A Domain Test for Lie Color Algebras

Abstract

Lie color algebras are generalizations of Lie superalgebras and graded Lie algebras. The properties of a Lie color algebra can often be related directly to the ring structure of its universal enveloping algebra. We study the effects of torsion elements and torsion subspaces. Let [Formula: see text] denote a Lie color algebra. If [Formula: see text] is homogeneous and torsion then x2 = 0 in [Formula: see text]. If no homogeneous element of [Formula: see text] is torsion, then [Formula: see text] so [Formula: see text] is semiprime. In this case we can give a test which uses Gröbner basis methods to determine when [Formula: see text] is a domain. This is applied in an example to show [Formula: see text] may be a domain even if [Formula: see text] contains torsion elements and torsion subspaces.

Gradings and symmetries on Heisenberg type algebras

“…Lie color algebras were introduced in [36] as a generalization of Lie superalgebras and hence of Lie algebras. This kind of algebras has attracted the interest of several authors in the last years, (see [10,33,35,43,44]), being also remarkable the important role they play in theoretical physic, specially in conformal field theory and supersymmetries [39,42].…”

Poisson color algebras of arbitrary degree

“…On the other hand, we also recall that Lie color algebras were introduced in [23] as a generalization of Lie superalgebras and hence of Lie algebras. Since then, this kind of algebras has been an object of constant interest in mathematics, (see [21,22,32,33,34] for recent references), being also valuable the important role they play in theoretical physics, especially in conformal field theory and supersymmetries ([4, 16, 27, 30]). Definition 2.…”

On the structure of semi-invariant polynomials in Ore extensions

“…We also write ad (1,σ ) (b) as ad σ (b). 2465 Let S be an automorphism of R and D an S-derivation of R. The Ore extension of R by the S-derivation D, denoted by R[ X; S, D], is the ring freely generated by the ring R adjoined by an indeterminate X subjected to the commuting rule Xr = S(r)X + D(r) for r ∈ R.Ore extensions, since their discovery in [20], have been very important in constructing interesting mathematical objects, see for example [1,2,[4][5][6][7][8][9][10][11][12][13][15][16][17][18][19][21][22][23][24][25]. In modern algebras, this elementary construction is important in particular due to the fact that the quantum Borel subalgebra in DrinfeldJimbo quantization is an iterated Ore extension, see for example [4].…”