Date: Thu, 12 Aug 1993 16:59:28 -0400 From: chase@math.cornell.edu (Stephen Chase)
Although I accept the fact that the notion of Lie algebroid is what matters for the Lie theory of Lie groupoids, I think there are a couple of reasons why the general concept of Lie pseudo-algebra should not be dismissed.
I certainly agree that Lie pseudo-algebras deserve independent study. I have a survey article "Generalized Lie theories: Lie algebroids and Lie pseudo-algebras as algebraic invariants in differential geometry" in near-final stage, with a bibliog of about 100 references; this is available to people who have an interest in the subject. The orientation is towards the use of Lie algebroids and Lie pseudo-algebras as a unifying concept in first-order differential geometry, but the bibliog is intended to be complete and covers purely algebraic work that I know of.
Another important example arises from a Lie k-algebra L acting on a commutative k-algebra K by derivations: (K,K#L) is then a Lie pseudo-algebra, where K#L is the tensor product of K with L over k with a suitable twisted bracket operation.
That is, with the exception of the appendix on groupoid schemes and Hopf algebroids in Ravenel's 1986 book on stable homotopy.
From this perspective, the point of the # construction is that it enables general Lie pseudo-algebra morphisms to be reduced to the algebra-preserving case. In (K,K#L) above, L itself can be a Lie pseudo-algebra. Now if L --> L' is a morphism of Lie pseudo-algebras over a morphism of k-algebras K --> K',
If one considers Lie algebroids over a fixed base manifold, or if one considers Lie pseudo-algebras over a fixed (commutative) algebra, then everything in both cases is quite similar to the case of finite-dimensional Lie algebras over a field, at least as regards basic definitions and constructions. If one allows arbitrary base manifolds or base algebras things become very different. Firstly, morphisms of Lie algebroids and morphisms of Lie pseudo-algebras no longer correspond: the concept of morphism of Lie algebroid which arises by differentiating a morphism of Lie groupoids does not corrrespond to the natural concept of morphism of Lie pseudo-algebra. Higgins and I (Math. Proc. Camb. Phil. Soc., to appear) defined concepts of comorphism, so that a comorphism of Lie algebroids induces a morphism of their Lie pseudo-algebras, and a morphism of Lie algebroids induces a comorphism of Lie pseudo-algebras; this in particular enables a duality to be defined in both categories. there is an action of L on K', and the morphism can be lifted to (K',K'#L), and is now a morphism over K'. [MPCPS, cited above.] In the Lie algebroid context the analogue of the # construction was found independently and called an "action Lie algebroid"; it is the infinitesimal analogue of the Ehresmann construction of a groupoid from an action of a group(oid) and its characterization in terms of covering morphisms. See J. Alg. 129, 1990, 194--230 and refs there. Incidentally, morphisms of Lie pseudo-algebras also arise in Poisson geometry: a Poisson morphism induces not a morphism of the cotangent Lie algebroids but a morphism of the Lie pseudo-algebras of 1-forms. Quotients of Lie algebroids in the base-varying case are more complicated and need a kind of reduction process. I am not sure what the situation is with quotients of Lie pseudo-algebras. See J. Alg. 129, 1990, 194--230 again. The corresponding general quotient for (Lie) groupoids [Jour LMS, 42, 1990, 101--110] should have implications for Ravenel's 1986 Appendix. Kirill Mackenzie +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++