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Quantum supergroups of \(\text{GL}(n| m)\) type: Differential forms, Koszul complexes, and Berezinians. (English) Zbl 0905.16018

The authors study differential Hopf algebras generated by sets of matrix elements, generalizing the construction of a bialgebra generated by a single set of matrix elements (without a differential) as studied by M. Takeuchi [Isr. J. Math. 72, No. 1/2, 232-251 (1990; Zbl 0723.17013)] and the second author [J. Algebra 158, No. 2, 375-399 (1993; Zbl 0791.17015)]. The generalization under review is a universally coacting bialgebra preserving several algebras generated by a set of coordinates (as in the articles cited above), but where the data are morphisms in the category of graded differential complexes. A Hecke matrix is defined in this graded differential setting and used to construct a quantum supergroup whose algebra of functions is a \(Z\)-graded differential coquasitriangular Hopf algebra. A differential Hopf algebra quotient of \(H\) involves differential forms. Koszul complexes for Hecke matrices are introduced, which involve two differentials, whose anticocommutator is called the Laplacian. The cohomology space of one of the differentials is called the Berizinian, generalizing the determinant. The authors prove that the Berizinian of a Hecke sum is the tensor product of the Berizinians. The Berizinian is used to define the quantum superdeterminant, and this is calculated in several examples. In particular, in the algebra \(\Omega\) of differential forms on the standard quantum group \(\text{GL}(n| m)\), the superdeterminant equals 1, confirming that \(\Omega\) has no central group-like elements, which explains why it has not been possible to construct a differential calculus on special linear groups with the same dimension as the classical case. Finally the authors examine the Woronowicz differential calculus on quantum groups [S. L. Woronowicz, Commun. Math. Phys. 122, No. 1, 125-170 (1989; Zbl 0751.58042)]. In particular, they show that each first-order differential calculus can be extended to a differential Hopf algebra.

MSC:

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
17B37 Quantum groups (quantized enveloping algebras) and related deformations
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
17C70 Super structures
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
46L85 Noncommutative topology
46L87 Noncommutative differential geometry
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