<p>We study the action of Dynkin diagram automorphisms <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation> on generalized Gaudin algebras, focusing in particular on the big Gaudin algebra <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {B}(\mathfrak {g}) \subset (U(\mathfrak {g}) \otimes S(\mathfrak {g}))^{\mathfrak {g}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">B</mi> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="fraktur">g</mi> <mo stretchy="false">)</mo> </mrow> <mo>⊂</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="fraktur">g</mi> <mo stretchy="false">)</mo> </mrow> <mo>⊗</mo> <mi>S</mi> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="fraktur">g</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mi mathvariant="fraktur">g</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> and its evaluated versions <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {B}^\lambda (\mathfrak {g})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="script">B</mi> </mrow> <mi>λ</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="fraktur">g</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {B}_\chi (\mathfrak {g})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">B</mi> <mi>χ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="fraktur">g</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We show isomorphisms between the coinvariants of the generalized Gaudin algebras associated with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathfrak {g}^\vee $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="fraktur">g</mi> </mrow> <mo>∨</mo> </msup> </math></EquationSource> </InlineEquation> and the generalized Gaudin algebras associated with <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathfrak {g}_\sigma ^\vee $</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="fraktur">g</mi> <mi>σ</mi> <mo>∨</mo> </msubsup> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathfrak g_\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="fraktur">g</mi> <mi>σ</mi> </msub> </math></EquationSource> </InlineEquation> is the fixed point subalgebra. In particular, we get an isomorphism <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {B}^\lambda (\mathfrak {g}^\vee )_\sigma \simeq \mathcal {B}^\lambda (\mathfrak {g}^\vee _\sigma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="script">B</mi> </mrow> <mi>λ</mi> </msup> <msub> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="fraktur">g</mi> </mrow> <mo>∨</mo> </msup> <mo stretchy="false">)</mo> </mrow> <mi>σ</mi> </msub> <mo>≃</mo> <msup> <mrow> <mi mathvariant="script">B</mi> </mrow> <mi>λ</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msubsup> <mrow> <mi mathvariant="fraktur">g</mi> </mrow> <mi>σ</mi> <mo>∨</mo> </msubsup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for any <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>-invariant dominant weight <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>, which allows us to reprove Jantzen’s twining formula. Our approach relies on interpreting generalized Gaudin algebras via spaces of opers, which explains the appearance of the Langlands duals in our results and in Jantzen’s twining formula.</p>

Dynkin Automorphism Actions on Gaudin Algebras

期刊论文
发表日期：

2025年5月10日

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Vladyslav Zveryk

摘要

We study the action of Dynkin diagram automorphisms $\sigma $ $σ$ on generalized Gaudin algebras, focusing in particular on the big Gaudin algebra $\mathcal {B}(\mathfrak {g}) \subset (U(\mathfrak {g}) \otimes S(\mathfrak {g}))^{\mathfrak {g}}$ $B (g) \subset {(U (g) \otimes S (g))}^{g}$ and its evaluated versions $\mathcal {B}^\lambda (\mathfrak {g})$ $B^{λ} (g)$ and $\mathcal {B}_\chi (\mathfrak {g})$ $B_{χ} (g)$ . We show isomorphisms between the coinvariants of the generalized Gaudin algebras associated with $\mathfrak {g}^\vee $ $g^{\lor}$ and the generalized Gaudin algebras associated with $\mathfrak {g}_\sigma ^\vee $ $g_{σ}^{\lor}$ , where $\mathfrak g_\sigma $ $g_{σ}$ is the fixed point subalgebra. In particular, we get an isomorphism $\mathcal {B}^\lambda (\mathfrak {g}^\vee )_\sigma \simeq \mathcal {B}^\lambda (\mathfrak {g}^\vee _\sigma )$ $B^{λ} {(g^{\lor})}_{σ} ≃ B^{λ} (g_{σ}^{\lor})$ for any $\sigma $ $σ$ -invariant dominant weight $\lambda $ $λ$ , which allows us to reprove Jantzen’s twining formula. Our approach relies on interpreting generalized Gaudin algebras via spaces of opers, which explains the appearance of the Langlands duals in our results and in Jantzen’s twining formula.