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nLab > Latest Changes: Lie algebra

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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeJul 25th 2014
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Author: Urs
Format: MarkdownItexadded to _[[Lie algebra]]_ a brief paragraph _[general abstract perspective](http://ncatlab.org/nlab/show/Lie%20algebra#GeneralAbstractPerspective)_ to go along with [this MO reply](http://mathoverflow.net/a/177072/381)

added to Lie algebra a brief paragraph general abstract perspective to go along with this MO reply
- CommentRowNumber2.
- CommentAuthorTodd_Trimble
- CommentTimeJul 30th 2014
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Author: Todd_Trimble
Format: MarkdownItexI was also looking at Qiaochu's answer in that MO thread, which I found very nice. Turning to the nLab, I found related material under [[differential graded coalgebra]] and [[differential graded Hopf algebra]], which looks nice, but I wanted to understand some of this at a simpler level first. Is there an adjunction $U \dashv P$ where $P$ takes a cocommutative Hopf algebra to the space of primitives (which I guess is a LIe algebra), and $U$ takes a Lie algebra to its universal enveloping algebra, viewed as a Hopf algebra?

I was also looking at Qiaochu’s answer in that MO thread, which I found very nice. Turning to the nLab, I found related material under differential graded coalgebra and differential graded Hopf algebra, which looks nice, but I wanted to understand some of this at a simpler level first.

Is there an adjunction $U \dashv P$ where $P$ takes a cocommutative Hopf algebra to the space of primitives (which I guess is a LIe algebra), and $U$ takes a Lie algebra to its universal enveloping algebra, viewed as a Hopf algebra?
- CommentRowNumber3.
- CommentAuthorTodd_Trimble
- CommentTimeJul 31st 2014
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Author: Todd_Trimble
Format: MarkdownItexThe answer to my last question is a straightforward 'yes'. It's amazing that such an easy and basic result isn't seen in the accounts I've looked at. I've added some more material to [[primitive elements]]. (The previous version was somewhat flawed because we really need to work in the context of *unital* coalgebras. It's a little surprising how many authors forget to mention this!)

The answer to my last question is a straightforward ’yes’. It’s amazing that such an easy and basic result isn’t seen in the accounts I’ve looked at.

I’ve added some more material to primitive elements. (The previous version was somewhat flawed because we really need to work in the context of unital coalgebras. It’s a little surprising how many authors forget to mention this!)
- CommentRowNumber4.
- CommentAuthorTim_Porter
- CommentTimeJul 31st 2014
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Author: Tim_Porter
Format: MarkdownItexI think I had seen something like that in some of the rational homotopy theory stuff and further back some related ideas in Quillen's rational homotopy theory paper, but was unable to check so did not mention it. Certainly something like that is mentioned in Tanré's lecture notes I seem to remember.

I think I had seen something like that in some of the rational homotopy theory stuff and further back some related ideas in Quillen’s rational homotopy theory paper, but was unable to check so did not mention it. Certainly something like that is mentioned in Tanré’s lecture notes I seem to remember.
- CommentRowNumber5.
- CommentAuthorMike Shulman
- CommentTimeAug 1st 2014
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Author: Mike Shulman
Format: MarkdownItex@Todd #3: The remarks before Theorem 22.3.1 in *More Concise Algebraic Topology* come tantalizingly close: > A quick calculation shows that the $R$-module $P A$ of primitive elemnts of a Hopf algebra $A$ is a Lie subalgebra. The universal property of $U(P A)$ thus gives a natural map of Hopf algebras $g:U(P A)\to A$.... Let $\mathcal{L}$ and $\mathcal{P H}$ denote the categories of Lie algebras and primitive Hopf algebras over $R$. We have functors $U:\mathcal{L}\to \mathcal{P H}$ and $P:\mathcal{P H} \to \mathcal{L}$, a natural inclusion $L\subset P(U L)$, and a natural epimorphism $g:U(P A) \to A$... This much would be true over any commutative ring $R$, but when $R$ is a field of characteristic zero we have the following result. > **Theorem 22.3.1**. The functors $U$ and $P$ are inverse equivalences of categories.

@Todd #3: The remarks before Theorem 22.3.1 in More Concise Algebraic Topology come tantalizingly close:

A quick calculation shows that the $R$ -module $P A$ of primitive elemnts of a Hopf algebra $A$ is a Lie subalgebra. The universal property of $U(P A)$ thus gives a natural map of Hopf algebras $g:U(P A)\to A$ …. Let $\mathcal{L}$ and $\mathcal{P H}$ denote the categories of Lie algebras and primitive Hopf algebras over $R$ . We have functors $U:\mathcal{L}\to \mathcal{P H}$ and $P:\mathcal{P H} \to \mathcal{L}$ , a natural inclusion $L\subset P(U L)$ , and a natural epimorphism $g:U(P A) \to A$ … This much would be true over any commutative ring $R$ , but when $R$ is a field of characteristic zero we have the following result.

