Is there any Lie algebra structure on the sheaf of sections of adjoint bundle












3














Let $X$ be an irreducible smooth projective variety over $mathbb{C}$.
Let $G$ be an affine algebraic group over $mathbb{C}$.
Let $p : E_G longrightarrow X$ be a holomorphic principal $G$-bundle on $X$. Let $ad(E_G) = E_G times^G mathfrak{g}$ be the adjoint vector bundle of $E_G$ associated to the adjoint representation $ad : G longrightarrow End(mathfrak{g})$ of $G$ on its Lie algebra $mathfrak{g}$. The fibers of $ad(E_G)$ are $mathbb{C}$-linearly isomorphic to $mathfrak{g}$.
Consider $ad(E_G)$ as a sheaf of $mathcal{O}_X$-modules on $X$.



Question: Is there any $mathcal{O}_X$-bilinear homomorphism
$[,] : ad(E_G)times ad(E_G) to ad(E_G)$ giving a Lie algebra structure on the sheaf $ad(E_G)$?






ag.algebraic-geometry principal-bundles







share|cite|improve this question



























    3














    Let $X$ be an irreducible smooth projective variety over $mathbb{C}$.
    Let $G$ be an affine algebraic group over $mathbb{C}$.
    Let $p : E_G longrightarrow X$ be a holomorphic principal $G$-bundle on $X$. Let $ad(E_G) = E_G times^G mathfrak{g}$ be the adjoint vector bundle of $E_G$ associated to the adjoint representation $ad : G longrightarrow End(mathfrak{g})$ of $G$ on its Lie algebra $mathfrak{g}$. The fibers of $ad(E_G)$ are $mathbb{C}$-linearly isomorphic to $mathfrak{g}$.
    Consider $ad(E_G)$ as a sheaf of $mathcal{O}_X$-modules on $X$.



    Question: Is there any $mathcal{O}_X$-bilinear homomorphism
    $[,] : ad(E_G)times ad(E_G) to ad(E_G)$ giving a Lie algebra structure on the sheaf $ad(E_G)$?






    ag.algebraic-geometry principal-bundles







    share|cite|improve this question

























      3












      3








      3







      Let $X$ be an irreducible smooth projective variety over $mathbb{C}$.
      Let $G$ be an affine algebraic group over $mathbb{C}$.
      Let $p : E_G longrightarrow X$ be a holomorphic principal $G$-bundle on $X$. Let $ad(E_G) = E_G times^G mathfrak{g}$ be the adjoint vector bundle of $E_G$ associated to the adjoint representation $ad : G longrightarrow End(mathfrak{g})$ of $G$ on its Lie algebra $mathfrak{g}$. The fibers of $ad(E_G)$ are $mathbb{C}$-linearly isomorphic to $mathfrak{g}$.
      Consider $ad(E_G)$ as a sheaf of $mathcal{O}_X$-modules on $X$.



      Question: Is there any $mathcal{O}_X$-bilinear homomorphism
      $[,] : ad(E_G)times ad(E_G) to ad(E_G)$ giving a Lie algebra structure on the sheaf $ad(E_G)$?






      ag.algebraic-geometry principal-bundles







      share|cite|improve this question













      Let $X$ be an irreducible smooth projective variety over $mathbb{C}$.
      Let $G$ be an affine algebraic group over $mathbb{C}$.
      Let $p : E_G longrightarrow X$ be a holomorphic principal $G$-bundle on $X$. Let $ad(E_G) = E_G times^G mathfrak{g}$ be the adjoint vector bundle of $E_G$ associated to the adjoint representation $ad : G longrightarrow End(mathfrak{g})$ of $G$ on its Lie algebra $mathfrak{g}$. The fibers of $ad(E_G)$ are $mathbb{C}$-linearly isomorphic to $mathfrak{g}$.
      Consider $ad(E_G)$ as a sheaf of $mathcal{O}_X$-modules on $X$.



