An example of a braided locally compact group
نویسنده
چکیده
The notions of the crossed product and the braided category are discussed within the theory of C∗-algebras. Like in the purely algebraic approach of S. Majid these notions are used to generalize the quantum group theory. We give an example of a braided group. It coincides with the closed subset C = {z : |z| = . . . , q−2, q−1, 1, q, q2, . . . , 0} of the complex plane equipped with the q2braiding: z2z1 = qz1z2, z2z1 = qz1z2. The deformation parameter 0 < q < 1. C is closed under braided addition. The braided group C is selfdual and the universal bicharacter describing the selfduality is found. 0 Introduction In the usual notion of the tensor product A⊗B of two algebras A and B one assumes that the copies A and B in A⊗B do commute: for any a ∈ A and b ∈ B, j2(b)j1(a) = j1(a)j2(b), where j1(a) = a⊗IB and j2(b) = IA⊗b. Replacing this simple low by a more complicated one we arrive to the notion of crossed product of algebras. For instance, considering Z2-graded algebras and introducing in the right hand side of the above equation ± sign depending on the grades of homogeneous elements a and b we arrive to the notion of supersymmetrical tensor product. S. Majid proposed to use the rather general commutation rule: j2(b)j1(a) = ∑ j1(ak)j2(bk), where ∑ ak⊗ bk = R(b⊗ a) and the mapping R : B⊗A −→ A⊗B is given in advance. Considering a collection of algebras equipped with the mappings R he arrived to the notion of braided category (to have the associativity of the crossed product one has to assume that the mappings R satisfy the braid equation). Next S. Majid used this ∗Supported by Komitet Badań Naukowych, grant No 2 P301 020 07 and by the Commission of European Communities Contract No. CIPA – CT92 – 3006.
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