theory for color Lie algebras has been introduced in [44] (see also [43] and [31]). The theory becomes analogous to the cohomology theory of the usual Lie algebras [12] and Lie superalgebras, i.e.…”

“…Since the 1970s, generalized (color) Lie algebras have been an object of constant interest in both mathematics and physics [1,2,7,8,11,13,[15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][37][38][39][40][41][42][43][44][45][46][47][48][49][50]. The main goal of the paper is to describe the cohomology with scalar coefficients of three-dimensional (three generators) color Lie algebras.…”