Notes on quantum groups and quantum de Rham complexes
type: Item
instance of: scholarly article
title: Notes on quantum groups and quantum de Rham complexes
zbMATH Open document ID: 0810.17003
DOI: 10.1007/BF01017077
published in: Theoretical and Mathematical Physics
publication date: 1994-07-13T00:00:00Z
review text: 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 polynomia…
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 46L85
Mathematics Subject Classification ID: 46L87
zbMATH DE Number: 589549
zbMATH Keywords: universal differential graded algebra
zbMATH Keywords: skew products
zbMATH Keywords: quantum groups
zbMATH Keywords: quadratic algebras
zbMATH Keywords: quantum planes
zbMATH Keywords: quantum de Rham complex
zbMATH Keywords: cyclic cohomology
author: Yuri I. Manin
reviewed by: Vladimir G. Pestov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 589549