Realization of graded-simple algebras as loop algebras

By B. Allison, S. Berman, J. Faulkner, A. Pianzola. 2005

72 citations recorded by the source corpus.

Abstract Multiloop algebras determined by n commuting algebra automorphisms of finite order are natural generalizations of the classical loop algebras that are used to realize affine Kac-Moody Lie algebras. In this paper, we obtain necessary and sufficient conditions for a ℤ n -graded algebra to be realized as a multiloop algebra based on a finite dimensional simple algebra over an algebraically closed field of characteristic 0. We also obtain necessary and sufficient conditions for two such multiloop algebras to be graded-isomorphic, up to automorphism of the grading group. We prove these facts as consequences of corresponding results for a generalization of the multiloop construction. This more general setting allows us to work naturally and conveniently with arbitrary grading groups and arbitrary base fields. 2000 Mathematics Subject Classification: 16W50, 17B70; 17B65, 17B67.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/a3f4a9b883c0d1a23447636e67541e33e683f09e

How researchers see this paper

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In Algebra and Number Theory

1. Michael Lau
2. A. Pianzola
3. S. Berman
4. J. Faulkner
5. B. Allison
6. P. Gille
7. Yōji Yoshii
8. E. Neher

In Computational Theory and Mathematics

1. Michael Lau
2. A. Pianzola

In Geometry and Topology

1. Michael Lau
2. A. Pianzola
3. S. Berman
4. J. Faulkner
5. B. Allison
6. P. Gille
7. Yōji Yoshii
8. E. Neher

Citations in the corpus

• Multiloop realization of extended affine Lie algebras and Lie tori (2007, methodology): “We rst recall the denitions and the results that we will need from [ ABFP ] regarding multiloop algebras in general.”
• Representations of Multiloop Algebras (2008, result): “It has recently been shown that the centreless core of almost every extended affine Lie algebra is a multiloop algebra [Allison et al. 2009], using results of [Allison et al. 2008; Neher 2004].”
• Isotriviality and étale cohomology of Laurent polynomial rings (2008, background): “This is a very interesting class of algebras with intricate relations to the centreless cores of Extended Affine Lie Algebras (see [ 2 ] and [19] for details and further references).”
• Extended Affine Lie Algebras and Other Generalizations of Affine Lie Algebras – A Survey (2008, unknown): “For the description of slI(A) as a multiloop algebra ([ ABFP1 ]) it is important to know when is slI(A) finitely generated over its centroid.”