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“Exactness†of operadic cohomology
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There are two somewhat widely known theorems which say
if $A$ is a nonnegatively graded commutative algebra in char $0$, then forgetful map on operadic cohomology $H^*_Harr(A, A) to H^*_Hoch(A, A)$ is injective
if $L$ is a Lie algebra in char $0$, then $H^*(L, L) to H^*(L, UL)$ is injective
Are they related at all? Is it a part of more general phenomenon occuring for exact sequences of (probably Koszul) operads?
(Also, it's obvious that latter theorem does not care about grading at all, just because $L to UL$ is split injection of modules. Is it true for former one? I gess not.)
homological-algebra operads
asked 3 hours ago
Denis T.
861514
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5
down vote
favorite
There are two somewhat widely known theorems which say
if $A$ is a nonnegatively graded commutative algebra in char $0$, then forgetful map on operadic cohomology $H^*_Harr(A, A) to H^*_Hoch(A, A)$ is injective
if $L$ is a Lie algebra in char $0$, then $H^*(L, L) to H^*(L, UL)$ is injective
Are they related at all? Is it a part of more general phenomenon occuring for exact sequences of (probably Koszul) operads?
(Also, it's obvious that latter theorem does not care about grading at all, just because $L to UL$ is split injection of modules. Is it true for former one? I gess not.)
homological-algebra operads
asked 3 hours ago
Denis T.
861514
add a comment |Â
up vote
5
down vote
favorite
up vote
5
down vote
favorite
There are two somewhat widely known theorems which say
if $A$ is a nonnegatively graded commutative algebra in char $0$, then forgetful map on operadic cohomology $H^*_Harr(A, A) to H^*_Hoch(A, A)$ is injective
if $L$ is a Lie algebra in char $0$, then $H^*(L, L) to H^*(L, UL)$ is injective
Are they related at all? Is it a part of more general phenomenon occuring for exact sequences of (probably Koszul) operads?
(Also, it's obvious that latter theorem does not care about grading at all, just because $L to UL$ is split injection of modules. Is it true for former one? I gess not.)
homological-algebra operads
asked 3 hours ago
Denis T.
861514
There are two somewhat widely known theorems which say
if $A$ is a nonnegatively graded commutative algebra in char $0$, then forgetful map on operadic cohomology $H^*_Harr(A, A) to H^*_Hoch(A, A)$ is injective
if $L$ is a Lie algebra in char $0$, then $H^*(L, L) to H^*(L, UL)$ is injective
Are they related at all? Is it a part of more general phenomenon occuring for exact sequences of (probably Koszul) operads?
(Also, it's obvious that latter theorem does not care about grading at all, just because $L to UL$ is split injection of modules. Is it true for former one? I gess not.)
homological-algebra operads
homological-algebra operads
asked 3 hours ago
Denis T.
861514
asked 3 hours ago
Denis T.
861514
asked 3 hours ago
Denis T.
861514
asked 3 hours ago
Denis T.
861514
asked 3 hours ago
Denis T.
861514
861514
add a comment |Â
add a comment |Â
1 Answer
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The second result you mention is significantly easier than the first. Indeed from the PBW theorem we know that $L$ is a direct summand of $UL$ as a Lie algebra, and the second result follows. But in fact there is a strong connection between the two results, since the first theorem may also be seen as a consequence of a sufficiently abstract version of the PBW theorem. This is much less well known and I haven't seen it written down anywhere.
(In particular the phenomenon is not something general for any sequence of Koszul operads - you really need a PBW theorem in the background.)
Here is how it goes. Via the map $mathsfLie to mathsfAss$ we may consider $mathsfAss$ as a bimodule over the Lie operad. Recall that if $P$ is an operad, then a $P$-bimodule is the same thing as a $P$-algebra in the category of right $P$-modules, so we may think of $mathsfLie to mathsfAss$ as a morphism of Lie algebras in a certain tensor category. In fact we may identify $mathsfAss$ with the universal enveloping algebra of $mathsfLie$ in this category, considered as a Lie algebra. By a general form of the PBW theorem (for Lie algebras in abstract symmetric tensor categories over a field of characteristic zero) it then follows that there exists a splitting $phi colon mathsfAss to mathsfLie$ in the category of bimodules over the Lie operad.
