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Quantum symmetry groups of finite spaces. (English) Zbl 1013.17008
The goal is to introduce and describe the quantum automorphism groups of finite spaces (i.e., finite dimensional \(C^\ast\)-algebras). This is done by a categorical method. For a \(C^\ast\)-algebra \(B\) first the category of left quantum transformation groups of \(B\) is introduced. The objects of the category are pairs \((A,\alpha)\) where \(A\) is a compact quantum group and \(\alpha:B\to B\otimes A\) is a right coaction satisfying some natural axioms. For a functional \(\phi\) on \(B\) one also introduces the category of quantum transformation groups of the pair \((B,\phi)\). The quantum automorphism group of \(B\) is defined as the universal final object (if it exists) in the category of quantum transformation groups of \(B\). Similarly one defines the quantum automorphism group of the pair \((B,\phi)\). These definitions are then applied to two types of spaces.
Firstly, \(B=X_n\) is the \(C^\ast\)-algebra of continuous functions on a finite set containing \(n\) elements. In this case the quantum automorphism group of \(B\) is proven to exist and it is described explicitly.
The second type of space considered is the \(C^\ast\)-algebra \(B=M_n(\mathbb C)\) formed by \(n\times n\) matrices. In this case the author shows the existence of the automorphism group of the pair \((B,Tr)\) where \(Tr\) is the trace functional.
The two cases are then combined to a general one when \(B=\bigoplus_{k=1}^mM_{n_k}(\mathbb C)\). According to the main result of the paper the quantum automorphism group of \(B\) exists if and only if \(B\) is the finite space \(X_m\). On the other hand, there exists the quantum automorphism group of the pair \((B,\psi)\) where \(\psi\) is a functional induced by the traces in each summand.
For earlier works see Yu. I. Manin, Quantum groups and noncommutative geometry, Univ. Montreal (1988; Zbl 0724.17006), S. Wang, Commun. Math. Phys. 167, 671-692 (1995; Zbl 0838.46057) and 178, 747-764 (1996; Zbl 0876.17021), and A. Van Daele and S. Wang, Int. J. Math. 7, 255-263 (1996; Zbl 0870.17011).

MSC:
17B37 Quantum groups (quantized enveloping algebras) and related deformations
58B32 Geometry of quantum groups
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
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