Is there a relationship between pre-Lie algebras and post-Lie algebra?
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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
asked Jan 2 at 12:40
DaisyDaisy
462413
$endgroup$
add a comment |
$begingroup$
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
asked Jan 2 at 12:40
DaisyDaisy
462413
$endgroup$
$begingroup$
Please type up the definitions using $LaTeX$, in an edit.
$endgroup$
– Shaun
Jan 2 at 12:49
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@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
add a comment |
$begingroup$
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
asked Jan 2 at 12:40
DaisyDaisy
462413
$endgroup$
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
asked Jan 2 at 12:40
DaisyDaisy
462413
asked Jan 2 at 12:40
DaisyDaisy
462413
asked Jan 2 at 12:40
DaisyDaisy
462413
asked Jan 2 at 12:40
DaisyDaisy
462413
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
add a comment |
$begingroup$
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
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
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*}
answered Jan 2 at 13:22
Dietrich BurdeDietrich Burde
80.2k647104
$endgroup$
$begingroup$
@thank you very much. Your works are perfect.
$endgroup$
– Daisy
Jan 4 at 12:28
add a comment |
Your Answer
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1 Answer
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1 Answer
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$begingroup$
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*}
answered Jan 2 at 13:22
Dietrich BurdeDietrich Burde
80.2k647104
$endgroup$
$begingroup$
@thank you very much. Your works are perfect.
$endgroup$
– Daisy
Jan 4 at 12:28
add a comment |
$begingroup$
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*}
answered Jan 2 at 13:22
Dietrich BurdeDietrich Burde
80.2k647104
$endgroup$
$begingroup$
@thank you very much. Your works are perfect.
$endgroup$
– Daisy
Jan 4 at 12:28
add a comment |
$begingroup$
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*}
answered Jan 2 at 13:22
Dietrich BurdeDietrich Burde
80.2k647104
$endgroup$
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*}
answered Jan 2 at 13:22
Dietrich BurdeDietrich Burde
80.2k647104
edited Jan 2 at 13:44
answered Jan 2 at 13:22
Dietrich BurdeDietrich Burde
80.2k647104
answered Jan 2 at 13:22
Dietrich BurdeDietrich Burde
80.2k647104
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
add a comment |
$begingroup$
@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
add a comment |
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$begingroup$
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