Theorem 22.3.1. The functors $U$ and $P$ are inverse equivalences of categories.
- CommentRowNumber6.
- CommentAuthorTodd_Trimble
- CommentTimeAug 1st 2014
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Author: Todd_Trimble
Format: MarkdownItexYes, that's just it! "Tantalizingly close." (Admittedly, this is the closest I've seen yet, so thanks for mentioning! To me it makes sense just to come out and state the adjunction, which holds in maximal generality, and then make a more refined analysis that examine the unit and counit separately under more specific hypotheses.)

Yes, that’s just it! “Tantalizingly close.” (Admittedly, this is the closest I’ve seen yet, so thanks for mentioning! To me it makes sense just to come out and state the adjunction, which holds in maximal generality, and then make a more refined analysis that examine the unit and counit separately under more specific hypotheses.)
- CommentRowNumber7.
- CommentAuthorMike Shulman
- CommentTimeAug 1st 2014
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Author: Mike Shulman
Format: MarkdownItexYeah, to me too.

Yeah, to me too.
- CommentRowNumber8.
- CommentAuthorzskoda
- CommentTimeAug 1st 2014
- (edited Aug 1st 2014)
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Author: zskoda
Format: MarkdownItexThe commutative triangle of functors involving two adjunctions, one mentioned above, is stated at the beginning of the paper * [[Jean-Louis Loday]], [[Teimuraz Pirashvili]], _The tensor category of linear maps_, Georg. Math. J. [vol. 5](http://www.mat.ub.edu/EMIS/journals/GMJ/vol5/contents.htm), n.3 (1998) 263--276, [pdf.gz](http://www.mat.ub.edu/EMIS/journals/GMJ/vol5/v5n3-6.pdf.gz), [pdf](http://www.mat.ub.edu/EMIS/journals/GMJ/vol5/v5n3-6.pdf) quoted at [[Loday-Pirashvili category]].
The commutative triangle of functors involving two adjunctions, one mentioned above, is stated at the beginning of the paper
- Jean-Louis Loday, Teimuraz Pirashvili, The tensor category of linear maps, Georg. Math. J. vol. 5, n.3 (1998) 263–276, pdf.gz, pdf
quoted at Loday-Pirashvili category.
- CommentRowNumber9.
- CommentAuthorTodd_Trimble
- CommentTimeAug 1st 2014
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Author: Todd_Trimble
Format: MarkdownItexNice -- thank you Zoran!

Nice – thank you Zoran!
- CommentRowNumber10.
- CommentAuthorUrs
- CommentTimeNov 27th 2019
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Author: Urs
Format: MarkdownItextouched the section _[Definition -- Internal to a general linear category](https://ncatlab.org/nlab/show/Lie+algebra#InAGeneralLinearCategory)_ (added formatting, missing words, hyperlinks) Will be splitting this off now as a stand-alone entry _[[Lie algebra object]]_. <a href="https://ncatlab.org/nlab/revision/diff/Lie+algebra/46">diff</a>, <a href="https://ncatlab.org/nlab/revision/Lie+algebra/46">v46</a>, <a href="https://ncatlab.org/nlab/show/Lie+algebra">current</a>

touched the section Definition – Internal to a general linear category

(added formatting, missing words, hyperlinks)

Will be splitting this off now as a stand-alone entry Lie algebra object.

diff, v46, current
- CommentRowNumber11.
- CommentAuthorUrs
- CommentTimeNov 27th 2019
- PermaLink
Author: Urs
Format: MarkdownItextouched the section _[Definition -- Internal to a general linear category](https://ncatlab.org/nlab/show/Lie+algebra#InAGeneralLinearCategory)_ (added formatting, missing words, hyperlinks) Will be splitting this off now as a stand-alone entry _[[Lie algebra object]]_. <a href="https://ncatlab.org/nlab/revision/diff/Lie+algebra/46">diff</a>, <a href="https://ncatlab.org/nlab/revision/Lie+algebra/46">v46</a>, <a href="https://ncatlab.org/nlab/show/Lie+algebra">current</a>

touched the section Definition – Internal to a general linear category

(added formatting, missing words, hyperlinks)

Will be splitting this off now as a stand-alone entry Lie algebra object.

diff, v46, current
- CommentRowNumber12.
- CommentAuthorUrs
- CommentTimeJan 22nd 2021
- (edited Jan 22nd 2021)
- PermaLink
Author: Urs
Format: MarkdownItexIn private email somebody asks: Historically, who was the first to think of the Lie bracket as an _abstract_ algebraic operation, not defined through a commutator? <a href="https://ncatlab.org/nlab/revision/diff/Lie+algebra/50">diff</a>, <a href="https://ncatlab.org/nlab/revision/Lie+algebra/50">v50</a>, <a href="https://ncatlab.org/nlab/show/Lie+algebra">current</a>

