We have monads = triples for both assoc algebras and Lie algebras but monad structure is associative
>1)has any work been done on Lie analog of a monad? >(an `infinitesimal ' monad) As far as I know, there is presently no infinitesimal analog of (proper) monoids. The reason might be that infinitesimal transformations are infinitely near the identity and very often the set of inversible transformations is an open neighbourhood of the identity.
2) has any work been done on the Lie analog of a category? i.e. skew-comm `comnposition' satisfying Jacobi when all three are defined??
jim stasheff
There is a theory of Lie groupoids which I will discuss briefly. By definition, a Lie groupoid is a groupoid G=(G1,G0,d0,d1,m,u) in the category of smooth manifolds. In this definition, we need to take a pullback in the category of manifolds. The domain and codomain maps of G (ie d0 and d1) are better be submersions if we wish this pull-back to exist. The following infinitesimal structure can be associated to a Lie groupoid. It is a pair (A,D) where A is a commutative R-algebra and D is a Lie algebra (over R) such that 1) D is acting on A (as derivations): X(fg) = X(f)g + fX(g) , [X,Y](f) = X(Y(f)) - Y(X(f)) 2) D is equipped with an A-module structure such that (fX)(g) = fX(g) and [X,fY] = X(f)Y + f[X,Y] (for any f,g in A and X,Y in D) The pair (A,D) associated to G is the following: A is the algebra of smooth functions on G0. D is the set of infinitesimal one parameter deformations of u considered as a section of d0. In other words, if T(G1) is the tangent bundle of G1 and V is the subbundle of T(G1) consisting of vectors killed by d0, then D is the module of global sections of the pull-back of V along u. The mapping d1 defines a map from D to global sections of T(G0), from which we obtain the action of D on A. The Lie algebra structure on D is a bit harder to describe. The best way is to exibit a group H of which D is (morally) the Lie algebra. In analogy with the Lie algebra of vector fields on a manifold which is the infinitesimal portion of the group of diffeomorphisms of the manifold. H is the set of pairs (f,s) where f:G0-->G0 is a diffeomorphism and s:G0-->G1 is section of d0 such that (d1)s = f. The composite (f,s)*(g,t) is the pair (fg,w) where for every x in G0 we have w(x)=s(g(x))t(x). (this definition is reminescent of the wreath product). Remark: the group H fits in a crossed module stucture H-->a(G) where a(G) is the group of automorphism of the groupoid G. Actually, this crossed module is equivalent to the group of automorphism of G as a group object in Cat. It is quite clear that D is the infinitesimal counterpart of H (a pair (f,s) in H is determined by s). Lie theory produces a one to one correspondance between finite dimensional Lie algebras and simply connected Lie groups. More precisely, the left adjoint to the functor {Lie groups}--> {finite dim Lie algebras} defines an equivalence between the category of finite dimensional Lie algebras and the category of simply connected Lie groups. There is an analogous theory for Lie groupoids and pairs (A,D). It was developped rigorously only quite recently by Jean Pradine (verbatim). But functoriality is awkard if we insist in representing the infinitesimal structures by pairs (A,D) as above. However, functoriality is natural if D is dualised. Putting F = D* we obtain a sequence A-->F-->F^2 where d:A-->F is defined by d(f)(X) = X(f) and d:F-->F^2 by d(h)(X,Y)= X(h(Y)) - Y(h(X)) - h([X,Y]) (here F^2 is the exterior square of F). These two differential operators can be extended uniquely to an (anti-) derivation d of the full exterior algebra E(F) of the A-module F: A-->F-->F^2-->F^3-->... We have d(vw) = d(v)w +(+-1)vd(w) and also dd=0. In other words, we have a kind of de Rham complex structure on the exterior algebra of the A-module F. Conversely, it is easy to see that a triple (A,F,d) where d is (anti-) derivation of degre 1 on the exterior algebra of a projective module of finite rank F is obtained as above from a unique pair (A,D) with D=F*. There is an obvious category of the triples (A,F,d) (let me call them de Rham complexes): a morphism (A,F,d)-->(B,M,d) is a pair (f,g) where f:A-->B is an algebra map and g:F-->M is A-module map such that the corresponding homomorphism of exterior algebras E(F)-->E(M) commutes with the "exterior" derivatives. It is easy to see that we have a contravariant functor {Lie groupoids}--->{de Rham complexes} and it follows from the work of Pradine that it has a left adjoint Exp (the exponential functor) Exp:{de Rham complexes}--->{Lie groupoids} when it is restricted to complexes (A,F,D) in which A is an algebra of smooth functions on a manifold. The functor Exp is full and faithful. Its image should consist of some kind of simply connected Lie groupoids. I guess (I should really read Pradine ...) these groupoids are those for which the fibers of d0 are simply connected. ?? An interesting example is given by the ordinary de Rham complex DR(M) of a manifold M. We have Exp(DR(M))= pi1(M) where pi1(M) denotes the homotopy groupoid of M. The set M1 of arrows in pi1(M) is the set of homotopy classes of paths in M (end points fixed). The fiber of d0:M1-->M at x is the universal cover of M constructed from the basepoint x. Another interesting example is obtained by considering an integrable differential system on a manifold M. It is a set J of 1-forms such that d(J) is contained in the ideal of DR(M) generated by J. In other words, J generates a differential ideal and DR(M)/J (a short hand for DR(M)/JDR(M)) has the structure of a de Rham complex. The Lie groupoid Exp(DR(M)/J) has a simple geometric interpretation. According to a theorem of Frobenius J is the set of 1-forms vanishing on the leaves of a foliation on M. The arrows of Exp(DR(M)/J) are homotopy classes of paths lying in the leaves. The groupoid Exp(DR(M)/J) was used by A.Connes in his work on foliations. Best wishes, andre joyal +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++