Explicitly, the fact that $phi$ is a morphism of bimodules means that we have two commutative pentagons. The first says that the composition
$$ mathsfLie circ mathsfAss to mathsfAss circ mathsfAss to mathsfAss stackrelphito mathsfLie$$
coincides with
$$ mathsfLie circ mathsfAss stackrelmathrmidcirc phito mathsfLie circ mathsfLie to mathsfLie.$$
The other says the same thing except with $mathsfLie$ acting on $mathsfAss$ on the right.
Now let $A$ be a $C_infty$-algebra. The $C_infty$-structure is given by a Maurer--Cartan element $mu$ in the pre-Lie algebra $mathfrak g := mathrmHom_mathbb S(mathsfcoLie,mathsfEnd_A)$, which is essentially the Harrison chain complex. The pre-Lie product $f star g$ of two Harrison chains is given by the composition
$$ mathsfcoLie to mathsfcoLie circ_(1) mathsfcoLie stackrelf circ gto mathsfEnd_A circ_(1) mathsfEnd_A to mathsfEnd_A.$$
We also have the associative version $mathfrak h := mathrmHom_mathbb S(mathsfcoAss,mathsfEnd_A)$ with analogously defined pre-Lie product. Via the map $mathsfcoAss to mathsfcoLie$ we can think of $mathfrak g$ as a pre-Lie subalgebra of $mathfrak h$. Now the dual of $phi$ induces a map $phi^ast colon mathfrak h to mathfrak g$, which is not in general a morphism of pre-Lie algebras. However, the fact that $phi$ is a map of infinitesimal bimodules is exactly the condition needed for $phi^ast$ to satisfy a "projection formula": for $f in mathfrak g subset mathfrak h$ and $g in mathfrak h$ we have $phi^ast(f star g) = f star phi^ast(g)$ and $phi^ast(g star f) = phi^ast(g) star f$. Indeed if we write out the definitions of the pre-Lie product then the conditions we need for the projection formula to hold become exactly that we have a map of infinitesimal bi-comodules $mathsfcoLie to mathsfcoAss$.
In particular we have the Harrison differential on $mathfrak g$ given by $df = f star mu - (-1)^vert fvertmu star f$ and the analogous Hochschild differential on $mathfrak h$. The previous paragraph says in particular that the splitting $mathfrak h to mathfrak g$ is compatible with these differentials, so that Harrison chains are a direct summand of Hochschild chains.
answered 1 hour ago
Dan Petersen
24.7k269133
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
5
down vote
The second result you mention is significantly easier than the first. Indeed from the PBW theorem we know that $L$ is a direct summand of $UL$ as a Lie algebra, and the second result follows. But in fact there is a strong connection between the two results, since the first theorem may also be seen as a consequence of a sufficiently abstract version of the PBW theorem. This is much less well known and I haven't seen it written down anywhere.
(In particular the phenomenon is not something general for any sequence of Koszul operads - you really need a PBW theorem in the background.)
Here is how it goes. Via the map $mathsfLie to mathsfAss$ we may consider $mathsfAss$ as a bimodule over the Lie operad. Recall that if $P$ is an operad, then a $P$-bimodule is the same thing as a $P$-algebra in the category of right $P$-modules, so we may think of $mathsfLie to mathsfAss$ as a morphism of Lie algebras in a certain tensor category. In fact we may identify $mathsfAss$ with the universal enveloping algebra of $mathsfLie$ in this category, considered as a Lie algebra. By a general form of the PBW theorem (for Lie algebras in abstract symmetric tensor categories over a field of characteristic zero) it then follows that there exists a splitting $phi colon mathsfAss to mathsfLie$ in the category of bimodules over the Lie operad.
Explicitly, the fact that $phi$ is a morphism of bimodules means that we have two commutative pentagons. The first says that the composition
$$ mathsfLie circ mathsfAss to mathsfAss circ mathsfAss to mathsfAss stackrelphito mathsfLie$$
coincides with
$$ mathsfLie circ mathsfAss stackrelmathrmidcirc phito mathsfLie circ mathsfLie to mathsfLie.$$
The other says the same thing except with $mathsfLie$ acting on $mathsfAss$ on the right.
