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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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