A modified BLM approach to quantum affine {\mathfrak{gl}_n} |
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Authors: | Jie Du Qiang Fu |
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Institution: | (1) Department of Mathematics, Nara Women’s University, Kitauoyahigashi-machi, Nara 630-8506, Japan |
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Abstract: | We introduce a spanning set of Beilinson–Lusztig–MacPherson type, {A(j, r)}
A,j
, for affine quantum Schur algebras
S\vartriangle(n, r){{{\boldsymbol{\mathcal S}}_\vartriangle}(n, r)} and construct a linearly independent set {A(j)}
A,j
for an associated algebra
^(K)]\vartriangle(n){{{\boldsymbol{\widehat{\mathcal K}}}_\vartriangle}(n)} . We then establish explicitly some multiplication formulas of simple generators
E\vartriangleh,h+1(0){E^\vartriangle_{h,h+1}}(\mathbf{0}) by an arbitrary element A(j) in
^(K)]\vartriangle(n){{\boldsymbol{\widehat{{{\mathcal K}}}}_\vartriangle(n)}} via the corresponding formulas in
S\vartriangle(n, r){{{\boldsymbol{\mathcal S}}_\vartriangle(n, r)}} , and compare these formulas with the multiplication formulas between a simple module and an arbitrary module in the Ringel–Hall
algebras
\mathfrak H\vartriangle(n){{{\boldsymbol{\mathfrak H}_\vartriangle(n)}}} associated with cyclic quivers. This allows us to use the triangular relation between monomial and PBW type bases for
\mathfrak H\vartriangle(n){{\boldsymbol{\mathfrak H}}_\vartriangle}(n) established in Deng and Du (Adv Math 191:276–304, 2005) to derive similar triangular relations for
S\vartriangle(n, r){{{\boldsymbol{\mathcal S}}_\vartriangle}(n, r)} and
^(K)]\vartriangle(n){{\boldsymbol{\widehat{\mathcal K}}}_\vartriangle}(n) . Using these relations, we then show that the subspace
\mathfrak A\vartriangle(n){{{\boldsymbol{\mathfrak A}}_\vartriangle}(n)} of
^(K)]\vartriangle(n){{\boldsymbol{\widehat{{{\mathcal K}}}}_\vartriangle}(n)} spanned by {A(j)}
A,j
contains the quantum enveloping algebra
U\vartriangle(n){{{\mathbf U}_\vartriangle}(n)} of affine type A as a subalgebra. As an application, we prove that, when this construction is applied to quantum Schur algebras S(n,r){\boldsymbol{\mathcal S}(n,r)} , the resulting subspace
\mathfrak A\vartriangle(n){{{{\boldsymbol{\mathfrak A}}_\vartriangle}(n)}} is in fact a subalgebra which is isomorphic to the quantum enveloping algebra of
\mathfrakgln{\mathfrak{gl}_n} . We conjecture that
\mathfrak A\vartriangle(n){{{{{\boldsymbol{\mathfrak A}}_\vartriangle}(n)}}} is a subalgebra of
^(K)]\vartriangle(n){{\boldsymbol{\widehat{{{\mathcal K}}}}_\vartriangle}(n)} . |
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Keywords: | |
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