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I was reading the abstract of a recent preprint (Division Algebras and Supersymmetry III by Juhn Huerta), and I wondered if something much simpler than what he was talking about had been worked on: have Lie $2$-groups been applied to the resolution of differential equations, in the same manner that Lie groups originated from the study of differential equations?

In other words, do Lie 2-groups arise as symmetries for (certain kinds of) differential equations, and can these in turn be used for the integration/resolution of those same differential equations? If they do not, then in what setting can a 2-group be understood as a symmetry (if any), and to what 'use' can this information be put?

My motivation here is to expand the toolset I can use to solve problems in classical analysis (like differential equations), and not to explore the other areas where Lie groups have developed in to (like Lie algebras and their classification, etc). For the purposes of this question, these issues are out-of-scope. In-scope are applications (to classical analysis) of generalizations going all the way to $\infty$-Lie groupoids.

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It seems to me that the role of symmetries for solving differential equations is not much different from the role of symmetries for solving systems of algebraic equations. So maybe (at least to me) it would make sense to first ask the easier question: what are "higher" symmetries of a variety, and how can they be used to solve algebraic equations? – Michael Bächtold Sep 21 '11 at 9:46
@Michael: Do you have references for the explicit use of symmetries for solving systems of algebraic equations? The most trivial ones (i.e. when a system factors) are used, but work with other symmetries actually being leveraged is rarer. By this I mean that I have not seen symmetries show up in Groebner basis algorithms very often, except in the 4 papers I linked to here:…. What did I miss? – Jacques Carette Sep 21 '11 at 18:23
@Michael: Lets my comment is mis-interpreted - I agree with you, I just don't know if this work has already been done for symmetries in the multivariate case. – Jacques Carette Sep 21 '11 at 18:25
Jacques, sorry I don't have any references for the use of symmetries in solving algebraic equations and my comment was more of a spontaneous reaction after being intrigued by your question and Urs answer (which is also beyond my current understanding). My vague reasoning was that algebraic equations are a very degenerate case of differential equation (zero independent variables, or zero order). So if the question makes sense for differential equations it should also make sense for algebraic ones. – Michael Bächtold Sep 21 '11 at 20:48
The link between differential equations and algebraic equations is very subtle - see for example the preprint by Chen and Kauers for a hint. And algebraic equations are not 'degenerate' differential equations as they are non-linear in the dependent variable, while DEs are 'linear' (in an appropriate sense). Of course one can get a DE from an algebraic equation, but not in a (useful) unique way. [Read the paper linked above before commenting on that point, please!] – Jacques Carette Sep 22 '11 at 0:39
up vote 10 down vote accepted

A well-studied special case of higher symmetries of differential equations is that of differential equations that arise as Euler-Lagrange equations of local action functionals. The symmetries and symmetries-of-symmetries and symmetries-of-symmetries-of-symmetries of such a system of equations form an $\infty$-groupoid whose infinitesimal version is encoded by the corresponding BRST complex -- which is the Chevalley-Eilenberg algebra of the corresponding L-∞ algebroid. In simple cases (or else locally) this is the global quotient by a smooth ∞-group: the "ghosts" in the BRST complex are the cotangents to the local symmetries, the "ghosts-of-ghosts" are the cotangents to the local symmetries-of-symmetries, and so on.

For instance

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This is a better-than-hoped for answer for half of my question, thank you. Any idea about the other half: do we actually know how to use these generalized symmetries to 'integrate' (in the sense used in Peter Olver's excellent book) these fields? – Jacques Carette Sep 19 '11 at 23:55
The (higher) space of solutions of these differential equations is the derived critical locus ( of the corresponding action functional, which is the space of solutions, up to homotopy, modulo the higher gauge symmetries, up to homotopy. This higher derived space is famously modeled, dually, by the full BV-BRST complex ( Physicists have been using this since the 80s without knowing about higher and derived geometry. But that's precisely what it is about. – Urs Schreiber Sep 20 '11 at 6:27
Two further remarks: 1) I had said that the above comment applies to Euler-Lagrange equations of an action functional. But in fact the BV-BRST formalism works more generally, for arbitrary differential equations. Some references in this direction are listed here:… 2) the super-L-infinity algebra symmetries that John Huerta discusses were discovered by D'Auria and Fre as the symmetries of the differential equations of higher supergravity: – Urs Schreiber Sep 20 '11 at 7:22
I am sure that the answer to my question is indeed buried in there - it is just quite beyond my current state of knowledge of these matters for me to understand that answer. Sigh. I'll accept the answer. I guess what I really meant to ask is: have these led to concrete algorithms to solve certain equations (in ``closed form''!), like 1-Lie groups have? I am having fun trying to understand enough of the background to understand your answer, but eventually that is what I am after. – Jacques Carette Sep 20 '11 at 12:05
I am not aware of concrete algorithms for solving differential equations coming out of this formalism, sorry. This formalism has a different aim: it is all about providing concrete prescriptions for obtaining the canonical symplectic form on the space of solutions modulo gauge transformation (for the case of Euler-Lagrange PDEs, at least). But maybe there is more to be said than has been worked out. – Urs Schreiber Sep 20 '11 at 12:46

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