Let H be a Hopf algebra with a bijective antipode, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\mathcal {G}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">G</mi> </math></EquationSource> </InlineEquation> an H-dimodule Lie algebra and A a <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$({\mathcal {G}},H)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">G</mi> <mo>,</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-dimodule algebra. Assume that there is an H-colinear algebra map <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϕ</mi> </math></EquationSource> </InlineEquation> from H to <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$A^{\mathcal {G}}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>A</mi> <mi mathvariant="script">G</mi> </msup> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$Im \phi \subseteq Z(A)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>I</mi> <mi>m</mi> <mi>ϕ</mi> <mo>⊆</mo> <mi>Z</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Under some assumptions, we give the Fundamental theorem for <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$(A,{\mathcal {G}},H)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi mathvariant="script">G</mi> <mo>,</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-dimodules. We also prove the Fundamental theorem for Yetter–Drinfeld <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$(A,{\mathcal {G}},H)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi mathvariant="script">G</mi> <mo>,</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-modules when H is cocommutative, <InlineEquation ID="IEq10"> <EquationSource Format="TEX">${\mathcal {G}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">G</mi> </math></EquationSource> </InlineEquation> is a Yetter–Drinfeld H-module Lie algebra and A is a Yetter–Drinfeld <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$({\mathcal {G}},H)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">G</mi> <mo>,</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-module algebra. We particuliarise our results to the Poisson setting.

Fundamental theorem of $(A,{\mathcal {G}},H)$-dimodules

期刊论文
发表日期：

2026年7月6日

查看全文

Thomas Guédénon

摘要

Let H be a Hopf algebra with a bijective antipode, ${\mathcal {G}}$ $G$ an H-dimodule Lie algebra and A a $({\mathcal {G}},H)$ $(G, H)$ -dimodule algebra. Assume that there is an H-colinear algebra map $\phi $ $ϕ$ from H to $A^{\mathcal {G}}$ $A^{G}$ such that $Im \phi \subseteq Z(A)$ $I m ϕ \subseteq Z (A)$ . Under some assumptions, we give the Fundamental theorem for $(A,{\mathcal {G}},H)$ $(A, G, H)$ -dimodules. We also prove the Fundamental theorem for Yetter–Drinfeld $(A,{\mathcal {G}},H)$ $(A, G, H)$ -modules when H is cocommutative, ${\mathcal {G}}$ $G$ is a Yetter–Drinfeld H-module Lie algebra and A is a Yetter–Drinfeld $({\mathcal {G}},H)$ $(G, H)$ -module algebra. We particuliarise our results to the Poisson setting.

Fundamental theorem of \((A,{\mathcal {G}},H)\)-dimodules

摘要