Graded G-structures

Abstract

In this communication I would like to give a progress report on a program to construct graded geometric structures on graded differential manifolds. Most of the results so far are on the formal algebraic level and many important problems of analysis remain. Kostant I has given a definition of a graded differential manifold of dimension (n,m) and this is discussed in these proceedings by M. Batchelor. I only mention here that it consists of a pair (M,A) where M is an ordinary C ~ manifold of dimension n and A is a sheaf of Z2-graded commutative algebras whose underlying graded vector space V = V 0 + V I has dim V 0 = n, dim V 1 = m. A graded Lie group (G,A) then is a pair where G is a Lie group and the restricted dual A ° (vanishes on some ideal of finite codimension) has the structure of a graded Hopf algebra and can be represented as A ° = R(G) ^ E(G) where R(G) is the group ring, E(G) is the universal enveloping algebra of the graded Lie algebra G = G O + G I with G O the Lie algebra of G, and ^ denotes semidirect product. In analogy with the usual situation, graded frames can be introduced and the graded frame bundle constructed. This consists of a principal fibre bundle L(M,G) with group G = GL(n)×GL(m) such that the graded tangent bundle T(M,A) defined by Kostant is an associated vector bundle for L(M,G). The graded frame bundle (not really a bundle) is then given by the graded manifold (L(M,G), A8 H) where H ° = R(G) ^ E(G) , G = GL(n)× GL(m), G = gl(n,m) = End V. Now if G'C G is a closed Lie subgroup of G with a subHopf algebra H' C H , then (L(M,G'), A8 H') is called a graded G-struc ture if L(M,G') is reduced subbundle of L(M,G). There is a natural right action R a of H' defined on (L(M,G'), ASH'). Associated with the graded tangent bundle of a G-structure, there is the tangent sheaf derA + derH (der A = derivations of A), and this is a free A@ H module. A complement H to derH is called a horizontal subspace, and as usual H defines a connection if R * H = H. As underlying graded vector spaa ces H is isomorphic to V. In analogy with the ordinary ease, we study graded G-structures by studying graded Spencer cohomology. Let GC gl(n,m) = V8 V* be a graded Lie algebra. We define G (1) i i I I the first prolongation of G by all T E Hom(V,G) which satisfy T(u)v =(-i) luILvi T(V)U where u,v are homogeneous elements in V of degree lul, Ivl respectively. The k th prolongation is defined inductively by G (k) = (G(k-l)) (I) • We construct Z 2 gravector spaces C k+l'l = G (k) O Al(V *) where AI(V *) = AI(V~)0 Sl(V~) is the graded ded exterior algebra over V*. The cochain map ~:C k+l'/ >?,/+i is defined ~ by