      Question: Is there any $mathcal{O}_X$-bilinear homomorphism
      $[,] : ad(E_G)times ad(E_G) to ad(E_G)$ giving a Lie algebra structure on the sheaf $ad(E_G)$?






      ag.algebraic-geometry principal-bundles




      ag.algebraic-geometry principal-bundles






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 10 at 15:37









      Anonymous

      1286




      1286






















          2 Answers
          2






          active

          oldest

          votes


















          5














          A principal $G$-bundle gives a monoidal functor from the category of representations of $G$ to the category of vector bundles. In particular, it takes the morphism
          $$
          [-,-] colon mathfrak{g} otimes mathfrak{g} to mathfrak{g}
          $$
          of $G$-representations (for the adjoint action) to a morphism of vector bundles
          $$
          [-,-] colon ad(E_G) otimes ad(E_G) to ad(E_G).
          $$
          By functoriality, it is skew-symmetric and satisfies the Jacobi identity, hence provides the sheaf $ad(E_G)$ with a Lie algebra structure.






          share|cite|improve this answer





















          • Can you please show how the Jacobi identity follows from functoriality?
            – Vít Tuček
            Dec 10 at 16:25










          • @VítTuček: The Jacobian identity says that the sum of three maps $ad(E_G) otimes ad(E_G) otimes ad(E_G) to ad(E_G)$ vanishes. These maps come from three maps $mathfrak{g} otimes mathfrak{g} otimes mathfrak{g} to mathfrak{g}$. The sum of the latter maps is zero, hence so is the sum of the former maps.
            – Sasha
            Dec 10 at 16:41










          • So monoidal functors are automatically additive?
            – Vít Tuček
            Dec 10 at 17:25










          • No, certainly not. You need additivity and the functor also needs to be symmetric monoidal, not just monoidal, to preserve skew-symmetry and the Jacobi identity.
            – Qiaochu Yuan
            Dec 10 at 21:38



















          2














          Yes. It boils down to natural isomorphism $ad(E_G) otimes ad(E_G) simeq E_G times^G (mathfrak{g}otimes mathfrak{g})$ which allows you to compose tensor product of sections with the bracket on $mathfrak{g}otimes mathfrak{g}$.






          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',
            autoActivateHeartbeat: false,
            convertImagesToLinks: true,
            noModals: true,
            showLowRepImageUploadWarning: true,
            reputationToPostImages: 10,
            bindNavPrevention: true,
            postfix: "",
            imageUploader: {
            brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
            contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
            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%2f317337%2fis-there-any-lie-algebra-structure-on-the-sheaf-of-sections-of-adjoint-bundle%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown

























            2 Answers
            2






            active

            oldest

            votes








            2 Answers
            2






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            5














            A principal $G$-bundle gives a monoidal functor from the category of representations of $G$ to the category of vector bundles. In particular, it takes the morphism
            $$
            [-,-] colon mathfrak{g} otimes mathfrak{g} to mathfrak{g}
            $$
            of $G$-representations (for the adjoint action) to a morphism of vector bundles
            $$
            [-,-] colon ad(E_G) otimes ad(E_G) to ad(E_G).
            $$
            By functoriality, it is skew-symmetric and satisfies the Jacobi identity, hence provides the sheaf $ad(E_G)$ with a Lie algebra structure.






            share|cite|improve this answer





















            • Can you please show how the Jacobi identity follows from functoriality?
              – Vít Tuček
              Dec 10 at 16:25










            • @VítTuček: The Jacobian identity says that the sum of three maps $ad(E_G) otimes ad(E_G) otimes ad(E_G) to ad(E_G)$ vanishes. These maps come from three maps $mathfrak{g} otimes mathfrak{g} otimes mathfrak{g} to mathfrak{g}$. The sum of the latter maps is zero, hence so is the sum of the former maps.
              – Sasha
              Dec 10 at 16:41










            • So monoidal functors are automatically additive?
              – Vít Tuček
              Dec 10 at 17:25










            • No, certainly not. You need additivity and the functor also needs to be symmetric monoidal, not just monoidal, to preserve skew-symmetry and the Jacobi identity.
              – Qiaochu Yuan
              Dec 10 at 21:38
















            5














            A principal $G$-bundle gives a monoidal functor from the category of representations of $G$ to the category of vector bundles. In particular, it takes the morphism
            $$
            [-,-] colon mathfrak{g} otimes mathfrak{g} to mathfrak{g}
            $$
            of $G$-representations (for the adjoint action) to a morphism of vector bundles
            $$
            [-,-] colon ad(E_G) otimes ad(E_G) to ad(E_G).
            $$
            By functoriality, it is skew-symmetric and satisfies the Jacobi identity, hence provides the sheaf $ad(E_G)$ with a Lie algebra structure.






