Is there a relationship between pre-Lie algebras and post-Lie algebra?












4












$begingroup$


You can find a short survey in this paper https://arxiv.org/pdf/1712.09415.pdf on Post-Lie algebras. I am interested in them because both of them can be constructed on rooted trees. But I don't know ''is there a relationship between pre-Lie algebras and post-Lie algebra''?






abstract-algebra lie-algebras







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$endgroup$












  • $begingroup$
    Please type up the definitions using $LaTeX$, in an edit.
    $endgroup$
    – Shaun
    Jan 2 at 12:49












  • $begingroup$
    @Shaun, I think someone (don't know pre-Lie and post-Lie) can't know the relation just from the defintions. I also think the definitoins are superfluous for the one who know their relations.
    $endgroup$
    – Daisy
    Jan 4 at 14:36
















4












$begingroup$


You can find a short survey in this paper https://arxiv.org/pdf/1712.09415.pdf on Post-Lie algebras. I am interested in them because both of them can be constructed on rooted trees. But I don't know ''is there a relationship between pre-Lie algebras and post-Lie algebra''?






abstract-algebra lie-algebras







share|cite|improve this question









$endgroup$












  • $begingroup$
    Please type up the definitions using $LaTeX$, in an edit.
    $endgroup$
    – Shaun
    Jan 2 at 12:49












  • $begingroup$
    @Shaun, I think someone (don't know pre-Lie and post-Lie) can't know the relation just from the defintions. I also think the definitoins are superfluous for the one who know their relations.
    $endgroup$
    – Daisy
    Jan 4 at 14:36














4












4








4





$begingroup$


You can find a short survey in this paper https://arxiv.org/pdf/1712.09415.pdf on Post-Lie algebras. I am interested in them because both of them can be constructed on rooted trees. But I don't know ''is there a relationship between pre-Lie algebras and post-Lie algebra''?






abstract-algebra lie-algebras







share|cite|improve this question









$endgroup$




You can find a short survey in this paper https://arxiv.org/pdf/1712.09415.pdf on Post-Lie algebras. I am interested in them because both of them can be constructed on rooted trees. But I don't know ''is there a relationship between pre-Lie algebras and post-Lie algebra''?






abstract-algebra lie-algebras




abstract-algebra lie-algebras






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Jan 2 at 12:40









DaisyDaisy

462413




462413












  • $begingroup$
    Please type up the definitions using $LaTeX$, in an edit.
    $endgroup$
    – Shaun
    Jan 2 at 12:49












  • $begingroup$
    @Shaun, I think someone (don't know pre-Lie and post-Lie) can't know the relation just from the defintions. I also think the definitoins are superfluous for the one who know their relations.
    $endgroup$
    – Daisy
    Jan 4 at 14:36


















  • $begingroup$
    Please type up the definitions using $LaTeX$, in an edit.
    $endgroup$
    – Shaun
    Jan 2 at 12:49












  • $begingroup$
    @Shaun, I think someone (don't know pre-Lie and post-Lie) can't know the relation just from the defintions. I also think the definitoins are superfluous for the one who know their relations.
    $endgroup$
    – Daisy
    Jan 4 at 14:36
















$begingroup$
Please type up the definitions using $LaTeX$, in an edit.
$endgroup$
– Shaun
Jan 2 at 12:49






$begingroup$
Please type up the definitions using $LaTeX$, in an edit.
$endgroup$
– Shaun
Jan 2 at 12:49














$begingroup$
@Shaun, I think someone (don't know pre-Lie and post-Lie) can't know the relation just from the defintions. I also think the definitoins are superfluous for the one who know their relations.
$endgroup$
– Daisy
Jan 4 at 14:36




$begingroup$
@Shaun, I think someone (don't know pre-Lie and post-Lie) can't know the relation just from the defintions. I also think the definitoins are superfluous for the one who know their relations.
$endgroup$
– Daisy
Jan 4 at 14:36










1 Answer
1






active

oldest

votes


















5












$begingroup$

Yes, there are relations between both. It depends on which level you are looking for such relations. In our article



Affine actions on Lie groups and post-Lie algebra structures



we explain geometric and algebraic relations, pre-Lie algebra structures being a special case of post-Lie algebra structures. The references mention work by Vallette and Loday, who give many other viewpoits, i.e., by operad theory, rooted trees etc.
The operads PreLie and PostLie arise in the context of Manin's black product, and in other topics, e.g. homology of generalized partition posets and renormalization theory.



Manin's black product ${mathcal {P}} bullet {mathcal {Q}}$ of
binary quadratic operads has the operad ${mathcal {Lie}}$ as neutral element, i.e.,
$$
{mathcal {P}}={mathcal {Lie}}bullet {mathcal {P}}={mathcal {P}}bullet {mathcal {Lie}}.
$$



$bullet$ The bi-successor and tri-successor of a quadratic operad ${mathcal {P}}$
are given by
begin{align*}
Bi ({mathcal {P}}) & = {mathcal {PreLie}}bullet {mathcal {P}}, \
Tri ({mathcal {P}}) & = {mathcal {PostLie}}bullet {mathcal {P}}.
end{align*}






share|cite|improve this answer











$endgroup$













  • $begingroup$
    @thank you very much. Your works are perfect.
    $endgroup$
    – Daisy
    Jan 4 at 12:28











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1 Answer
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active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









5












$begingroup$

Yes, there are relations between both. It depends on which level you are looking for such relations. In our article



Affine actions on Lie groups and post-Lie algebra structures



we explain geometric and algebraic relations, pre-Lie algebra structures being a special case of post-Lie algebra structures. The references mention work by Vallette and Loday, who give many other viewpoits, i.e., by operad theory, rooted trees etc.
The operads PreLie and PostLie arise in the context of Manin's black product, and in other topics, e.g. homology of generalized partition posets and renormalization theory.



