# Examples of Lie Algebroids

The concept of a Lie Algebroid is given an important geometric meaning in the framework of Generalized Complex Geometry. For reference, the (barebones) definition of a Lie Algebroid is a vector bundle $E \rightarrow M$ with a Lie Bracket $[\cdot,\cdot]$ on $\Gamma(E)$ and a map $\rho :E \rightarrow TM$ such that $\forall f \in C^{\infty}(M)$ we have,

$$[X,fY] = \rho(X)[f] \cdot Y + f[X,Y]$$

In Generalized Complex Geometry, one is interested in the bundle $E = TM \oplus T^* M$ and a Lie Algebroid known as the Courant Algebroid. Having just read M. Gualtieri's Thesis as well Hitchin's Lectures on Generalized Complex Geometry, I was wondering if there are any "non-standard" examples of Lie Algebroids (hopefully with some interesting geometric interpretation!). By non-standard, I mean any of the examples on Wikipedia's page for Lie Algebroids (these are the same examples in Gualtieri's thesis) or the Courant Algebroid.

The main motivation for this question is to see if there are any other (perhaps natural) vector bundles that contain information about a Generalized Complex Geometry. For instance, I would be really happy to find an answer to the question,

Does the bundle $E = NM \oplus N^* M$, where $NM$ is the normal bundle and $N^*M$ is the conormal bundle, tell us anything about the topology/geometry of generalized complex manifolds?

Thanks!

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The broad question at the end is perhaps not suited to MO, but the examples is ok. My only feeling is that perhaps, as there is not one right answer, this should be a community wiki question. – David Roberts May 12 '11 at 3:01
One place where Lie algebroids occur naturally is Poisson geometry (work of Alan Weinstein). The algebroid structure is defined on sections of $T^\ast M$ extending the formula $[df,dg] = d\textrm{Poisson bracket of} f,g$ to all forms, and the anchor map associates to $\omega$ the vector field $X$ for which $X f = \pi (df, \omega)$, $\pi$ being the Poisson bivector field. I don't know if this is connected to generalized complex geometry in any way. – Dima Shlyakhtenko May 12 '11 at 5:37
@Dima: Thanks for the example. I believe that there is a relationship with Generalized Complex Geometry. Gualtieri has a paper (arxiv.org/abs/0710.2719) that gives an explicit construction of the Generalized Kähler Structure on a Compact, Poission Manifold. Moreover, the survey paper by Zabine (arxiv.org/abs/0906.1056) concisely gives these results. I'm not sure, however, if this is a "new" Lie Algebroid (I think that the set of Poisson structures is a subset of the set of Generalized Kähler Structures) in that the bundle E is not $TM \oplus T^* M$ (or a subbundle) – Tarun Chitra May 12 '11 at 6:41
a small correction: Courant algebroids are in fact not Lie algebroids (depending on the definition, the bracket is either not skew-symmetric or doesn't satisfy Jacobi identity) – Pavol S. May 12 '11 at 8:12
@Trial: Thanks! Yes, I suppose I was assuming that there existed an integrable Generalized Complex Structure, in which case we get a Lie Bialgebroid (see page 48 of Gualtieri's Thesis). However, I'm still interested in the case of a generic Lie Algebroid – Tarun Chitra May 12 '11 at 8:29

Here is an exotic Lie algebroid structure. Consider an inclusion of algebraic varieties $i:X\hookrightarrow Y$. Then we have a short exact sequence in $\mathcal O_X$-mod: $$0\to T_X\to i^*T_Y\to N\to 0,$$ where $N$ is the normal bundle. Then in the derived category of $X$ we have a map $N[-1]\to T_X$. Now observe that $N[-1]=T_{X/Y}$ is itself equipped with a Lie bracket. It is a Lie algebroid object in the derived catgory of $X$. This Lie algebroid is actually the Lie algebroid of the derived self-intersection $X\times^{{\bf R}}_YX$ of $X$ into $Y$, which is a (derived) groupoid in an obvious way.