Now let $A$ be a $C_infty$-algebra. The $C_infty$-structure is given by a Maurer--Cartan element $mu$ in the pre-Lie algebra $mathfrak g := mathrmHom_mathbb S(mathsfcoLie,mathsfEnd_A)$, which is essentially the Harrison chain complex. The pre-Lie product $f star g$ of two Harrison chains is given by the composition
$$ mathsfcoLie to mathsfcoLie circ_(1) mathsfcoLie stackrelf circ gto mathsfEnd_A circ_(1) mathsfEnd_A to mathsfEnd_A.$$
We also have the associative version $mathfrak h := mathrmHom_mathbb S(mathsfcoAss,mathsfEnd_A)$ with analogously defined pre-Lie product. Via the map $mathsfcoAss to mathsfcoLie$ we can think of $mathfrak g$ as a pre-Lie subalgebra of $mathfrak h$. Now the dual of $phi$ induces a map $phi^ast colon mathfrak h to mathfrak g$, which is not in general a morphism of pre-Lie algebras. However, the fact that $phi$ is a map of infinitesimal bimodules is exactly the condition needed for $phi^ast$ to satisfy a "projection formula": for $f in mathfrak g subset mathfrak h$ and $g in mathfrak h$ we have $phi^ast(f star g) = f star phi^ast(g)$ and $phi^ast(g star f) = phi^ast(g) star f$. Indeed if we write out the definitions of the pre-Lie product then the conditions we need for the projection formula to hold become exactly that we have a map of infinitesimal bi-comodules $mathsfcoLie to mathsfcoAss$.
In particular we have the Harrison differential on $mathfrak g$ given by $df = f star mu - (-1)^vert fvertmu star f$ and the analogous Hochschild differential on $mathfrak h$. The previous paragraph says in particular that the splitting $mathfrak h to mathfrak g$ is compatible with these differentials, so that Harrison chains are a direct summand of Hochschild chains.
answered 1 hour ago
Dan Petersen
24.7k269133
add a comment |Â
up vote
5
down vote
The second result you mention is significantly easier than the first. Indeed from the PBW theorem we know that $L$ is a direct summand of $UL$ as a Lie algebra, and the second result follows. But in fact there is a strong connection between the two results, since the first theorem may also be seen as a consequence of a sufficiently abstract version of the PBW theorem. This is much less well known and I haven't seen it written down anywhere.
(In particular the phenomenon is not something general for any sequence of Koszul operads - you really need a PBW theorem in the background.)
Here is how it goes. Via the map $mathsfLie to mathsfAss$ we may consider $mathsfAss$ as a bimodule over the Lie operad. Recall that if $P$ is an operad, then a $P$-bimodule is the same thing as a $P$-algebra in the category of right $P$-modules, so we may think of $mathsfLie to mathsfAss$ as a morphism of Lie algebras in a certain tensor category. In fact we may identify $mathsfAss$ with the universal enveloping algebra of $mathsfLie$ in this category, considered as a Lie algebra. By a general form of the PBW theorem (for Lie algebras in abstract symmetric tensor categories over a field of characteristic zero) it then follows that there exists a splitting $phi colon mathsfAss to mathsfLie$ in the category of bimodules over the Lie operad.
Explicitly, the fact that $phi$ is a morphism of bimodules means that we have two commutative pentagons. The first says that the composition
$$ mathsfLie circ mathsfAss to mathsfAss circ mathsfAss to mathsfAss stackrelphito mathsfLie$$
coincides with
$$ mathsfLie circ mathsfAss stackrelmathrmidcirc phito mathsfLie circ mathsfLie to mathsfLie.$$
The other says the same thing except with $mathsfLie$ acting on $mathsfAss$ on the right.
Now let $A$ be a $C_infty$-algebra. The $C_infty$-structure is given by a Maurer--Cartan element $mu$ in the pre-Lie algebra $mathfrak g := mathrmHom_mathbb S(mathsfcoLie,mathsfEnd_A)$, which is essentially the Harrison chain complex. The pre-Lie product $f star g$ of two Harrison chains is given by the composition
$$ mathsfcoLie to mathsfcoLie circ_(1) mathsfcoLie stackrelf circ gto mathsfEnd_A circ_(1) mathsfEnd_A to mathsfEnd_A.$$
We also have the associative version $mathfrak h := mathrmHom_mathbb S(mathsfcoAss,mathsfEnd_A)$ with analogously defined pre-Lie product. Via the map $mathsfcoAss to mathsfcoLie$ we can think of $mathfrak g$ as a pre-Lie subalgebra of $mathfrak h$. Now the dual of $phi$ induces a map $phi^ast colon mathfrak h to mathfrak g$, which is not in general a morphism of pre-Lie algebras. However, the fact that $phi$ is a map of infinitesimal bimodules is exactly the condition needed for $phi^ast$ to satisfy a "projection formula": for $f in mathfrak g subset mathfrak h$ and $g in mathfrak h$ we have $phi^ast(f star g) = f star phi^ast(g)$ and $phi^ast(g star f) = phi^ast(g) star f$. Indeed if we write out the definitions of the pre-Lie product then the conditions we need for the projection formula to hold become exactly that we have a map of infinitesimal bi-comodules $mathsfcoLie to mathsfcoAss$.
