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Left-Invariant Affine Structures on Reductive Lie Groups

✍ Scribed by Dietrich Burde

Publisher: Elsevier Science
Year: 1996
Tongue: English
Weight: 230 KB
Volume: 181
Category: Article
ISSN: 0021-8693
DOI: 10.1006/jabr.1996.0151

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✦ Synopsis

We describe left-invariant affine structures that is, left-invariant flat torsion-free . affine connections ٌ on reductive linear Lie groups G. They correspond bijectively to LSA-structures on the Lie algebra ᒄ of G. Here LSA stands for left-symmetric algebra. If ᒄ has trivial or one-dimensional center ᒗ then the affine representation ␣ s [ 1 of ᒄ, induced by any LSA-structure ᒄ on ᒄ is radiant, i.e., the

Here we have the associative LSA-structure given by ordinary matrix n Ž . multiplication corresponding to the bi-invariant affine structure on GL n , which was believed to be essentially the only possible LSA-structure on ᒄ ᒉ . We exhibit n interesting LSA-structures different from the associative one. They arise as certain deformations of the matrix algebra. Then we classify all LSA-structures on ᒄ ᒉ n using a result of Baues. For n s 2 we compute all structures explicitly over the complex numbers.

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