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This is not particularly an answer to your question. Beyond Lie algebroids arising from Poisson structures (which Dima Shlyakhtenko has already mentioned in the comments), I think the Wikipedia list is the standard one. You really should think of Lie algebroids as combining the behavior of (1) bundles of Lie algebras (2) infinitesimal actions (3) foliations.

Instead, I'd like to sketch a maybe better way of thinking about Lie and Courant algebroids and their generalizations. I know very little about the Hitchin+Gualtieri+... picture of geometry. (Although they'll be giving a masterclass together in June in Aarhus, Denmark, so maybe you're planning on going?) So my sketch may or may not have much overlap with their generalizations. By I think it is a dramatic improvement on the very 20th-century approach given on Wikipedia. (Already the definitions I will give are of a very familiar flavor to 20th-century phycisits. The 21st-century improvement is to see Q-manifolds as objects not of a 1-category but of an $\infty$-category, and to think of them as some part (the "derived" part?) of the theory of $\infty$-stacks of manifolds.)

Recall that the category of $\mathbb Z$-graded vector spaces, whose objects are formal direct sums $\bigoplus [n]V_n$ where the $V_n$ are vector spaces (and I use the notation that $[n]$ is a chosen one-dimensional vector space "in degree $n$", and I generally drop $\otimes$ from the notation, so that $[n]V = [n]\otimes V$) has a monoidal structure in an obvious way, and has a non-obvious symmetric structure whereby the braiding $[1]\otimes [1] \to [1]\otimes [1]$ is minus the identity. Let $M_0$ be a finite-dimensional manifold and $V$ a finite-dimensional graded vector space. Then you can form a sheaf, called $\mathcal C^\infty(M_0 \times V)$ of graded-commutative algebras over $M_0$, which assigns to $U \subseteq M_0$ the algebra $\mathcal C^\infty(U) \otimes \widehat{\operatorname{Sym}}(V^\ast)$, where $V^\ast$ is the dual graded vector bundle to $V$, and $\operatorname{Sym}$ is taken in the signed sense, and $\widehat{\operatorname{Sym}}$ denotes completion with respect to the polynomial degree. I will define a $\mathbb Z$-graded manifold to be any sheaf of graded-commutative algebras which is locally isomorphic to a sheaf of the form $\mathcal C^\infty(M\times V)$. In particular, the "total space" of any graded vector bundle is of this type.

The category of graded manifolds is pretty similar to the category of manifolds, and pretty much all of the usual calculus carries over. See for example Rajan Mehta's thesis.

Once we are in graded land, you can talk about fairly right and interesting structures. To make some numbers cleaner, I will use the opposite of the usual convention, and use homological gradings. Then a Q-structure ("Q" is short for "qohomological") on a graded manifold $M$ a vector field $Q$ on $M$ with grading $|Q| = -1$, satisfying $[Q,Q] = 0$, where $[,]$ is the Schouten-Nijenhuis bracket (commutator of vector fields). Notice that this is a nontrivial condition because of the gradings and sign conventions. (More precisely, what is a "vector field with grading $-1$"? Let $M$ be our graded manifold and $\mathrm T M$ its tangent bundle. $M$ also has a trivial vector bundle $[n]\times M \to M$ for each $n \in \mathbb Z$. A vector field with grading $n$ is morphism $[n]\times M \to \mathrm T M$ of vector bundles over $M$.)

