scholarly journals The Weil algebra and the Van Est isomorphism

2011 ◽  
Vol 61 (3) ◽  
pp. 927-970 ◽  
Author(s):  
Camilo Arias Abad ◽  
Marius Crainic
Keyword(s):  
Weil Algebra

The Weil Algebra

1999 ◽  
pp. 33-40
Author(s):  
Victor W. Guillemin ◽  
Shlomo Sternberg ◽  
Jochen Brüning
Keyword(s):  
Weil Algebra

Circle Actions

2020 ◽  
pp. 161-166
Author(s):  
Loring W. Tu
Keyword(s):  
Differential Form ◽  
Circle Action ◽  
Graded Algebra ◽  
Graded Algebras ◽  
Circle Actions ◽  
Algebra Isomorphism ◽  
Weil Algebra ◽  
Differential Graded Algebras ◽  
Cartan Model

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.

scholarly journals The Weil Algebra of a Double Lie Algebroid

2020 ◽  
Author(s):  
Eckhard Meinrenken ◽  
Jeffrey Pike
Keyword(s):  
Vector Fields ◽  
Vector Bundles ◽  
Lie Algebroid ◽  
Weil Algebra ◽  
Algebra Of Functions ◽  
Double Vector Bundles ◽  
Definition Of ◽  
Deformation Complex ◽  
Double Vector Bundle ◽  
Gerstenhaber Bracket

Abstract 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 Weil Algebra and the Weil Model

2020 ◽  
pp. 151-160
Author(s):  
Loring W. Tu
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Total Space ◽  
Algebra Homomorphism ◽  
Graded Algebra ◽  
Algebraic Models ◽  
Weil Algebra ◽  
Differential Graded Algebra ◽  
Definite Sense ◽  
Universal Bundle

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.

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