Representations of affine quantum function algebras

Bharath Narayanan

doi:10.1016/j.jalgebra.2003.06.002

Representations of affine quantum function algebras

Journal of Algebra ◽

10.1016/j.jalgebra.2003.06.002 ◽

2004 ◽

Vol 272 (2) ◽

pp. 775-800

Author(s):

Bharath Narayanan

Keyword(s):

Function Algebras ◽

Quantum Function

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Links between cofinite prime ideals in quantum function algebras

Israel Journal of Mathematics ◽

10.1007/bf02773644 ◽

1997 ◽

Vol 100 (1) ◽

pp. 285-308 ◽

Cited By ~ 1

Author(s):

Ian M. Musson

Keyword(s):

Prime Ideals ◽

Function Algebras ◽

Quantum Function

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PRESENTATION BY BOREL SUBALGEBRAS AND CHEVALLEY GENERATORS FOR QUANTUM ENVELOPING ALGEBRAS

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091504000689 ◽

2006 ◽

Vol 49 (2) ◽

pp. 291-308 ◽

Cited By ~ 2

Author(s):

Fabio Gavarini

Keyword(s):

Integral Form ◽

R Matrix ◽

Generators And Relations ◽

Enveloping Algebras ◽

Function Algebras ◽

Alternative Approach ◽

Q Matrix ◽

Lower Triangular ◽

Standard Presentation ◽

Quantum Function

AbstractWe provide an alternative approach to the Faddeev–Reshetikhin–Takhtajan presentation of the quantum group $\uqg$, with $L$-operators as generators and relations ruled by an $R$-matrix. We look at $\uqg$ as being generated by the quantum Borel subalgebras $U_q(\mathfrak{b}_+)$ and $U_q(\mathfrak{b}_-)$, and use the standard presentation of the latter as quantum function algebras. When $\mathfrak{g}=\mathfrak{gl}_n$, these Borel quantum function algebras are generated by the entries of a triangular $q$-matrix. Thus, eventually, $U_q(\mathfrak{gl}_n)$ 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 $\Bbbk[q,q^{-1}]$ the unrestricted $\Bbbk [q,q^{-1}]$-integral form of $U_q(\mathfrak{gl}_n)$ 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 $\mathfrak{g}=\mathfrak{sl}_n$ too.

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