            share|cite|improve this answer





















            • Can you please show how the Jacobi identity follows from functoriality?
              – Vít Tuček
              Dec 10 at 16:25










            • @VítTuček: The Jacobian identity says that the sum of three maps $ad(E_G) otimes ad(E_G) otimes ad(E_G) to ad(E_G)$ vanishes. These maps come from three maps $mathfrak{g} otimes mathfrak{g} otimes mathfrak{g} to mathfrak{g}$. The sum of the latter maps is zero, hence so is the sum of the former maps.
              – Sasha
              Dec 10 at 16:41










            • So monoidal functors are automatically additive?
              – Vít Tuček
              Dec 10 at 17:25










            • No, certainly not. You need additivity and the functor also needs to be symmetric monoidal, not just monoidal, to preserve skew-symmetry and the Jacobi identity.
              – Qiaochu Yuan
              Dec 10 at 21:38














            5












            5








            5






            A principal $G$-bundle gives a monoidal functor from the category of representations of $G$ to the category of vector bundles. In particular, it takes the morphism
            $$
            [-,-] colon mathfrak{g} otimes mathfrak{g} to mathfrak{g}
            $$
            of $G$-representations (for the adjoint action) to a morphism of vector bundles
            $$
            [-,-] colon ad(E_G) otimes ad(E_G) to ad(E_G).
            $$
            By functoriality, it is skew-symmetric and satisfies the Jacobi identity, hence provides the sheaf $ad(E_G)$ with a Lie algebra structure.






            share|cite|improve this answer












            A principal $G$-bundle gives a monoidal functor from the category of representations of $G$ to the category of vector bundles. In particular, it takes the morphism
            $$
            [-,-] colon mathfrak{g} otimes mathfrak{g} to mathfrak{g}
            $$
            of $G$-representations (for the adjoint action) to a morphism of vector bundles
            $$
            [-,-] colon ad(E_G) otimes ad(E_G) to ad(E_G).
            $$
            By functoriality, it is skew-symmetric and satisfies the Jacobi identity, hence provides the sheaf $ad(E_G)$ with a Lie algebra structure.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Dec 10 at 15:51









            Sasha

            20.2k22654




            20.2k22654












            • Can you please show how the Jacobi identity follows from functoriality?
              – Vít Tuček
              Dec 10 at 16:25










            • @VítTuček: The Jacobian identity says that the sum of three maps $ad(E_G) otimes ad(E_G) otimes ad(E_G) to ad(E_G)$ vanishes. These maps come from three maps $mathfrak{g} otimes mathfrak{g} otimes mathfrak{g} to mathfrak{g}$. The sum of the latter maps is zero, hence so is the sum of the former maps.
              – Sasha
              Dec 10 at 16:41










            • So monoidal functors are automatically additive?
              – Vít Tuček
              Dec 10 at 17:25










            • No, certainly not. You need additivity and the functor also needs to be symmetric monoidal, not just monoidal, to preserve skew-symmetry and the Jacobi identity.
              – Qiaochu Yuan
              Dec 10 at 21:38


















            • Can you please show how the Jacobi identity follows from functoriality?
              – Vít Tuček
              Dec 10 at 16:25










            • @VítTuček: The Jacobian identity says that the sum of three maps $ad(E_G) otimes ad(E_G) otimes ad(E_G) to ad(E_G)$ vanishes. These maps come from three maps $mathfrak{g} otimes mathfrak{g} otimes mathfrak{g} to mathfrak{g}$. The sum of the latter maps is zero, hence so is the sum of the former maps.
              – Sasha
              Dec 10 at 16:41










            • So monoidal functors are automatically additive?
              – Vít Tuček
              Dec 10 at 17:25










            • No, certainly not. You need additivity and the functor also needs to be symmetric monoidal, not just monoidal, to preserve skew-symmetry and the Jacobi identity.
              – Qiaochu Yuan
              Dec 10 at 21:38
















            Can you please show how the Jacobi identity follows from functoriality?
            – Vít Tuček
            Dec 10 at 16:25




            Can you please show how the Jacobi identity follows from functoriality?
            – Vít Tuček
            Dec 10 at 16:25












            @VítTuček: The Jacobian identity says that the sum of three maps $ad(E_G) otimes ad(E_G) otimes ad(E_G) to ad(E_G)$ vanishes. These maps come from three maps $mathfrak{g} otimes mathfrak{g} otimes mathfrak{g} to mathfrak{g}$. The sum of the latter maps is zero, hence so is the sum of the former maps.
            – Sasha
            Dec 10 at 16:41




