New Simple Infinite-Dimensional Lie Algebras of Characteristic 0

✍ Scribed by J.Marshall Osborn

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 198 KB
• Volume: 185
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1996.0352

✦ Synopsis

We construct here several classes of simple Lie algebras of characteristic Ž . 0 which include the Virasoro algebra without central charge and the graded Lie algebras of Cartan type. Our construction is motivated by our w x recent construction of simple locally Novikov algebras in 5 .

Our simple Lie algebras occur as subalgebras of certain algebras which w x have been defined by Kawamoto 4 . We give a construction of these algebras in Section 2, and we also define there certain subalgebras. These Ž . algebras include the algebras W n of Cartan type as well as the Virasoro algebra. The class of all the algebras defined in Section 2 we denote by W U . Occurring as subalgebras of the algebras of this class are three classes that we call S U , H U , and K U , since they contain respectively the algebras of Cartan type S, H, and K. These three classes are discussed respectively in Sections 3, 4, 5. In the case of the class S U we find some new simple algebras which have no analogue in the usual theory of Lie algebras of Cartan type. Finally we end with a few remarks in Section 6 looking toward a structure theory for simple infinite-dimensional Lie algebras.

2. LIE ALGEBRAS OF TYPE W U

The algebras of Kawamoto that we define here are less general in two ways than the class that Kawamoto defines. First, we shall restrict ourselves to simple algebras, and second, we shall suppose that the torus is finite dimensional. The latter assumption is not essential for everything 820