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nLab > Latest Changes: L-infinity-algebra

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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeNov 19th 2009
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Author: Urs
Format: Markdownexpanded [[L-infinity-algebra]] as indicated <a href="http://golem.ph.utexas.edu/category/2009/11/courant_algebroids_from_catego.html#c029383">on the nCafe, here</a>

expanded L-infinity-algebra as indicated on the nCafe, here
- CommentRowNumber2.
- CommentAuthorUrs
- CommentTimeAug 8th 2011
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Author: Urs
Format: MarkdownItexI have finally added the statement that every $L_\infty$-algebra is quasi-isomorphic to a dg-Lie algebra (in char 0), together with a pointer to Kriz-May. Also at [[dg-Lie algebra]]

I have finally added the statement that every $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>$ -algebra is quasi-isomorphic to a dg-Lie algebra (in char 0), together with a pointer to Kriz-May. Also at dg-Lie algebra
- CommentRowNumber3.
- CommentAuthorzskoda
- CommentTimeAug 8th 2011
- (edited Aug 8th 2011)
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Author: zskoda
Format: MarkdownItexUrs, do you have any thoughts in the parallel discussion on the very related question I opened under [Magnus infinity algebra](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=2998) ?

Urs, do you have any thoughts in the parallel discussion on the very related question I opened under Magnus infinity algebra ?
- CommentRowNumber4.
- CommentAuthorUrs
- CommentTimeSep 11th 2011
- (edited Sep 11th 2011)
- PermaLink
Author: Urs
Format: MarkdownItexI have changed at [[L-infinity algebra]] the "skew symmetric" brackets to "graded-symmetric", for clarity. I am having an email exchange with somebody trying to learn the basics of $L_\infty$-algebras. This highlights what is of course evident: the whole discussion here could do with more examples and more details. I don't have the time to look into this. But maybe I can convince my correspondent to include discussion of his newly gained insights into the $n$Lab entry, for the sake of those students coming after him...

I have changed at L-infinity algebra the “skew symmetric” brackets to “graded-symmetric”, for clarity.

I am having an email exchange with somebody trying to learn the basics of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>$ -algebras. This highlights what is of course evident: the whole discussion here could do with more examples and more details.

I don’t have the time to look into this. But maybe I can convince my correspondent to include discussion of his newly gained insights into the $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>$ Lab entry, for the sake of those students coming after him…
- CommentRowNumber5.
- CommentAuthorGuest
- CommentTimeSep 12th 2011
- PermaLink
Author: Guest
Format: Text@urs: I have changed at L-infinity algebra the "skew symmetric" brackets to "graded-symmetric", for clarity. I don't have time to check whether you are writing at the level of the underlying gr vector space or at the Chevalley Eilenberg shifted level graded symmetric is fine after the shift graded skew-symmetric at the unshifted level
@urs: I have changed at L-infinity algebra the "skew symmetric" brackets to "graded-symmetric", for clarity.

I don't have time to check whether you are writing at the level of the underlying gr vector space
or at the Chevalley Eilenberg shifted level
graded symmetric is fine after the shift
graded skew-symmetric at the unshifted level
- CommentRowNumber6.
- CommentAuthorUrs
- CommentTimeSep 12th 2011
- (edited Sep 12th 2011)
- PermaLink
Author: Urs
Format: MarkdownItexSince it says "$\mathbb{N}_+$-graded" it's shifted, hence "graded-symmetric" is correct. Maybe I find the time to expand and polish the whole entry. Or maybe somebody else does.

Since it says “ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ℕ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">\mathbb{N}_+</annotation></semantics></math>$ -graded” it’s shifted, hence “graded-symmetric” is correct.

Maybe I find the time to expand and polish the whole entry. Or maybe somebody else does.
- CommentRowNumber7.
- CommentAuthorUrs
- CommentTimeSep 12th 2011
- PermaLink
Author: Urs
Format: MarkdownItexOkay, I have filled in the signs of the "$L_\infty$-Jacobi identity" and then added a further section [Reformulation in terms of semifree dg-algebra -- Details](http://ncatlab.org/nlab/show/L-infinity-algebra#DGAlgebraDetails) with an explicit pedestrian walk through the proof that an $L_\infty$-algebra structure on $\mathfrak{g}$ is the same as a dg-algebra structure on $\wedge \mathfrak{g}^*$ (for the degreewise finite dimensional case).

Okay, I have filled in the signs of the “ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>$ -Jacobi identity” and then added a further section

Reformulation in terms of semifree dg-algebra – Details with an explicit pedestrian walk through the proof that an $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>$ -algebra structure on $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math>$ is the same as a dg-algebra structure on $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∧</mo><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\wedge \mathfrak{g}^*</annotation></semantics></math>$ (for the degreewise finite dimensional case).
- CommentRowNumber8.
- CommentAuthorUrs
- CommentTimeMay 28th 2015
- PermaLink
Author: Urs
Format: MarkdownItexI have substantially polished the section _[Definition -- In terms of higher brackets](http://ncatlab.org/nlab/show/L-infinity-algebra#DefinitionViaHigherBrackets)_.