            @VítTuček: The Jacobian identity says that the sum of three maps $ad(E_G) otimes ad(E_G) otimes ad(E_G) to ad(E_G)$ vanishes. These maps come from three maps $mathfrak{g} otimes mathfrak{g} otimes mathfrak{g} to mathfrak{g}$. The sum of the latter maps is zero, hence so is the sum of the former maps.
            – Sasha
            Dec 10 at 16:41












            So monoidal functors are automatically additive?
            – Vít Tuček
            Dec 10 at 17:25




            So monoidal functors are automatically additive?
            – Vít Tuček
            Dec 10 at 17:25












            No, certainly not. You need additivity and the functor also needs to be symmetric monoidal, not just monoidal, to preserve skew-symmetry and the Jacobi identity.
            – Qiaochu Yuan
            Dec 10 at 21:38




            No, certainly not. You need additivity and the functor also needs to be symmetric monoidal, not just monoidal, to preserve skew-symmetry and the Jacobi identity.
            – Qiaochu Yuan
            Dec 10 at 21:38











            2














            Yes. It boils down to natural isomorphism $ad(E_G) otimes ad(E_G) simeq E_G times^G (mathfrak{g}otimes mathfrak{g})$ which allows you to compose tensor product of sections with the bracket on $mathfrak{g}otimes mathfrak{g}$.






            share|cite|improve this answer


























              2














              Yes. It boils down to natural isomorphism $ad(E_G) otimes ad(E_G) simeq E_G times^G (mathfrak{g}otimes mathfrak{g})$ which allows you to compose tensor product of sections with the bracket on $mathfrak{g}otimes mathfrak{g}$.






              share|cite|improve this answer
























                2












                2








                2






                Yes. It boils down to natural isomorphism $ad(E_G) otimes ad(E_G) simeq E_G times^G (mathfrak{g}otimes mathfrak{g})$ which allows you to compose tensor product of sections with the bracket on $mathfrak{g}otimes mathfrak{g}$.






                share|cite|improve this answer












                Yes. It boils down to natural isomorphism $ad(E_G) otimes ad(E_G) simeq E_G times^G (mathfrak{g}otimes mathfrak{g})$ which allows you to compose tensor product of sections with the bracket on $mathfrak{g}otimes mathfrak{g}$.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Dec 10 at 16:24









                Vít Tuček

                4,99711748




                4,99711748






























                    draft saved

                    draft discarded




















































                    Thanks for contributing an answer to MathOverflow!


                    • Please be sure to answer the question. Provide details and share your research!

                    But avoid …



                    • Asking for help, clarification, or responding to other answers.

                    • Making statements based on opinion; back them up with references or personal experience.


                    Use MathJax to format equations. MathJax reference.


                    To learn more, see our tips on writing great answers.





                    Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


                    Please pay close attention to the following guidance:


                    • Please be sure to answer the question. Provide details and share your research!

                    But avoid …



                    • Asking for help, clarification, or responding to other answers.

                    • Making statements based on opinion; back them up with references or personal experience.


                    To learn more, see our tips on writing great answers.




                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function () {
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f317337%2fis-there-any-lie-algebra-structure-on-the-sheaf-of-sections-of-adjoint-bundle%23new-answer', 'question_page');
                    }
                    );

                    Post as a guest















                    Required, but never shown





















































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown

































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown







                    -

                    Popular posts from this blog

                    flock() on closed filehandle LOCK_FILE at /usr/bin/apt-mirror

                    Image
                    0 I recently upgraded an Ubuntu 16.04 virtual machine to 18.04. I regularly use this VM to run apt-mirror at school on their high-speed internet, then rsync these files to my rack-mount server at home for updating multiple computers at home without wasting the tiny amount of bandwidth I have there. Since updating I can't get apt-mirror to run on my VM. It keeps giving the error: flock() on closed filehandle LOCK_FILE at /usr/bin/apt-mirror line 214 immediately followed by: apt-mirror is already running, exiting at /usr/bin/apt-mirror line 217 apt-mirror is NOT running, and does not have a cron job. I've tried rebooting, and even looked for the old-style lockfile to delete, but I guess that method is no longer used. Is there some way to force unlock this so I can run apt-mirror? UPDATE: So... I f
                    Read more