Similarly, a degree-$n$ P-structure on $M$ ("P" for "Poisson"; abbreviate "$P_n$-structure") is a degree-$n$ section of $P$ of $\mathrm T^{\otimes 2}M$ satisfying $[P,P] = 0$ (Schouten-Nijenhuis bracket) and either symmetry or antisymmetry depending on the parity of $n$; better is to say that it's a section of $[n] ([-n]\mathrm T)^{\wedge 2} M$ whose corresponding section of $\mathrm T^{\otimes 2}$ satisfies $[P,P] = 0$. Note that, as a section of $\mathrm T^{\otimes 2}$, $P$ defines a map of vector bundles $\mathrm T^\ast \to \mathrm T$; I say that $P$ is symplectic if this map has an inverse $\omega : \mathrm T \to \mathrm T^\ast$.

The most important example: let $A \to X$ be a classical vector bundle. Then the following data are equivalent: (1) a Lie algebroid structure on $A \to X$, (2) a $Q$-structure on the total space of $[1]A$, (3) a $P_1$-structure on the total space of $[1]A^\ast$. Since (1)->(2) is covariant, I like to take (2) as the definition of (1); it explains what are the correct morphisms of Lie algebroids. (Actually, the category of classical Lie algebroids is only a full subcategory of the category of Q-manifolds; it is full on those manifolds whose algebras of functions are generated in gradings $0,-1$, so that the "geometric" gradings of the manifold are precisely $0,1$.)

Another important example: the total space of $[n]\mathrm T^* M$ is $P_n$ (symplectic) for any graded manifold $M$. When $n = 0$, this is the symplectic structure you already know; when $n=\pm 1$, the P structure is the Schouten-Nijenhuis bracket.

Anyway, one can string adjectives together, talking about e.g. $QP_n$-manifolds, which would be a manifold with a vector field $Q$, a bivector field $P$, satisfying $|Q| = -1$ and $|P| = n$, and satisfing $[Q,Q] = 0$, $[P,P] = 0$, and $[Q,P] = 0$. Or you could have two (commuting) Q-structures. And so on.

If memory serves, a Courant algebroid is the following (the best reference is Roytenberg's thesis), in the same way that a Lie algebroid is a Q-manifold (so as with Lie algebroids, Courant algebroids are the following plus restrictions on the gradings of the generators). A Courant algebroid is a $QP_2$ manifold whose Poisson structure is symplectic. Important examples include $[2]\mathrm T^\ast M$ for $M$ a Q-manifold, and in particular $[2]\mathrm T^\ast [1]A$ for $A \to X$ a Lie algebroid (extend the Q-structure on $M$ to a Q-structure on $[2]\mathrm T^\ast M$ by taking Lie derivatives in the $Q$-direction). But more generally: if $M$ is a $P_n$ manifold, then the Poisson structure on $M$ can be encoded as a Q-structure on $[n+1]\mathrm T^\ast M$. So if $N$ is $QP_1$, then $[2]\mathrm T^\ast M$ has a Q-structure from $Q|_M$, a Q-structure from the $P_1$-structure on $N$, and they commute, so add them and get a $Q$-structure on $[2]\mathrm T^\ast N$, and it still commutes with the canonical symplectic structure. (You can tell apart the two $Q$ structures if you remember the euler vector field that encodes that $[2]\mathrm T^\ast N$ is the total space of a vector bundle over $N$.) Finally we get the main example: if $A\to X$ is a classical Lie bialgebroid, then $[1]A$ is $QP_1$, so $[2]\mathrm T^\ast[1]A$ is Courant. (Note that there exists an isomorphism $[2]\mathrm T^\ast[1]A \cong [2]\mathrm T^\ast[1]A^\ast$ of $QP_2$ manifolds over $X$, and that $M = [2]\mathrm T^\ast[1]A$ as a graded vector bundles over $X$ is of the form $M \cong [2]\mathrm T^\ast X \oplus [1] (A\oplus A^\ast)$, and so you can give equivalent data to $M$ by giving $A\oplus A^\ast$ some interesting structure, which is the usual way of defining/building Courant algebroids.)

