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Deformation Theory Seminar

Wednesday, November 7, 2012 - 2:00pm

Marty Karel

Rutgers - Camden

Location

University of Pennsylvania

DRL 2C8

POSTPONED TO DATE UNCERTAIN DUE TO NOR'EASTER

Following several papers of J. Huebschmann, we discuss basics of Lie-Rinehart algebras, which are algebraic structures analogous to the pair $(C^{\infty}(N), Vect(N))$ for a smooth manifold $N$. Namely, a Lie-Rinehart algebra (or pair) consists of a commutative associative algebra $A$ with unity and a Lie algebra $L$, each acting on the other in a natural way, with the actions required to be compatible. Given $A$, one has the example $(A, Der(A))$ with the obvious actions. Lie-Rinehart algebras provide a categorical language to express a problem for quantization: show that descent after quantization is the same as quantization after descent. In case $(A, \{\dot, \dot\})$ is a Poisson algebra and $D_{A}$ is the $A$-module of formal differentials of $A,$ then $Der(A) = Hom_{A}(D_{A}, A)$ and there is a natural skew-symmetric bilinear form $\pi$ (the Poisson 2-form for the Poisson algebra). The adjoint of $\pi$ is a morphism $D_{A} ightarrow Der(A)$ of $A$-modules and turns the pair $(A, D_{A})$ into a LIe-Rinehart algebra, which captures important infinitesimal information about the Poisson algebra.