Quantum deformation of classical groups. (English) Zbl 0806.16040
Let \(A(G)\) be the coordinate algebra of a Lie group \(G\). A quantum deformation \(A(G_ q)\) of \(A(G)\) is a one-parameter family of Hopf algebras whose representation theories are the same as those of \(A(G)\). The author constructs the quantum deformations of \(A(O(N))\), \(A(SO(N))\) and \(A(Sp(N))\) by using Jimbo’s solutions of the Yang-Baxter equation and determines their Peter-Weyl decompositions. For this purpose, he investigates the so-called quantum matrix bialgebras, and defines Hopf algebras \(A(G_ q)\) as quotients of corresponding quantum matrix bialgebras. Also, he studies group-like elements of these quantum matrix bialgebras and gives a new realization of the universal \(R\)-matrix.
Reviewer: Ye Jiachen (Shanghai)
MSC:
| 16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 16S80 | Deformations of associative rings |
| 17B60 | Lie (super)algebras associated with other structures (associative, Jordan, etc.) |
| 22E70 | Applications of Lie groups to the sciences; explicit representations |
| 81R05 | Finite-dimensional groups and algebras motivated by physics and their representations |
| 81U20 | \(S\)-matrix theory, etc. in quantum theory |
Keywords:
coordinate algebra; Lie group; quantum deformation; one-parameter family of Hopf algebras; Yang-Baxter equation; Peter-Weyl decompositions; quantum matrix bialgebras; group-like elements; universal \(R\)-matrixReferences:
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