Classification of bicovariant differential calculi on quantum groups of type A,B, C and D |
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Authors: | Konrad Schmüdgen Axel Schüler |
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Institution: | (1) Fachbereich Mathematik/Informatik, Universität Leipzig, Augustusplatz 10, D-04109 Leipzig, Germany |
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Abstract: | Under the assumptions thatq is not a root of unity and that the differentialsdu
j
i
of the matrix entries span the left module of first order forms, we classify bicovariant differential calculi on quantum groupsA
n–1
,B
n
,C
n
andD
n
. We prove that apart one dimensional differential calculi and from finitely many values ofq, there are precisely2n such calculi on the quantum groupA
n–1
=SL
q
(n) forn 3. All these calculi have the dimensionn
2. For the quantum groupsB
n
,C
n
andD
n
we show that except for finitely manyq there exist precisely twoN
2-dimensional bicovariant calculi forN 3, whereN=2n+1 forB
n
andN=2n forC
n
,D
n
. The structure of these calculi is explicitly described and the corresponding ad-invariant right ideals of ker are determined. In the limitq 1 two of the 2n calculi forA
n–1
and one of the two calculi forB
n
,C
n
andD
n
contain the ordinary classical differential calculus on the corresponding Lie group as a quotient. |
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Keywords: | |
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