Zoran Skoda
abstract Brno 2016

Correcting coexponential map for Lie algebroids

Lie algebroids and their higher categorical analogues, L-infinity algebroids, represent infinitesimal symmetries and under some cohomological restrictions integrate to global symmetry objects (Lie groupoids and higher analogues). Their behaviour is less regular than that of Lie algebras, for example the appropriate generalization of the symmetrization or coexponential map from the symmetric to the universal enveloping algebra does not respect the coalgebra structure with consequences on physical applications like the construction of the corresponding noncommutative phase spaces. In the first part of the talk I will present a way to correct the coexponential map (using a connection on a Lie algebroid) so that the corrected map respects the coalgebra structure. This part is a joint work with G. Sharygin (Moscow). I will also sketch a relationship between this problem and the study of realizations of sections of the Lie algebroid by vector fields in suitable coordinates (in special cases the exponential map for Lie groupoids plays a role). In the second part I will present a related work in progress on enlarging the universal enveloping of the Lie algebroid by including certain automorphism operators so that the whole structure enhances to a quantum groupoid in certain precise sense. This quantum groupoid is motivated by constructions of new noncommutative spaces of relevance to mathematical physics and by deformation theory.


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