## Departmental PhD Thesis Exam – Jeffrey Pike

Friday, August 14, 2020
11:00 a.m.

PhD Candidate:  Jeffrey Pike
Supervisor:   Eckhard Meinrenken
Thesis title:  Weil Algebras for Double Lie Algebroids

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Given a double vector bundle D → M, we define a bigraded bundle of algebras W(D) → M called the ‘Weil algebra bundle’. The space 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′, D′′ obtained from D by duality operations. We show that VB-algebroid structures on D are equivalent to horizontal or vertical differentials on two of the Weil algebras and a Gerstenhaber bracket on the third. 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 = T A is the tangent prolongation of a Lie algebroid, we find that 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 multivector fields, and 2-term representations up to homotopy, all have natural interpretations in terms of our Weil algebras.

A copy of the thesis can be found here: ut-thesis