Circle Actions

This chapter focuses on circle actions. Specifically, it specializes the Weil algebra and the Weil model to a circle action. In this case, all the formulas simplify. The chapter derives a simpler complex, called the Cartan model, which is isomorphic to the Weil model as differential graded algebras. It considers the theorem that for a circle action, there is a graded-algebra isomorphism. Under the isomorphism F, the Weil differential δ‎ corresponds to a differential called the Cartan differential. An element of the Cartan model is called an equivariant differential form or equivariant form for a circle action on the manifold M.

The Weil Algebra and the Weil Model

This chapter evaluates the Weil algebra and the Weil model. The Weil algebra of a Lie algebra g is a g-differential graded algebra that in a definite sense models the total space EG of a universal bundle when g is the Lie algebra of a Lie group G. The Weil algebra of the Lie algebra g and the map f is called the Weil map. The Weil map f is a graded-algebra homomorphism. The chapter then shows that the Weil algebra W(g) is a g-differential graded algebra. The chapter then looks at the cohomology of the Weil algebra; studies algebraic models for the universal bundle and the homotopy quotient; and considers the functoriality of the Weil model.

The Weil Algebra of a Double Lie Algebroid

Given a double vector bundle $D\to M$, we define a bigraded bundle of algebras $W(D)\to M$ called the “Weil algebra bundle”. The space ${\mathcal{W}}(D)$ of sections of this algebra bundle ”realizes” the algebra of functions on the supermanifold $D[1,1]$. We describe in detail the relations between the Weil algebra bundles of $D$ and those of the double vector bundles $D^{\prime},\ D^{\prime\prime}$ obtained from $D$ by duality operations. We show that ${\mathcal{V}\mathcal{B}}$-algebroid structures on $D$ are equivalent to horizontal or vertical differentials on two of the Weil algebras and a Gerstenhaber bracket on the 3rd. Furthermore, Mackenzie’s definition of a double Lie algebroid is equivalent to compatibilities between two such structures on any one of the three Weil algebras. In particular, we obtain a ”classical” version of Voronov’s result characterizing double Lie algebroid structures. In the case that $D=TA$ is the tangent prolongation of a Lie algebroid, we find that ${\mathcal{W}}(D)$ is the Weil algebra of the Lie algebroid, as defined by Mehta and Abad–Crainic. We show that the deformation complex of Lie algebroids, the theory of IM forms and IM multi-vector fields, and 2-term representations up to homotopy all have natural interpretations in terms of our Weil algebras.

The Cubic Dirac Operator for Infinite-Dimensonal Lie Algebras

Let be an infinite-dimensional graded Lie algebra, with dim , equipped with a non-degenerate symmetric bilinear form B of degree 0. The quantum Weil algebra is a completion of the tensor product of the enveloping and Clifford algebras of g. Provided that the Kac–Peterson class of g vanishes, one can construct a cubic Dirac operator D 2 , whose square is a quadratic Casimir element. We show that this condition holds for symmetrizable Kac– Moody algebras. Extending Kostant's arguments, one obtains generalized Weyl–Kac character formulas for suitable “equal rank” Lie subalgebras of Kac–Moody algebras. These extend the formulas of G. Landweber for affine Lie algebras.

Pairings in Hopf cyclic cohomology of algebras and coalgebras with coefficients

This paper is concerned with the theory of cup products in the Hopf cyclic cohomology of algebras and coalgebras. We show that the cyclic cohomology of a coalgebra can be obtained from a construction involving the noncommutative Weil algebra. Then we introduce the notion of higher -twisted traces and use a generalization of the Quillen and Crainic constructions (see [14] and [3]) to define the cup product. We discuss the relation of the cup product above and S-operations on cyclic cohomology. We show that the product we define can be realized as a combination of the composition product in bivariant cyclic cohomology and a map from the cyclic cohomology of coalgebras to bivariant cohomology. In the last section, we briefly discuss the relation of our constructions with that in [9]. More precisely, we propose still another construction of such pairings which can be regarded as an intermediate step between the “Crainic” pairing and that of [9]. We show that it coincides with what in [9] and as far its relation to Crainic's construction is concerned, we reduce the question to a discussuion of a certain map in cohomology (see the question at the end of section 5).The results of the current paper were announced in [12].