Quantum duality principle and quantum symmetric pairs

<jats:p> The <jats:italic>quantum duality principle</jats:italic> (QDP) by Drinfeld predicts a connection between the <jats:italic>quantized universal enveloping algebras</jats:italic> and the <jats:italic>quantized coordinate algebras</jats:italic> , where the underlying classical objects are related by the duality in Poisson geometry. The current paper gives an explicit formulization of the QDP for quantum symmetric pairs. </jats:p> <jats:p> Let <jats:inline-formula> <jats:alternatives> <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mrow> <mml:mi mathvariant="fraktur">g</mml:mi> </mml:mrow> </mml:math> <jats:tex-math>$\mathfrak {g}$</jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" content-type="simple" xlink:href="S2050509426102370_inline1.png"> <jats:alt-text content-type="machine-generated">German g</jats:alt-text> </jats:inline-graphic> </jats:alternatives> </jats:inline-formula> be a complex semi-simple Lie algebra, equipped with the standard Lie bialgebra structure. Let <jats:inline-formula> <jats:alternatives> <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>θ</mml:mi> </mml:math> <jats:tex-math>$\theta $</jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" content-type="simple" xlink:href="S2050509426102370_inline2.png"> <jats:alt-text content-type="machine-generated">theta</jats:alt-text> </jats:inline-graphic> </jats:alternatives> </jats:inline-formula> be a Lie algebra involution on <jats:inline-formula> <jats:alternatives> <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mrow> <mml:mi mathvariant="fraktur">g</mml:mi> </mml:mrow> </mml:math> <jats:tex-math>$\mathfrak {g}$</jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" content-type="simple" xlink:href="S2050509426102370_inline3.png"> <jats:alt-text content-type="machine-generated">German g</jats:alt-text> </jats:inline-graphic> </jats:alternatives> </jats:inline-formula> and denote by <jats:inline-formula> <jats:alternatives> <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mrow> <mml:mi mathvariant="fraktur">k</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="fraktur">g</mml:mi> </mml:mrow> <mml:mi>θ</mml:mi> </mml:msup> </mml:math> <jats:tex-math>$\mathfrak {k}=\mathfrak {g}^\theta $</jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" content-type="simple" xlink:href="S2050509426102370_inline4.png"> <jats:alt-text content-type="machine-generated">German k equals German g Superscript theta</jats:alt-text> </jats:inline-graphic> </jats:alternatives> </jats:inline-formula> the fixed point subalgebra. The quantum symmetric pair <jats:inline-formula> <jats:alternatives> <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi mathvariant="normal">U</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="normal">U</mml:mi> </mml:mrow> <mml:mi>ı</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:math> <jats:tex-math>$(\mathrm {U},\mathrm {U}^\imath )$</jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" content-type="simple" xlink:href="S2050509426102370_inline5.png"> <jats:alt-text content-type="machine-generated">left parenthesis normal upper U comma normal upper U Superscript modifying above dotless i Baseline right parenthesis</jats:alt-text> </jats:inline-graphic> </jats:alternatives> </jats:inline-formula> is originally defined to be a quantization of the symmetric pair of the universal enveloping algebras <jats:inline-formula> <jats:alternatives> <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mo stretchy="false">(</mml:mo> <mml:mi>U</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi mathvariant="fraktur">g</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>U</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi mathvariant="fraktur">k</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:math> <jats:tex-math>$(U(\mathfrak {g}),U(\mathfrak {k}))$</jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" content-type="simple" xlink:href="S2050509426102370_inline6.png"> <jats:alt-text content-type="machine-generated">left parenthesis upper U left parenthesis German g right parenthesis comma upper U left parenthesis German k right parenthesis right parenthesis</jats:alt-text> </jats:inline-graphic> </jats:alternatives> </jats:inline-formula> . In this paper, we show that an explicit specialization of <jats:inline-formula> <jats:alternatives> <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <m