Discrete Cocompact Subgroups of the Four-Dimensional Nilpotent Connected Lie Group and Their Group C*-Algebras
✍ Scribed by Paul Milnes; Samuel Walters
- Publisher
- Elsevier Science
- Year
- 2001
- Tongue
- English
- Weight
- 153 KB
- Volume
- 253
- Category
- Article
- ISSN
- 0022-247X
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✦ Synopsis
Let G 4 be the unique, connected, simply connected, four-dimensional, nilpotent Lie group. In this paper, the discrete cocompact subgroups H of G 4 are classified and shown to be in 1-1 correspondence with triples p 1 p 2 p 3 ∈ 3 that satisfy p 2 p 3 > 0 and a certain restriction on p 1 . The K-groups of the group C * -algebra C * H are computed and shown to involve all three parameters. Furthermore, for each such subgroup H, the set of faithful simple quotients (i.e., those generated by a faithful representation of H) of the group C * -algebra C * H is shown to be independent of p 1 and p 3 and to be in 1-1 correspondence with the irrational θ's in 0 1/2 . The other infinite-dimensional simple quotients of C * H (those generated by a representation of H that is not faithful) are shown to be isomorphic to matrix algebras over irrational rotation algebras.