I have substantially polished the section Definition – In terms of higher brackets.
- CommentRowNumber9.
- CommentAuthorUrs
- CommentTimeJan 12th 2016
- (edited Jan 12th 2016)
- PermaLink
Author: Urs
Format: MarkdownItexJim Stasheff suggested that the entry _[[L-infinity algebra]]_ ought to contain discussion of literature of $L_\infty$-algebras appearing in physics. I gather he would like to use such a list for his talk at "Higher structures in geometry and physics" in Bonn next week. Since we didn't have such a list yet, I have now [added one](https://ncatlab.org/nlab/show/L-infinity-algebra#ReferencesInPhysics). Naturally the examples that come to my mind tend to be those that I have worked on myself. Please feel invited to add more.

Jim Stasheff suggested that the entry L-infinity algebra ought to contain discussion of literature of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>$ -algebras appearing in physics. I gather he would like to use such a list for his talk at “Higher structures in geometry and physics” in Bonn next week. Since we didn’t have such a list yet, I have now added one. Naturally the examples that come to my mind tend to be those that I have worked on myself. Please feel invited to add more.
- CommentRowNumber10.
- CommentAuthorUrs
- CommentTimeJan 15th 2016
- PermaLink
Author: Urs
Format: MarkdownItexTogether with Jim Stasheff (via email) we have been further expanding the [list of references](https://ncatlab.org/nlab/show/L-infinity-algebra#ReferencesInPhysics) for $L_\infty$-algebras in physics. For ease of editing and linking, I have now split it off as a separate entry _[[L-infinity algebras in physics]]_.

Together with Jim Stasheff (via email) we have been further expanding the list of references for $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>$ -algebras in physics.

For ease of editing and linking, I have now split it off as a separate entry L-infinity algebras in physics.
- CommentRowNumber11.
- CommentAuthorUrs
- CommentTimeJan 20th 2016
- PermaLink
Author: Urs
Format: MarkdownItexHere are now the slides from Jim's talk yesterday: [Stasheff 16](https://ncatlab.org/nlab/show/L-infinity+algebras+in+physics#Stasheff16). Jim asks me to add that a more complete preprint is in preparation, and that meanwhile comments on the slides are most welcome.

Here are now the slides from Jim’s talk yesterday: Stasheff 16.

Jim asks me to add that a more complete preprint is in preparation, and that meanwhile comments on the slides are most welcome.
- CommentRowNumber12.
- CommentAuthorUrs
- CommentTimeJan 11th 2017
- PermaLink
Author: Urs
Format: MarkdownItexI have polished a little the section _[Definition -- In terms of higher brackets](https://ncatlab.org/nlab/show/L-infinity-algebra#DefinitionViaHigherBrackets)_, at _[[L-infinity algebra]]_. And in the section _[In terms of semifree differential coalgebra](https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInSemifreeDgCoalgebra)_ I have added text at the beginning highlighting that this reformulation was shown already in the original articles on $L_\infty$-algebras.

I have polished a little the section Definition – In terms of higher brackets, at L-infinity algebra.

And in the section In terms of semifree differential coalgebra I have added text at the beginning highlighting that this reformulation was shown already in the original articles on $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>$ -algebras.
- CommentRowNumber13.
- CommentAuthorUrs
- CommentTimeMar 8th 2017
- PermaLink
Author: Urs
Format: MarkdownItexadded a remark ([here](https://ncatlab.org/nlab/show/L-infinity-algebra#IndConilpotency)) amplifying the ind-conilpotency of $CE_\bullet(\mathfrak{g})$

added a remark (here) amplifying the ind-conilpotency of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>CE</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CE_\bullet(\mathfrak{g})</annotation></semantics></math>$
- CommentRowNumber14.
- CommentAuthorUrs
- CommentTimeApr 12th 2018
- PermaLink
Author: Urs
Format: MarkdownItexadded references [Buijs-Felix-Murillo 12](https://ncatlab.org/nlab/show/L-infinity-algebra#BuijsFelixMurillo12) and [Buijs-Murillo 12](https://ncatlab.org/nlab/show/L-infinity-algebra#BuijsMurillo12) on $L_\infty$-algebras as models for [[rational homotopy theory]] <a href="https://ncatlab.org/nlab/revision/diff/L-infinity-algebra/90">diff</a>, <a href="https://ncatlab.org/nlab/revision/L-infinity-algebra/90">v90</a>, <a href="https://ncatlab.org/nlab/show/L-infinity-algebra">current</a>

added references Buijs-Felix-Murillo 12 and Buijs-Murillo 12 on $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>$ -algebras as models for rational homotopy theory

diff, v90, current
- CommentRowNumber15.
- CommentAuthornLab edit announcer
- CommentTimeSep 10th 2018
- PermaLink
Author: nLab edit announcer
Format: MarkdownItexI have added the reference for the survey posted today on arxiv by Stasheff, but am not sure if the formatting (that I used in editing this article) is appropriate. Anonymous <a href="https://ncatlab.org/nlab/revision/diff/L-infinity-algebra/94">diff</a>, <a href="https://ncatlab.org/nlab/revision/L-infinity-algebra/94">v94</a>, <a href="https://ncatlab.org/nlab/show/L-infinity-algebra">current</a>

I have added the reference for the survey posted today on arxiv by Stasheff, but am not sure if the formatting (that I used in editing this article) is appropriate.