In particular we have the Harrison differential on $mathfrak g$ given by $df = f star mu - (-1)^vert fvertmu star f$ and the analogous Hochschild differential on $mathfrak h$. The previous paragraph says in particular that the splitting $mathfrak h to mathfrak g$ is compatible with these differentials, so that Harrison chains are a direct summand of Hochschild chains.
answered 1 hour ago
Dan Petersen
24.7k269133
add a comment |Â
up vote
5
down vote
up vote
5
down vote
The second result you mention is significantly easier than the first. Indeed from the PBW theorem we know that $L$ is a direct summand of $UL$ as a Lie algebra, and the second result follows. But in fact there is a strong connection between the two results, since the first theorem may also be seen as a consequence of a sufficiently abstract version of the PBW theorem. This is much less well known and I haven't seen it written down anywhere.
(In particular the phenomenon is not something general for any sequence of Koszul operads - you really need a PBW theorem in the background.)
Here is how it goes. Via the map $mathsfLie to mathsfAss$ we may consider $mathsfAss$ as a bimodule over the Lie operad. Recall that if $P$ is an operad, then a $P$-bimodule is the same thing as a $P$-algebra in the category of right $P$-modules, so we may think of $mathsfLie to mathsfAss$ as a morphism of Lie algebras in a certain tensor category. In fact we may identify $mathsfAss$ with the universal enveloping algebra of $mathsfLie$ in this category, considered as a Lie algebra. By a general form of the PBW theorem (for Lie algebras in abstract symmetric tensor categories over a field of characteristic zero) it then follows that there exists a splitting $phi colon mathsfAss to mathsfLie$ in the category of bimodules over the Lie operad.
Explicitly, the fact that $phi$ is a morphism of bimodules means that we have two commutative pentagons. The first says that the composition
$$ mathsfLie circ mathsfAss to mathsfAss circ mathsfAss to mathsfAss stackrelphito mathsfLie$$
coincides with
$$ mathsfLie circ mathsfAss stackrelmathrmidcirc phito mathsfLie circ mathsfLie to mathsfLie.$$
The other says the same thing except with $mathsfLie$ acting on $mathsfAss$ on the right.
Now let $A$ be a $C_infty$-algebra. The $C_infty$-structure is given by a Maurer--Cartan element $mu$ in the pre-Lie algebra $mathfrak g := mathrmHom_mathbb S(mathsfcoLie,mathsfEnd_A)$, which is essentially the Harrison chain complex. The pre-Lie product $f star g$ of two Harrison chains is given by the composition
$$ mathsfcoLie to mathsfcoLie circ_(1) mathsfcoLie stackrelf circ gto mathsfEnd_A circ_(1) mathsfEnd_A to mathsfEnd_A.$$
We also have the associative version $mathfrak h := mathrmHom_mathbb S(mathsfcoAss,mathsfEnd_A)$ with analogously defined pre-Lie product. Via the map $mathsfcoAss to mathsfcoLie$ we can think of $mathfrak g$ as a pre-Lie subalgebra of $mathfrak h$. Now the dual of $phi$ induces a map $phi^ast colon mathfrak h to mathfrak g$, which is not in general a morphism of pre-Lie algebras. However, the fact that $phi$ is a map of infinitesimal bimodules is exactly the condition needed for $phi^ast$ to satisfy a "projection formula": for $f in mathfrak g subset mathfrak h$ and $g in mathfrak h$ we have $phi^ast(f star g) = f star phi^ast(g)$ and $phi^ast(g star f) = phi^ast(g) star f$. Indeed if we write out the definitions of the pre-Lie product then the conditions we need for the projection formula to hold become exactly that we have a map of infinitesimal bi-comodules $mathsfcoLie to mathsfcoAss$.
