Differential Hopf Algebras on Quantum Groups of Type A
نویسنده
چکیده
Let A be a Hopf algebra and Γ be a bicovariant first order differential calculus over A. It is known that there are three possibilities to construct a differential Hopf algebra Γ = Γ/J that contains Γ as its first order part. Corresponding to the three choices of the ideal J , we distinguish the ‘universal’ exterior algebra, the ‘second antisymmetrizer’ exterior algebra, and Woronowicz’ external algebra, respectively. Let Γ be one of the N-dimensional bicovariant first order differential calculi on the quantum group GLq(N) or SLq(N), and let q be a transcendental complex number. For Woronowicz’ external algebra we determine the dimension of the space of left-invariant and of bi-invariant k-forms, respectively. Bi-invariant forms are closed and represent different de Rham cohomology classes. The algebra of bi-invariant forms is graded anti-commutative. For N ≥ 3 the three differential Hopf algebras coincide. However, in case of the 4D±-calculi on SLq(2) the universal differential Hopf algebra is strictly larger than Woronowicz’ external algebra. The bi-invariant 1-form is not closed.
منابع مشابه
Reiter’s Properties for the Actions of Locally Compact Quantum Goups on von Neumann Algebras
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