Anonymous

diff, v94, current
- CommentRowNumber16.
- CommentAuthorGuest
- CommentTimeSep 27th 2019
- PermaLink
Author: Guest
Format: TextI'm unsure as I'm using this page to learn what a L-infinity algebra is, but shouldn't the definition 3.2 "In terms of higher brackets" of a L-infinity algebra use a n-tuple and a (i,n-i) unshuffle instead of a (n+1)-tuple and a (i,j) unshuffle?
I'm unsure as I'm using this page to learn what a L-infinity algebra is, but shouldn't the definition 3.2 "In terms of higher brackets" of a L-infinity algebra use a n-tuple and a (i,n-i) unshuffle instead of a (n+1)-tuple and a (i,j) unshuffle?
- CommentRowNumber17.
- CommentAuthormrmuon
- CommentTimeNov 26th 2019
- PermaLink
Author: mrmuon
Format: MarkdownItexThe statement of the strong homotopy Jacobi identity contains some errors. Note that there are $n$ $v$'s in the equation. This is correct, since a composition of $l_j$ with $l_i$ should take $n=i+j-1$ arguments. For this reason, it should say "for all $n$-tuples" (not $n+1$). I believe the inner sum should be over $(i,j-1)$-unshuffles, since $(i,j)$-unshuffles would permute $n+1$ objects. A similar error occurs when Loday and Valette try to state this identity for the operad. I am actually struggling to find the correct identity written down with signs anywhere.

The statement of the strong homotopy Jacobi identity contains some errors. Note that there are $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math>$ ’s in the equation. This is correct, since a composition of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>l</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">l_j</annotation></semantics></math>$ with $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>l</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">l_i</annotation></semantics></math>$ should take $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mi>i</mi><mo>+</mo><mi>j</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n=i+j-1</annotation></semantics></math>$ arguments. For this reason, it should say “for all $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>$ -tuples” (not $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math>$ ). I believe the inner sum should be over $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i,j-1)</annotation></semantics></math>$ -unshuffles, since $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i,j)</annotation></semantics></math>$ -unshuffles would permute $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math>$ objects.

A similar error occurs when Loday and Valette try to state this identity for the operad. I am actually struggling to find the correct identity written down with signs anywhere.
- CommentRowNumber18.
- CommentAuthorUrs
- CommentTimeNov 26th 2019
- PermaLink
Author: Urs
Format: MarkdownItexI don't have the leisure to check now. But if/where it has "$n+1$" when it should read "$n$" and you are sure you know what you are doing: Please just fix it! (Just hit "edit" at the bottom of the page).

I don’t have the leisure to check now. But if/where it has “ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math>$ ” when it should read “ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>$ ” and you are sure you know what you are doing: Please just fix it! (Just hit “edit” at the bottom of the page).
- CommentRowNumber19.
- CommentAuthormrmuon
- CommentTimeNov 26th 2019
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Author: mrmuon
Format: MarkdownItexI corrected the aforementioned error. This is my first edit, so I hope I haven't broken anything. <a href="https://ncatlab.org/nlab/revision/diff/L-infinity-algebra/97">diff</a>, <a href="https://ncatlab.org/nlab/revision/L-infinity-algebra/97">v97</a>, <a href="https://ncatlab.org/nlab/show/L-infinity-algebra">current</a>

I corrected the aforementioned error. This is my first edit, so I hope I haven’t broken anything.

diff, v97, current
- CommentRowNumber20.
- CommentAuthorUrs
- CommentTimeNov 26th 2019
- PermaLink
Author: Urs
Format: MarkdownItexLooks good. Thanks!

Looks good. Thanks!
- CommentRowNumber21.
- CommentAuthorDavidRoberts
- CommentTimeJul 24th 2021
- (edited Jul 24th 2021)
- PermaLink
Author: DavidRoberts
Format: MarkdownItex>etc (where the $\vee$ is just a funny way to write the wedge $\wedge$, in order to remind us that:…) remind us what, exactly? >work out or see the references for the signs and prefacors better to see this on the actual page >exercise for the reader: spell this all out more in detail with all the signs and everyrthing. Possibly by looking it up in the references given below ditto

etc (where the $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∨</mo></mrow><annotation encoding="application/x-tex">\vee</annotation></semantics></math>$ is just a funny way to write the wedge $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∧</mo></mrow><annotation encoding="application/x-tex">\wedge</annotation></semantics></math>$ , in order to remind us that:…)

remind us what, exactly?

work out or see the references for the signs and prefacors

better to see this on the actual page

exercise for the reader: spell this all out more in detail with all the signs and everyrthing. Possibly by looking it up in the references given below

ditto

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nLab > Latest Changes: L-infinity-algebra