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The universal \(T\)-matrix. (English) Zbl 0822.17013
Sally, Paul J. jun. (ed.) et al., Mathematical aspects of conformal and topological field theories and quantum groups. AMS-IMS-SIAM summer research conference, June 13-19, 1992, Mount Holyoke College, South Hadley, MA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 175, 73-88 (1994).
Summary: The universal \(T\)-matrix of a quantum group is the Hopf algebra dual form, expressed in terms of the generators of the algebra and the generators of its dual. In the physical applications it is the familiar \(T\)-matrix of integrable models, here calculated in the structure, without specialization to a representation of the algebra of physical variables, nor to a representation of the auxiliary algebra. This article deals with some rather surprising facts that were discovered by examination of the formula for the universal \(T\)-matrix for \(U_ q (sl_ 2)\), among them the existence of a new series of quantum deformations of \(U(gl_ n)\) and a generalization of the quantum double. The new quantum groups have physical applications with essentially new features, principally arising from the fact that the dual Lie algebra (algebra of physical variables) is not solvable.
For the entire collection see [Zbl 0801.00049].

MSC:
17B37 Quantum groups (quantized enveloping algebras) and related deformations
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
70Sxx Classical field theories
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