Manin's black product ${mathcal {P}} bullet {mathcal {Q}}$ of
binary quadratic operads has the operad ${mathcal {Lie}}$ as neutral element, i.e.,
$$
{mathcal {P}}={mathcal {Lie}}bullet {mathcal {P}}={mathcal {P}}bullet {mathcal {Lie}}.
$$



$bullet$ The bi-successor and tri-successor of a quadratic operad ${mathcal {P}}$
are given by
begin{align*}
Bi ({mathcal {P}}) & = {mathcal {PreLie}}bullet {mathcal {P}}, \
Tri ({mathcal {P}}) & = {mathcal {PostLie}}bullet {mathcal {P}}.
end{align*}






share|cite|improve this answer











$endgroup$













  • $begingroup$
    @thank you very much. Your works are perfect.
    $endgroup$
    – Daisy
    Jan 4 at 12:28
















5












$begingroup$

Yes, there are relations between both. It depends on which level you are looking for such relations. In our article



Affine actions on Lie groups and post-Lie algebra structures



we explain geometric and algebraic relations, pre-Lie algebra structures being a special case of post-Lie algebra structures. The references mention work by Vallette and Loday, who give many other viewpoits, i.e., by operad theory, rooted trees etc.
The operads PreLie and PostLie arise in the context of Manin's black product, and in other topics, e.g. homology of generalized partition posets and renormalization theory.



Manin's black product ${mathcal {P}} bullet {mathcal {Q}}$ of
binary quadratic operads has the operad ${mathcal {Lie}}$ as neutral element, i.e.,
$$
{mathcal {P}}={mathcal {Lie}}bullet {mathcal {P}}={mathcal {P}}bullet {mathcal {Lie}}.
$$



$bullet$ The bi-successor and tri-successor of a quadratic operad ${mathcal {P}}$
are given by
begin{align*}
Bi ({mathcal {P}}) & = {mathcal {PreLie}}bullet {mathcal {P}}, \
Tri ({mathcal {P}}) & = {mathcal {PostLie}}bullet {mathcal {P}}.
end{align*}






share|cite|improve this answer











$endgroup$













  • $begingroup$
    @thank you very much. Your works are perfect.
    $endgroup$
    – Daisy
    Jan 4 at 12:28














5












5








5





$begingroup$

Yes, there are relations between both. It depends on which level you are looking for such relations. In our article



Affine actions on Lie groups and post-Lie algebra structures



we explain geometric and algebraic relations, pre-Lie algebra structures being a special case of post-Lie algebra structures. The references mention work by Vallette and Loday, who give many other viewpoits, i.e., by operad theory, rooted trees etc.
The operads PreLie and PostLie arise in the context of Manin's black product, and in other topics, e.g. homology of generalized partition posets and renormalization theory.



Manin's black product ${mathcal {P}} bullet {mathcal {Q}}$ of
binary quadratic operads has the operad ${mathcal {Lie}}$ as neutral element, i.e.,
$$
{mathcal {P}}={mathcal {Lie}}bullet {mathcal {P}}={mathcal {P}}bullet {mathcal {Lie}}.
$$



$bullet$ The bi-successor and tri-successor of a quadratic operad ${mathcal {P}}$
are given by
begin{align*}
Bi ({mathcal {P}}) & = {mathcal {PreLie}}bullet {mathcal {P}}, \
Tri ({mathcal {P}}) & = {mathcal {PostLie}}bullet {mathcal {P}}.
end{align*}






share|cite|improve this answer











$endgroup$



Yes, there are relations between both. It depends on which level you are looking for such relations. In our article



Affine actions on Lie groups and post-Lie algebra structures



we explain geometric and algebraic relations, pre-Lie algebra structures being a special case of post-Lie algebra structures. The references mention work by Vallette and Loday, who give many other viewpoits, i.e., by operad theory, rooted trees etc.
The operads PreLie and PostLie arise in the context of Manin's black product, and in other topics, e.g. homology of generalized partition posets and renormalization theory.



Manin's black product ${mathcal {P}} bullet {mathcal {Q}}$ of
binary quadratic operads has the operad ${mathcal {Lie}}$ as neutral element, i.e.,
$$
{mathcal {P}}={mathcal {Lie}}bullet {mathcal {P}}={mathcal {P}}bullet {mathcal {Lie}}.
$$



$bullet$ The bi-successor and tri-successor of a quadratic operad ${mathcal {P}}$
are given by
begin{align*}
Bi ({mathcal {P}}) & = {mathcal {PreLie}}bullet {mathcal {P}}, \
Tri ({mathcal {P}}) & = {mathcal {PostLie}}bullet {mathcal {P}}.
end{align*}







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Jan 2 at 13:44

























answered Jan 2 at 13:22









Dietrich BurdeDietrich Burde

80.2k647104




80.2k647104












  • $begingroup$
    @thank you very much. Your works are perfect.
    $endgroup$
    – Daisy
    Jan 4 at 12:28


















  • $begingroup$
    @thank you very much. Your works are perfect.
    $endgroup$
    – Daisy
    Jan 4 at 12:28
















$begingroup$
@thank you very much. Your works are perfect.
$endgroup$
– Daisy
Jan 4 at 12:28




$begingroup$
@thank you very much. Your works are perfect.
$endgroup$
– Daisy
Jan 4 at 12:28


















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