Simple Lie Color Algebras of Witt Type

✍ Scribed by D.S Passman

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 305 KB
• Volume: 208
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1998.7527

✦ Synopsis

Let K be a field and let : ⌫ = ⌫ ª K ⅷ be a bicharacter defined on the multiplicative group ⌫. We suppose that A is a ⌫-graded, associative K-algebra that is color commutative with respect to . Furthermore, let ⌬ be a nonzero ⌫-graded, K-vector space of color derivations of A and suppose that ⌬ is also color commutative with respect to the bicharacter . Then, with a rather natural definition, A m ⌬ s A⌬ becomes a Lie color algebra, and we obtain necessary K and sufficient conditions here for this Lie color algebra to be simple. With two minor exceptions when dim ⌬ s 1, simplicity occurs if and only if A is graded K ⌬-simple and A ⌬ m ⌬ s A ⌬ ⌬ acts faithfully as color derivations on A. ᮊ 1998 Academic Press 1. CONSTRUCTION OF THE LIE ALGEBRA w x

In a recent paper Pa , we took a ring theoretic approach to the construction of simple Lie algebras of Witt type. Here, we use similar methods to show that analogous results hold in the context of Lie color algebras. Since the color aspects of this construction may be new, we will include all definitions and carefully verify all required identities.

To start with, let K be a field and let ⌫ be a multiplicative abelian group. Recall that a bicharacter : ⌫ = ⌫ ª K ⅷ is a map that satisfies

x, y y, x s 1 for all x, y g ⌫ E1

Ž

. Ž . Ž .

x, yz s x, y x, z for all x, y, z g ⌫.