Covariant differential complexes on quantum linear groups

Abstract

We consider the possible covariant external algebra structures for Cartan's 1-forms on GL_q(N) and SL_q(N). We base upon the following natural postulates: 1. the invariant 1-forms realize an adjoint representation of quantum group; 2. all monomials of these forms possess the unique ordering. For the obtained external algebras we define the exterior derivative possessing the usual nilpotence condition, and the generally deformed version of Leibniz rules. The status of the known examples of GL_q(N)-differential calculi in the proposed classification scheme, and the problems of SL_q(N)-reduction are discussed.

23 pages

References in corpus (5)

• Differential Geometry on Linear Quantum Groups
• Bicovariant Quantum Algebras and Quantum Lie Algebras
• Deformed Traces and Covariant Quantum Algebras for Quantum Groups $G{L}_{qp}(2)$ and $G{L}_{qp}(1|1)$
• GLq(N)-Covariant Quantum Algebras and Covariant Differential Calculus
• Quantum group covariant noncommutative geometry

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• On construction of unitary quantum group differential calculus
• Maxwell operator on q-Minkowski space and q-hyperboloid algebras