Notes on quantum groups and quantum de Rham complexes
An approach initiated earlier by the same author [Ann. Inst. Fourier 37, No. 4, 191-205 (1987; Zbl 0648.16005)] (and also, as acknowledged in the present paper, suggested independently and simultaneously by \textit{G. Maltsiniotis} [C. R. Acad. Sci., Paris, Sér. I 311, 831-834 (1990; Zbl 0728.17010)]) treats (``algebraic'') quantum groups as ``groups of quantum isomorphisms'' of quantum linear spaces, through a universal coaction of the corresponding Hopf algebras on the deformed algebras of ``polynomial functions'' on quantum spaces. (That is, actions of quantum groups on quantum spaces.) In the author's notes ``Quantum groups and non- commutative geometry'' (CRM; Université de Montréal, 1988; Zbl 0724.17006)] this approach was elaborated in great detail for the quantum general linear group \(GL_ q(2)\) and then extended to a convenient general setting of quadratic algebras; further generalizations, along with some corrections of the initial approach, are to be found in the author's book ``Topics in noncommutative geometry'' (Princeton 1991; Zbl 0724.17007). However, in order to obtain a complete set of relations governing the quantum group \(GL_ q (2)\) (as well as more complicated ones), one has to consider simultaneously two actions of this quantum group: both on the ``bosonic'' and ``fermionic'' quantum planes, \(A_ q^{2| 0}\) and \(A_ q^{0 | 2}\), each of them separately producing only half of the relations needed. This strange peculiarity of the quantum situation (why should one invoke superspace in a quantum, yet purely bosonic, environment?) was explained by \textit{J. Wess} and \textit{B. Zumino} [Covariant differential calculus on the quantum hyperplane, Nucl. Phys. Proc. Suppl. B 18, 302-312 (1990)] who incorporated both quantum planes into a single new object, the quantum de Rham complex \(\Omega[A_ q^{2 | 0}]\) of the bosonic quantum plane \(A_ q^{2 | 0}\). As a vector space, \(\Omega[A_ q^{2| 0}]\) is obtained by tensoring the noncommutative polynomial algebras \(A_ q^{2 | 0}\) and \(A_ q^{0 | 2}\), with some additional cross-commutation relations imposed. From this viewpoint, the quantum group \(GL_ q (2)\) becomes the quantum automorphism group of the complex \(\Omega [A_ q^{2| 0}]\). In the present paper, the Wess-Zumino approach is extended to and fully merged into the general setting of quadratic algebras. In particular, the origin of the defining cross-commutation relations for \(\Omega [A_ q^{2| 0}]\) is explained in terms of a solution to a certain universal problem. (In this way, even the uniqueness of the relations is proved.) The relationship between the quantum and the classical de Rham complexes is studied, and cyclic cohomology of the emerging complexes is calculated in simplest cases. What is especially remarkable, is that this new insight into quantum groups leads to a substantial overlap and possible interplay with noncommutative differential geometry.
scientific article; zbMATH DE number 589549
scientific article