Mathematics – Quantum Algebra
Scientific paper
2004-03-22
Proceedings of the Edinburgh Mathematical Society 49 (2006), 291-308
Mathematics
Quantum Algebra
AMS-TeX, 17 pages. Final version appeared in "Proceedings of the Edinburgh Mathematical Society"
Scientific paper
10.1017/S0013091504000689
We provide an alternative approach to the Faddeev-Reshetikhin-Takhtajan presentation of the quantum group U_q(g), with L-operators as generators and relations ruled by an R-matrix. We look at U_q(g) as being generated by the quantum Borel subalgebras U_q(b_+) and U_q(b_-), and we use the standard presentation of the latters as quantum function algebras. When g = gl(n) these Borel quantum function algebras are generated by the entries of a triangular q-matrix, thus eventually U_q(gl(n)) is generated by the entries of an upper triangular and a lower triangular q-matrix, which share the same diagonal. The same elements generate over the ring of Laurent polynomials the unrestricted integer form of U_q(gl(n)) of De Concini and Procesi, which we present explicitly, together with a neat description of the associated quantum Frobenius morphisms at roots of 1. All this holds, mutatis mutandis, for g = sl(n) too.
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