                    Mangá

                    Image
                      Nota:  Não confundir com Manga. Mangás e Animes Mangá Mangaká • Tankōbon Film comic La nouvelle manga Mangá original em inglês Anime História Animação influenciada por animes • Filler • OVA • Seiyū Públicos Kodomo • Shōjo • Shōnen Josei • Seinen Gêneros Mahō shōjo • Mecha • Harém Ecchi • Hentai ( Yaoi • Yuri Futanari • Shotacon • Lolicon Toddlercon ) Termos Bishōjo • Chibi • Fan service Gaiden • Kawaii • Kaitō • Light novel Mahō shōnen • Moe (antropomorfismo • Kemonomimi Nekomimi ) • Super deformed Omake • Tsundere • Yonkoma Listas Mangás • Animes • Ecchi • Mangaka • Estúdios • Revistas Fãs ( otaku ) Anime Music Video Convenções • Cosplay Dōjinshi • Fansub • Fandub • Fujoshi OS-tan • Scanlation Portal de Anime e mangá v d e O mangá (português brasileiro) ou manga (português europeu) [ 1 ] (em japonês: 漫画 , manga ? , lit. “história em quadrinhos”) , é a
                    Read more

                     ⁒  ․,‪⁊‑⁙ ⁖, ⁇‒※‌, †,⁖‗‌⁝    ‾‸⁘,‖⁔⁣,⁂‾
”‑,‥–,‬ ,⁀‹⁋‴⁑ ‒ ,‴⁋”‼ ⁨,‷⁔„ ‰′,‐‚ ‥‡‎“‷⁃⁨⁅⁣,⁔
⁇‘⁔⁡⁏⁌⁡‿‶‏⁨ ⁣⁕⁖⁨⁩⁥‽⁀  ‴‬⁜‟ ⁃‣‧⁕‮ …‍⁨‴ ⁩,⁚⁖‫ ,‵ ⁀,‮⁝‣‣ ⁑  ⁂– ․, ‾‽ ‏⁁“⁗‸ ‾… ‹‡⁌⁎‸‘ ‡⁏⁌‪ ‵⁛ ‎⁨ ―⁦⁤⁄⁕

                    ⁉…‧ ⁩,“⁡‹⁃‽⁞•,⁠‚  ⁓⁎  ⁜ ‴ ⁥⁡⁎‭,‍‑⁁⁌⁦ ⁁‟⁙‥‒“⁈ ,‪⁧ ⁙‿⁇⁦,⁝‘ ⁎“⁘‫‪,‗‹‛‹⁒  ⁇⁄‗‒•“⁇   ‐ ⁂
‘‛, ‶⁦ ⁇ ​⁞‧ 
,⁏ …‹–⁗⁡⁘⁄ ‏‿‬‟″ ‏⁢⁦⁗⁚⁝‬⁢⁏,⁧″⁍′‖⁂※⁄‘– ‵⁡ ‭⁢⁚,⁅‧,‼‛⁗,⁡⁗“⁔⁃⁍‷⁏‗‱⁌‚  
•⁝ ‚‒ ‥‰  ′⁌, ⁄, ⁛—‭‑”⁂,⁑ ‡⁨‴⁖⁙…,‑, ,⁎,․‣‥  ‴‎⁎,″‛⁎‭‿ ⁤ ‷‾‼‖,⁠⁊‾,⁦“‗‑‌†,⁙,†⁝′ 
,―‚․‘, ⁢  ‿‌,⁀„‣,⁌,‭‖ ⁆“‪,―  ⁀ ⁃,
‷⁥,​,‟‗⁄⁉⁐⁗―
,‮⁢⁩⁆⁣‹‭⁅ ⁝‾‏‭⁖ ⁕⁄‷,†‟‒ ⁣‹,​…​,⁤⁂⁨‮⁎⁙–‟‼ ⁍‱‱,‽,‫‽‣ ⁒,⁨ ⁄ ⁝”„‿‑‎⁈‿
 ⁀‛‼‟ ⁑“,⁩​ ‒
–⁡…⁢ ⁂ ⁈⁎,‵‡,⁛,,‱⁧⁋,․‬⁄ “,⁄‴⁊  ‗―,‵⁙⁔—⁀ —  ⁀‬‬ ‒⁋‼⁀― ,⁌′ ‍‶ ’
                    Read more