A \(q\)-analogue of \(U(\mathfrak{gl}(N+1))\), Hecke algebra, and the Yang-Baxter equation. (English) Zbl 0602.17005

The following problem about the Yang-Baxter equation (YBE) is studied: given a classical r-matrix (= solution to the classical YBE), is it possible to find the corresponding quantum R-matrix (= solution to the quantum YBE) having the former as its classical limit ? Note that now the classical r-matrices associated with a simple Lie algebra \(\mathfrak g\) are classified. It has been realized that to construct R-matrices one has to “quantize” the Lie algebra structure itself. This was worked out by Drinfel’d for a class of rational solutions that generalize Yang’s one.
Here a \(q\)-analogue \(\hat U(\mathfrak g)\) of the universal enveloping algebra \(U(\mathfrak g)\) appears. In this letter, the structure and the representations of \(\hat U(\mathfrak g)\) in the case \(\mathfrak g=\mathfrak{gl}(N+1)\) are studied having in mind its applications to the YBE.


17B37 Quantum groups (quantized enveloping algebras) and related deformations
16T25 Yang-Baxter equations
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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