In particular we have the Harrison differential on $mathfrak g$ given by $df = f star mu - (-1)^vert fvertmu star f$ and the analogous Hochschild differential on $mathfrak h$. The previous paragraph says in particular that the splitting $mathfrak h to mathfrak g$ is compatible with these differentials, so that Harrison chains are a direct summand of Hochschild chains.
answered 1 hour ago
Dan Petersen
24.7k269133
The second result you mention is significantly easier than the first. Indeed from the PBW theorem we know that $L$ is a direct summand of $UL$ as a Lie algebra, and the second result follows. But in fact there is a strong connection between the two results, since the first theorem may also be seen as a consequence of a sufficiently abstract version of the PBW theorem. This is much less well known and I haven't seen it written down anywhere.
(In particular the phenomenon is not something general for any sequence of Koszul operads - you really need a PBW theorem in the background.)
Here is how it goes. Via the map $mathsfLie to mathsfAss$ we may consider $mathsfAss$ as a bimodule over the Lie operad. Recall that if $P$ is an operad, then a $P$-bimodule is the same thing as a $P$-algebra in the category of right $P$-modules, so we may think of $mathsfLie to mathsfAss$ as a morphism of Lie algebras in a certain tensor category. In fact we may identify $mathsfAss$ with the universal enveloping algebra of $mathsfLie$ in this category, considered as a Lie algebra. By a general form of the PBW theorem (for Lie algebras in abstract symmetric tensor categories over a field of characteristic zero) it then follows that there exists a splitting $phi colon mathsfAss to mathsfLie$ in the category of bimodules over the Lie operad.
Explicitly, the fact that $phi$ is a morphism of bimodules means that we have two commutative pentagons. The first says that the composition
$$ mathsfLie circ mathsfAss to mathsfAss circ mathsfAss to mathsfAss stackrelphito mathsfLie$$
coincides with
$$ mathsfLie circ mathsfAss stackrelmathrmidcirc phito mathsfLie circ mathsfLie to mathsfLie.$$
The other says the same thing except with $mathsfLie$ acting on $mathsfAss$ on the right.
Now let $A$ be a $C_infty$-algebra. The $C_infty$-structure is given by a Maurer--Cartan element $mu$ in the pre-Lie algebra $mathfrak g := mathrmHom_mathbb S(mathsfcoLie,mathsfEnd_A)$, which is essentially the Harrison chain complex. The pre-Lie product $f star g$ of two Harrison chains is given by the composition
$$ mathsfcoLie to mathsfcoLie circ_(1) mathsfcoLie stackrelf circ gto mathsfEnd_A circ_(1) mathsfEnd_A to mathsfEnd_A.$$
We also have the associative version $mathfrak h := mathrmHom_mathbb S(mathsfcoAss,mathsfEnd_A)$ with analogously defined pre-Lie product. Via the map $mathsfcoAss to mathsfcoLie$ we can think of $mathfrak g$ as a pre-Lie subalgebra of $mathfrak h$. Now the dual of $phi$ induces a map $phi^ast colon mathfrak h to mathfrak g$, which is not in general a morphism of pre-Lie algebras. However, the fact that $phi$ is a map of infinitesimal bimodules is exactly the condition needed for $phi^ast$ to satisfy a "projection formula": for $f in mathfrak g subset mathfrak h$ and $g in mathfrak h$ we have $phi^ast(f star g) = f star phi^ast(g)$ and $phi^ast(g star f) = phi^ast(g) star f$. Indeed if we write out the definitions of the pre-Lie product then the conditions we need for the projection formula to hold become exactly that we have a map of infinitesimal bi-comodules $mathsfcoLie to mathsfcoAss$.
In particular we have the Harrison differential on $mathfrak g$ given by $df = f star mu - (-1)^vert fvertmu star f$ and the analogous Hochschild differential on $mathfrak h$. The previous paragraph says in particular that the splitting $mathfrak h to mathfrak g$ is compatible with these differentials, so that Harrison chains are a direct summand of Hochschild chains.
answered 1 hour ago
Dan Petersen
24.7k269133
edited 12 mins ago
answered 1 hour ago
Dan Petersen
24.7k269133
answered 1 hour ago
Dan Petersen
24.7k269133
answered 1 hour ago
Dan Petersen
24.7k269133
24.7k269133
add a comment |Â
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