We give a q-analogue of Howe duality associated with a pair <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(\mathfrak {g},G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="fraktur">g</mi> <mo>,</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathfrak {g}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">g</mi> </math></EquationSource> </InlineEquation> is an orthosymplectic Lie superalgebra and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$G=O_\ell , Sp_{2\ell }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>=</mo> <msub> <mi>O</mi> <mi>ℓ</mi> </msub> <mo>,</mo> <mi>S</mi> <msub> <mi>p</mi> <mrow> <mn>2</mn> <mi>ℓ</mi> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation>. We define explicitly commuting actions of a quantized enveloping algebra of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathfrak {g}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">g</mi> </math></EquationSource> </InlineEquation> and the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\imath $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ı</mi> </math></EquationSource> </InlineEquation>quantum group of types AI and AII on a q-deformed supersymmetric space, and describe its semisimple decomposition whose classical limit recovers the <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$(\mathfrak {g},G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="fraktur">g</mi> <mo>,</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-duality. As special cases, we obtain q-analogues of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$(\mathfrak {g},G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="fraktur">g</mi> <mo>,</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-dualities on symmetric and exterior algebras for <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathfrak {g}=\mathfrak {so}_{2n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="fraktur">g</mi> <mo>=</mo> <msub> <mi mathvariant="fraktur">so</mi> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\mathfrak {sp}_{2n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="fraktur">sp</mi> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>.

q-deformed Howe duality for orthosymplectic Lie superalgebras

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2025年11月1日

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Jeong Bae,
Jae-Hoon Kwon

摘要

We give a q-analogue of Howe duality associated with a pair $(\mathfrak {g},G)$ $(g, G)$ , where $\mathfrak {g}$ $g$ is an orthosymplectic Lie superalgebra and $G=O_\ell , Sp_{2\ell }$ $G = O_{ℓ}, S p_{2 ℓ}$ . We define explicitly commuting actions of a quantized enveloping algebra of $\mathfrak {g}$ $g$ and the $\imath $ $ı$ quantum group of types AI and AII on a q-deformed supersymmetric space, and describe its semisimple decomposition whose classical limit recovers the $(\mathfrak {g},G)$ $(g, G)$ -duality. As special cases, we obtain q-analogues of $(\mathfrak {g},G)$ $(g, G)$ -dualities on symmetric and exterior algebras for $\mathfrak {g}=\mathfrak {so}_{2n}$ $g = {so}_{2 n}$ , $\mathfrak {sp}_{2n}$ ${sp}_{2 n}$ .