In private email somebody asks:

Historically, who was the first to think of the Lie bracket as an abstract algebraic operation, not defined through a commutator?

diff, v50, current
- CommentRowNumber13.
- CommentAuthorUrs
- CommentTimeMar 20th 2023
- PermaLink
Author: Urs
Format: MarkdownItexadded pointer to: * [[Hans Duistermaat]], [[Johan A. C. Kolk]], Chapter 1 of: *Lie groups*, Springer (2000) [[doi:10.1007/978-3-642-56936-4](https://doi.org/10.1007/978-3-642-56936-4)] <a href="https://ncatlab.org/nlab/revision/diff/Lie+algebra/53">diff</a>, <a href="https://ncatlab.org/nlab/revision/Lie+algebra/53">v53</a>, <a href="https://ncatlab.org/nlab/show/Lie+algebra">current</a>
added pointer to:
- Hans Duistermaat, Johan A. C. Kolk, Chapter 1 of: Lie groups, Springer (2000) [doi:10.1007/978-3-642-56936-4]
diff, v53, current
- CommentRowNumber14.
- CommentAuthornLab edit announcer
- CommentTimeJul 10th 2023
- PermaLink
Author: nLab edit announcer
Format: MarkdownItexadded pointer to * [[Peter Woit]], Ch. 5 of *Quantum Theory, Groups and Representations, an Introduction*, Springer 2017 [[doi:10.1007/978-3-319-64612-1](https://doi.org/10.1007/978-3-319-64612-1), ISBN:978-3-319-64610-7)] Quinn <a href="https://ncatlab.org/nlab/revision/diff/Lie+algebra/55">diff</a>, <a href="https://ncatlab.org/nlab/revision/Lie+algebra/55">v55</a>, <a href="https://ncatlab.org/nlab/show/Lie+algebra">current</a>
added pointer to
- Peter Woit, Ch. 5 of Quantum Theory, Groups and Representations, an Introduction, Springer 2017 [doi:10.1007/978-3-319-64612-1, ISBN:978-3-319-64610-7)]
Quinn

diff, v55, current
- CommentRowNumber15.
- CommentAuthorUrs
- CommentTimeSep 7th 2023
- PermaLink
Author: Urs
Format: MarkdownItexadded pointer to * [[Howard Georgi]], *Lie Algebras In Particle Physics*, Westview Press (1999), CRC Press (2019) [[doi:10.1201/9780429499210](https://doi.org/10.1201/9780429499210)] (here and in related entries) <a href="https://ncatlab.org/nlab/revision/diff/Lie+algebra/56">diff</a>, <a href="https://ncatlab.org/nlab/revision/Lie+algebra/56">v56</a>, <a href="https://ncatlab.org/nlab/show/Lie+algebra">current</a>
added pointer to
- Howard Georgi, Lie Algebras In Particle Physics, Westview Press (1999), CRC Press (2019) [doi:10.1201/9780429499210]
(here and in related entries)

diff, v56, current
- CommentRowNumber16.
- CommentAuthorUrs
- CommentTimeSep 1st 2024
- PermaLink
Author: Urs
Format: MarkdownItexadded pointer to: * [[Jean-Pierre Serre]]: *Lie Algebras and Lie Groups -- 1964 Lectures given at Harvard University*, Lecture Notes in Mathematics **1500** (1992) [[doi:10.1007/978-3-540-70634-2](https://doi.org/10.1007/978-3-540-70634-2)] (here and in related entries) <a href="https://ncatlab.org/nlab/revision/diff/Lie+algebra/60">diff</a>, <a href="https://ncatlab.org/nlab/revision/Lie+algebra/60">v60</a>, <a href="https://ncatlab.org/nlab/show/Lie+algebra">current</a>
added pointer to:
- Jean-Pierre Serre: Lie Algebras and Lie Groups – 1964 Lectures given at Harvard University, Lecture Notes in Mathematics 1500 (1992) [doi:10.1007/978-3-540-70634-2]
(here and in related entries)

diff, v60, current
- CommentRowNumber17.
- CommentAuthorUrs
- CommentTimeDec 8th 2024
- (edited Dec 8th 2024)
- PermaLink
Author: Urs
Format: MarkdownItexadded pointer to: * [[Nathan Jacobson]]: *Lie Algebras*, Dover Books (1979) [[ISBN:9780486638324](https://store.doverpublications.com/products/9780486638324)] <a href="https://ncatlab.org/nlab/revision/diff/Lie+algebra/65">diff</a>, <a href="https://ncatlab.org/nlab/revision/Lie+algebra/65">v65</a>, <a href="https://ncatlab.org/nlab/show/Lie+algebra">current</a>
added pointer to:
- Nathan Jacobson: Lie Algebras, Dover Books (1979) [ISBN:9780486638324]
diff, v65, current

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nLab > Latest Changes: Lie algebra