Differential Calculi on Some Quantum Prehomogeneous Vector Spaces

Details Differential Calculi on Some Quantum Prehomogeneous Vector Spaces Differential Calculi on Some Quantum Prehomogeneous Vector Spaces

2006-06-20

arxiv.org/abs/math/0606499v3

Mathematics

Quantum Algebra

LaTeX2e, 51 pages; changed content

Scientific paper

10.1063/1.2746131

This paper is devoted to study of differential calculi over quadratic algebras, which arise in the theory of quantum bounded symmetric domains. We prove that in the quantum case dimensions of the homogeneous components of the graded vector spaces of k-forms are the same as in the classical case. This result is well-known for quantum matrices. The quadratic algebras, which we consider in the present paper, are q-analogues of the polynomial algebras on prehomogeneous vector spaces of commutative parabolic type. This enables us to prove that the de Rham complex is isomorphic to the dual of a quantum analogue of the generalized Bernstein-Gelfand-Gelfand resolution.

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