1998/05/29 by Axel Schueler, A. Schueler, Schueler, Axel
Mathematics · Physics and Astronomy · #17B37 #58B30 #81R50 #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Algebra (math.QA) #math-ph #math.MP #math.QA #msc:17B37 #msc:58B30 #msc:81R50
paper · pdf · doi:10.48550/arxiv.math/9805139
30 pages
arxiv created 1998/05/29 · openalex publication_date 1998/05/29 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let A be a Hopf algebra and Gamma be a bicovariant first order differential calculus over A. It is known that there are three possibilities to construct a differential Hopf algebra Gammawedge that contains Gamma as its first order part; namely the universal exterior algebra, the second antisymmetrizer exterior algebra, and Woronowicz' external algebra. Now let A be one of the quantum groups GLq(N) or SLq(N). Let Gamma be one of the N2-dimensional bicovariant first order differential calculi over A and let q be transcendental. For Woronowicz' external algebra we determine the dimension of the space of left-invariant and of bi-invariant k-forms. Bi-invariant forms are closed and represent different de Rham cohomology classes. The algebra of bi-invariant forms is graded anti-commutative. For N>2 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.