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Hecke symmetries with central associated determinant. (Symétries de Hecke à determinant associé central.) (French) Zbl 1041.16034

The author proves the existence of a family of Hecke symmetries which are not deformations of the volte and with central associated determinant.
Let us recall that the first quantum groups \(U_q(\text{sl}(n))\) are defined in terms of Hecke symmetries. The corresponding Hilbert-Poincaré series coincide with the classical Hilbert-Poincaré series. D. I. Gurevich [in Algebra Anal. 2, No. 4, 119-148 (1990; Zbl 0713.17010)], gave an explicit construction of non classical Hecke symmetries, that is Hecke symmetries with central associated determinant. The corresponding Poincaré series are drastically different from the classical ones.
In this paper, after recalling the relevant notions, the author shows the existence of a family of Hecke symmetries with that property. The construction consists in gluing in a particular way two Hecke symmetries with rank 2. The required existence is then obtained recursively.
In the quasi-classical case, this construction gives the Hecke symmetries associated to the quantum groups \(U_q(\text{sl}(n))\).

MSC:

16W35 Ring-theoretic aspects of quantum groups (MSC2000)
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References:

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