“Exactness” of operadic cohomology

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
5
down vote

favorite
1












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.)










share|cite|improve this question

























    up vote
    5
    down vote

    favorite
    1












    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.)










    share|cite|improve this question























      up vote
      5
      down vote

      favorite
      1









      up vote
      5
      down vote

      favorite
      1






      1





      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.)










      share|cite|improve this question













      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






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked 3 hours ago









      Denis T.

      861514




      861514




















          1 Answer
          1






          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.






          share|cite|improve this answer






















            Your Answer





            StackExchange.ifUsing("editor", function ()
            return StackExchange.using("mathjaxEditing", function ()
            StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
            StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
            );
            );
            , "mathjax-editing");

            StackExchange.ready(function()
            var channelOptions =
            tags: "".split(" "),
            id: "504"
            ;
            initTagRenderer("".split(" "), "".split(" "), channelOptions);

            StackExchange.using("externalEditor", function()
            // Have to fire editor after snippets, if snippets enabled
            if (StackExchange.settings.snippets.snippetsEnabled)
            StackExchange.using("snippets", function()
            createEditor();
            );

            else
            createEditor();

            );

            function createEditor()
            StackExchange.prepareEditor(
            heartbeatType: 'answer',
            convertImagesToLinks: true,
            noModals: false,
            showLowRepImageUploadWarning: true,
            reputationToPostImages: 10,
            bindNavPrevention: true,
            postfix: "",
            imageUploader:
            brandingHtml: "",
            contentPolicyHtml: "",
            allowUrls: true
            ,
            noCode: true, onDemand: true,
            discardSelector: ".discard-answer"
            ,immediatelyShowMarkdownHelp:true
            );



            );













             

            draft saved


            draft discarded


















            StackExchange.ready(
            function ()
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f314250%2fexactness-of-operadic-cohomology%23new-answer', 'question_page');

            );

            Post as a guest






























            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.






            share|cite|improve this answer


























              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.






              share|cite|improve this answer
























                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.






                share|cite|improve this answer














                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.







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited 12 mins ago

























                answered 1 hour ago









                Dan Petersen

                24.7k269133




                24.7k269133



























                     

                    draft saved


                    draft discarded















































                     


                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function ()
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f314250%2fexactness-of-operadic-cohomology%23new-answer', 'question_page');

                    );

                    Post as a guest













































































                    Comments

                    Popular posts from this blog

                    What does second last employer means? [closed]

                    Installing NextGIS Connect into QGIS 3?

                    One-line joke