Classification of Bicovariant Differential Calculi on the Quantum Groups $SL_q(n+1)$ and $Sp_q(2n)$

Details Classification of Bicovariant Differential Calculi on the Quantum Groups $SL_q(n+1)$ and $Sp_q(2n)$ Classification of Bicovariant Differential Calculi on the Quantum Groups $SL_q(n+1)$ and $Sp_q(2n)$

1997-07-25

arxiv.org/abs/q-alg/9707032v2

Mathematics

Quantum Algebra

21 pages, LaTeX2e, uses amstex.sty; extended version. To appear in J. reine angew. Math. (Crelle's Journal)

Scientific paper

For transcendental values of $q$ all bicovariant first order differential calculi on the coordinate Hopf algebras of the quantum groups $SL_q(n+1)$ and $Sp_q(2n)$ are classified. It is shown that the irreducible bicovariant first order calculi are determined by an irreducible corepresentation of the quantum group and a complex number $\zeta$ such that $\zeta^{n+1}=1$ for $SL_q(n+1)$ and $\zeta^2=1$ for $Sp_q(2n)$. Any bicovariant calculus is inner and its quantum Lie algebra is generated by a central element. The main technical ingredient is a result of the Hopf algebra $R(G_q)^0$ for arbitrary simple Lie algebras.

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