Matrix pseudogroups associated with anti-commutative plane. (English) Zbl 0725.17016
The author realizes various algebras of functions on the \(2\times 2\) matrix groups for the deformation parameter \(q=-1\) in terms of \(C^*\)-algebras of functions, subalgebras of all continuous and vanishing at infinity maps of the corresponding classical group to a \(C^*\)-algebra. In particular, the algebra \(\mathcal H\) of polynomials on \(M_{-1}(2,\mathbb{C})\) is represented by a subalgebra of the algebra \(\mathrm{Pol}(M,M)\) of all polynomial maps of \(M = M(2,\mathbb{C})\) into itself, and the polynomial algebra \(\mathcal K\) of the anti-commutative plane (realized by a subalgebra of \(\mathrm{Pol}(C^2,M))\) is a subalgebra of \(\mathcal H\).
Reviewer: Jan Chrastina (Brno)
MSC:
46L89 | Other “noncommutative” mathematics based on \(C^*\)-algebra theory |
16T20 | Ring-theoretic aspects of quantum groups |
57T05 | Hopf algebras (aspects of homology and homotopy of topological groups) |
58B32 | Geometry of quantum groups |
20G42 | Quantum groups (quantized function algebras) and their representations |
References:
[1] | Woronowicz, S. L., Unbounded elements affiliated with C *-algebras and non-compact quantum groups, to appear in Comm. Math. Phys. · Zbl 0743.46080 |
[2] | Zakrzewski, S., Quantum and classical pseudogroups, to appear in Comm. Math. Phys. |
[3] | WoronowiczS. L., Pseudospaces, pseudogroups and Pontryagin duality, Proceedings of the International Conference on Mathematics and Physics, Lausanne 1979, Lecture Notes in Physics 116, Springer-Verlag, New York. |
[4] | Podle?P. and WoronowiczS. L., Quantum deformation of Lorentz group, Comm. Math. Phys. 130, 381-431 (1990). · Zbl 0703.22018 · doi:10.1007/BF02473358 |
[5] | WoronowiczS. L., Differential calculus on compact matrix pseudogroups (quantum groups), Comm. Math. Phys. 122, 125-170 (1989). · Zbl 0751.58042 · doi:10.1007/BF01221411 |
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