This post should be considered as part two of the earlier post Lie-Infinity Algebroids?

The term \(L_{\infty} \) -algebroid seems not to be very well established in the literature. A nice discussion of this can be found at the nLab.

To quickly recall, the definition I use is that the Q-manifold \((\Pi E, Q)\) is an \(L_{\infty} \) -algebroid, where \(E \rightarrow M \) is a vector bundle and \(Q \) is an arbitrary weight homological vector field. The weight is provided by the assignment of zero to the base coordinates and one to the fibre coordinates. If the homological vector field is of weight one, then we have a Lie algebroid.

It is by now quite well established that a Lie algebroid, as above is equivalently described by

i) A weight minus one Schouten structure on the total space \(\Pi E^{*}\).
ii) A weight minus one Poisson structure on the total space of \(E^{*}\).

In other words, Lie algebroids are equivalent to certain graded Schouten or Poisson algebras. Recall, a Schouten algebra is an odd version of a Poisson algebra. The point is ignoring all gradings and parity, we have a Lie algebra such that the Lie bracket satisfies a Leibniz rule over the product of elements of the Lie algebra. We need a notion of multiplication, in this case it is just the “point-wise” product of functions.

Thus, there is a close relation between Poisson/Schouten algebras (or manifolds) and Lie algebroids.

The natural question now is “does something similar happen for \(L_{\infty}\)-algebroids?”

The answer is “yes”, but we now have to consider homotopy versions of Schouten and Poisson algebras.

Definition: A homotopy Schouten/Poisson algebra is a suitably “superised” \(L_{\infty}\)-algebra (see here) such that the n-linear operations (“brackets”) satisfy a Leibniz over the supercommutative product of elements.

This definition requires that we don’t have just an underlying vector space structure, but that of a supercommutative algebra. I will assume we also have a unit. Though, I think that noncommutative and non-unital algebras are no problem. The point is, I have in mind (at least for now) algebras of functions over (graded) supermanifolds.

Theorem: Given an \(L_{\infty}\)-algebroid \((\Pi E, Q)\) one can canonically construct
i) A total weight one higher Schouten structure on the total space of \(\Pi E^{*}\).
ii) A total weight one higher Poisson structure on the total space of \(E^{*}\).

Proof and details of the assignment of weights can be found in [1].

So, the point is that there is a close relation between homotopy versions of Poisson/Schouten algebras \(L_{\infty}\)-algebroids. To my knowledge, this has not appeared in the literature before. The specific case of \(L_{\infty}\)-algebras (algebroids over a “point”) also seems not to be discussed in the literature before.

The way we interpret this is interesting. We think of a Lie algebroid as a generalisation of the tangent bundle and a Lie algabra. The homological vector field \(Q \) “mixes” the de Rham differential over a manifold and the Chevalley-Eilenberg differential of a Lie algebra \(\mathfrak{g}\). Furthermore, we have a Poisson bracket on \(C^{\infty}( E^{*})\) which “mixes” the canonical Poisson on \(T^{*}M\) with the Lie-Poisson bracket on \(\mathfrak{g}^{*}\). Similar statements hold for the Schouten bracket.

For \(L_{\infty}\)-algebroids the homological vector field again generalises the de Rham and Chevalley-Eilenberg differentials, but it is now inhomogeneous. It resembles a “mix or higher order BRST-like” operator [3]. A homotopy version of the Maurer-Cartan equation naturally appears here. It is clear that we can consider the homotopy Schouten/Poisson algabras associated with an \(L_{\infty}\)-algebra as playing the role of the Lie-Poisson algebras, however there is no obvious higher brackets to consider on the cotangent bundle. It is not clear to me what should replace the tangent bundle here, if anything.

Exactly what technical use the theorem above is awaits to be explored. There are some interesting related notion in Mehta [2], I have yet to fully assimilate them. Maybe more on that another time.

References
[1] From \(L_{\infty}\)-algebroids to higher Schouten/Poisson structures. Andrew James Bruce, arXiv:1007.1389 [math-ph]

[2]On homotopy Poisson actions and reduction of symplectic Q-manifolds. Rajan Amit Mehta, arXiv:1009.1280v1 [math.SG]

[3] Higher order BRST and anti-BRST operators and cohomology for compact Lie algebras. C. Chryssomalakos, J. A. de Azcarraga, A. J. Macfarlane, J. C. Perez Bueno, arXiv:hep-th/9810212v2