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SU(n)-connections and noncommutative differential geometry

✍ Scribed by Michel Dubois-Violette; Thierry Masson

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
912 KB
Volume
25
Category
Article
ISSN
0393-0440

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✦ Synopsis

We study the noncommutative differential geometry of the algebra of endomorphisms of any SU (n)-vector bundle. We show that ordinary connections on such SU (n)-vector bundles can be interpreted in a natural way as a noncommutative 1 -form on this algebra for the differential calculus based on derivations. We interpret the Lie algebra of derivations of the algebra of endomorphisms as a Lie algebroid. Then we look at noncommutative connections as generalizations of these usual connections.

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In the early days [ 1,2] a was taken to be A itself. Later [9, Chap. 31 examples where Z? formed a Lie algebra, or some other algebraic relationship such as [p, X] = 1 as in quantum mechanics, or xy = qyx as in q-deformed algebras, were studied. For each subspace B one could construct a co-frame. T