<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {U}(\widehat{\mathfrak {gl}}_{m|n})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">U</mi> <mo stretchy="false">(</mo> <msub> <mover accent="true"> <mi mathvariant="fraktur">gl</mi> <mo stretchy="false">^</mo> </mover> <mrow> <mi>m</mi> <mo stretchy="false">|</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be the universal enveloping superalgebra of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\widehat{\mathfrak {gl}}_{m|n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mover accent="true"> <mi mathvariant="fraktur">gl</mi> <mo stretchy="false">^</mo> </mover> <mrow> <mi>m</mi> <mo stretchy="false">|</mo> <mi>n</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>. There is a natural surjective superalgebra homomorphism <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\xi _r$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ξ</mi> <mi>r</mi> </msub> </math></EquationSource> </InlineEquation> from <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {U}(\widehat{\mathfrak {gl}}_{m|n})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">U</mi> <mo stretchy="false">(</mo> <msub> <mover accent="true"> <mi mathvariant="fraktur">gl</mi> <mo stretchy="false">^</mo> </mover> <mrow> <mi>m</mi> <mo stretchy="false">|</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> to the affine Schur superalgebra <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\widehat{\mathcal {S}}(m|n,r)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi mathvariant="script">S</mi> <mo stretchy="false">^</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">|</mo> <mi>n</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We give the extra defining relations needed to define the affine Schur superalgebra <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\widehat{\mathcal {S}}(m|n,r)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi mathvariant="script">S</mi> <mo stretchy="false">^</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">|</mo> <mi>n</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> as a quotient of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {U}(\widehat{\mathfrak {gl}}_{m|n})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">U</mi> <mo stretchy="false">(</mo> <msub> <mover accent="true"> <mi mathvariant="fraktur">gl</mi> <mo stretchy="false">^</mo> </mover> <mrow> <mi>m</mi> <mo stretchy="false">|</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.</p>

Presenting Affine Schur Superalgebras

期刊论文
发表日期：

2025年5月14日

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Qiang Fu,
Chengquan Sun

摘要

Let $\mathcal {U}(\widehat{\mathfrak {gl}}_{m|n})$ $U ({\hat{gl}}_{m | n})$ be the universal enveloping superalgebra of $\widehat{\mathfrak {gl}}_{m|n}$ ${\hat{gl}}_{m | n}$ . There is a natural surjective superalgebra homomorphism $\xi _r$ $ξ_{r}$ from $\mathcal {U}(\widehat{\mathfrak {gl}}_{m|n})$ $U ({\hat{gl}}_{m | n})$ to the affine Schur superalgebra $\widehat{\mathcal {S}}(m|n,r)$ $\hat{S} (m | n, r)$ . We give the extra defining relations needed to define the affine Schur superalgebra $\widehat{\mathcal {S}}(m|n,r)$ $\hat{S} (m | n, r)$ as a quotient of $\mathcal {U}(\widehat{\mathfrak {gl}}_{m|n})$ $U ({\hat{gl}}_{m | n})$ .