One final comparison: a $QP_1$ manifold whose P-structure is symplectic, if its algebra of functions is generated in non-positive gradings, is necessarily of the form $[1]\mathrm T^\ast X$ for $X$ a classical manifold, and the Q-structure exactly encodes a Poisson structure on $X$. So the usual definition of Courant algebroid is: what I gave above, and ask that the algebra of functions is supported only in nonpositive gradings --- i.e. the manifold itself has geometric gradings only in nonnegative gradings. Then it's clear that Courant algebroids are a generalization of Poisson manifolds. In the other direction, a $QP_0$ symplectic manifold with the same grading restrictions is necessarily a classical symplectic manifold with $Q=0$. Symplectic, Poisson, Courant, .... But from a graded-geometry point of view, restricting the gradings of the manifold is somewhat unnatural. You should instead see the sequence Symplectic, Poisson, Courant, ..., as being a sequence of full subcategories of some much richer categories.

I've mentioned already Mehta and Roytenberg as good references. Of course, canonical references also include Weinstein and collaborators, Severa, Cattaneo+Felder, anything with the string "AKSZ" in the title (including the original paper by A,K,S, and Z), and probably many others.

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Thanks a lot! The hint that you leave in your second-to-last paragraph is quite interesting since I've only thought of these brackets geometrically. My knowledge of Algebraic Geometry is weak, but is the "Z-graded manifold" simply a locally ringed space (or perhaps a stack?) – Tarun Chitra May 15 '11 at 5:46
@Tarun: In fact, in $\cal C^\infty$ land, it's enough to only consider global functions on $\mathbb Z$-graded manifolds. (But, just like deciding when a ring is $\cal C^\infty$ of a regular manifold is hard, so too for $\mathbb Z$-graded manifolds.) Yes, a good way to say "Z-graded manifold" is "locally ringed space that is locally isomorphic to Sym of a graded vector space". This is fine, provided that you imagine that all non-classical-manifold directions are "formal", i.e. impose that all formal power series in such variables converge. – Theo Johnson-Freyd May 15 '11 at 16:40
Lie algebroids and the like are a useful language for talking about some part of Stacks, but not all of them, and not quite functorially. Recall that any sufficiently geometric stack in manifolds is presented by a groupoid. Then there is a way to turn any groupoid into a Lie algebroid, for instance, and hence a Q-manifold; 2-groupoids turn into something like Courant algebroids, etc. But this "functor" sees only infinitesimal data about the stack, and is not functorial for general maps of stacks. (In particular, it does not take equivalent stacks to equivalent Q-manifolds.) – Theo Johnson-Freyd May 15 '11 at 16:44
What Q-manifolds can precisely talk about are things like the quotient of a manifold by a Lie algebra, whereas Stacks describe quotients by Lie groups. Q-manifolds can also perfectly talk about intersection theory. They cannot talk about "half a point", at least not if you restrict yourself to characteristic-$0$. I don't know if there's a good theory of "Q-schemes over $\mathbb Z$" that fully includes stacks. – Theo Johnson-Freyd May 15 '11 at 16:46

Strictly speaking, the "Courant algebroid" is not a Lie algebroid, but a Lie 2-algebroid . It is, after the Poisson Lie algebroid, the third in an infinite tower of higher Lie algebroids that deserve to be called symplectic Lie n-algebroids

http://nlab.mathforge.org/nlab/show/symplectic+Lie+n-algebroid

If you are therefore willing to consider more general Lie 2-algebroids

http://nlab.mathforge.org/nlab/show/Lie+infinity-algebroid

there is potentially a rich pool to choose from, depending on what you are after. For instance since generalized complex geometry is all about the infinitesimal description of circle 2-bundles (aka bundle gerbes) it might be interesting to consider more general nonabelian principal 2-bundles and their Atiyah Lie 2-algebroids.

Generally, the supply of (higher) Lie algebroids is vast. Asking for non-standard examples is a bit like asking for a non-standard example of a group or a Lie groupoid or such. This might have a good answer if one knew better what kind of questions it is supposed to be used for.

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