5
$\begingroup$

Can someone point me to a reference where the notion of "Lie crossed module" appeared for the first time?

I see many papers "recall" the definition of the Lie crossed module but, I do not see any mention of a "first-time" reference.

The definition of Lie crossed module I am referring to is mentioned as Definition 1.3 in

$\endgroup$
9
  • $\begingroup$ I googled "Lie crossed module" in quotes. The first paper on these on the arXiv I saw was here: arxiv.org/abs/1609.09297 In its first paragraph it attributes the concept to Kassel and Loday, giving a reference to their 1982 paper (in French) "Extensions centrales d’algebres de Lie." $\endgroup$
    – mme
    Commented Mar 5, 2021 at 16:44
  • $\begingroup$ I see that they mention about "crossed modules of Lie algebras". I don't know if they mean crossed module of Lie groups. $\endgroup$ Commented Mar 5, 2021 at 16:50
  • 1
    $\begingroup$ OK, though it seems the notation is ambiguous and that wasn't in your question. A similar process (now I started on google scholar instead of google) leads me to "Classification of principal bundles and lie groupoids with prescribed gauge group bundle", Kirill Mackenzie, as the earliest reference with a definition of crossed module of lie groups. At a glance I see nothing earlier and Mackenzie writes as if it is a novel concept. The concept was probably reinvented more than once. $\endgroup$
    – mme
    Commented Mar 5, 2021 at 17:12
  • $\begingroup$ @PraphullaKoushik: Your question does not mention "crossed modules of Lie groups" nor "crossed modules of Lie algebras", that is the ambiguity. No harm, but your comment of 2 hours ago comes a little strong. $\endgroup$
    – F Zaldivar
    Commented Mar 5, 2021 at 20:14
  • $\begingroup$ @FZaldivar If I read my comment now it does sound harsh. Apologies. I will delete it. $\endgroup$ Commented Mar 6, 2021 at 2:16

2 Answers 2

6
$\begingroup$

Crossed modules of Lie algebras are defined by Kassel and Loday in Definition A.1 of Extensions centrales d’algèbres de Lie (published 1982).

Crossed modules of Lie groups are defined by Mackenzie in Definition 1.5 of Classification of principal bundles and lie groupoids with prescribed gauge group bundle. Definition 3.3 there defines crossed modules of Lie groupoids (published 1989).

Crossed modules of local Lie groupoids are defined by Brown and İçen in Definition 4.1 of Towards a 2-dimensional notion of holonomy (published 2003).

$\endgroup$
5
  • $\begingroup$ Thank you. How can we be sure that he introduced it. I do not see any paper before the one by Mackenzie that talks about Lie crossed module. $\endgroup$ Commented Mar 6, 2021 at 2:22
  • $\begingroup$ @PraphullaKoushik: No references to prior sources is given for the definition, which means it was likely introduced in that paper. $\endgroup$ Commented Mar 6, 2021 at 2:42
  • $\begingroup$ Loday's paper cites an older paper of his on crossed modules of groups, so that thread doesn't go back further. The 1960 paper Analytic group kernels and Lie algebra kernels by Macauley (cited by Mackenzie) almost defines a crossed module, since it takes some of the theory from Eileberg and Mac Lane where they use "abstract kernels", secretly related to crossed modules, and develops it in the case of analytic groups (pretty much just Lie groups, really) $\endgroup$
    – David Roberts
    Commented Mar 6, 2021 at 7:17
  • $\begingroup$ @DavidRoberts do you want to make that as an answer adding some more details? $\endgroup$ Commented Mar 6, 2021 at 12:42
  • $\begingroup$ @PraphullaKoushik done $\endgroup$
    – David Roberts
    Commented Mar 7, 2021 at 2:12
4
$\begingroup$

For some more context, Kassel and Loday's 1982 paper defining crossed modules of Lie algebras cites an early paper by Loday that discusses crossed modules of (ordinary) groups, so it would appear that this citation trail won't give you crossed modules of Lie groups. Elsewhere I've seen crossed modules mentioned in conjunction with "abstract kernels", which date back to Eilenberg and Mac Lane's work on classifying nonabelian extensions of groups (which is secretly controlled by non-abelian cohomology, and the explicit group cohomology $H^3$—with values in an abelian group—that they use also helps classify 2-groups): Cohomology theory in abstract groups. II. Group extensions with a non-Abelian kernel.

Mackenzie in his 1989 paper defining crossed modules of Lie groups cites a 1960 paper on analytic abstract kernels:

Analytic groups here can be thought of as at least a special case of Lie groups (at the very least, classical groups are analytic), and the Macauley paper deals with constructions that turn up in the study of crossed modules of Lie groups, like induced maps to outer automorphism groups. So one could say that this latter paper almost invented crossed modules of Lie groups. But it's not too surprising that it doesn't, since even Eilenberg and Mac Lane's Annals paper involving abstract kernels (from 1947) don't cite Whitehead, even though they are even closer in subject matter (Mac Lane and Whitehead collaborated later in 1950, producing On the 3-type of a complex, and there crossed modules turn up). I think it unlikely someone working with Lie groups would have been familiar with work in algebraic homotopy theory or algebraic topology so as to even know of Whitehead's and/or Mac Lane–Whitehead's work.

If Mackenzie, who was an early proponent of Lie groupoids, could only cite someone who is a near-miss for crossed modules of Lie groups, then it's safe to say his paper is probably the earliest. The only others I could have imagined independently developing the notion would be Jean Pradines (another Lie groupoidist, from the Ehresmann school) or perhaps someone in Ronnie Brown's orbit arriving at the idea of a topological crossed module via an analogue of the Brown–Spencer theorem relating groupoids internal to $\mathbf{Grp}$ and crossed modules), but I briefly checked for the latter and couldn't find anything.

$\endgroup$
1
  • $\begingroup$ thank you....:) $\endgroup$ Commented Mar 7, 2021 at 3:26

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .