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Simple Lie algebras, involutive distributions of operator-valued evolutionary vector fields, and bi-differential affine connections

Arthemy Kiselev (joint with Johan W. van de Leur, Utrecht)


We introduce a well-defined notion of linear matrix operators in total derivatives, whose images in the Lie algebras of evolutionary vector fields on jet spaces are closed with respect to the commutation. This yields a generalization for the classical theory of recursion operators and Poisson structures for integrable systems. We explain how each operator transfers the commutation of the vector fields to Lie brackets with bi-differential structural constants on the quotient of its domain by the kernel.

Second, we associate such operators with 2D Toda systems related to simple complex Lie algebras. We reconstruct the operators that factor higher symmetries of the 2D Toda chains, and we calculate explicitly all the commutators, which are specified by the new Lie brackets on the domains of the operators. Thus we obtain a complete description of the higher symmetry algebras for these exponential systems.

Third, we endow the linear spaces of differential operators, such that the sum of their images is closed under the commutation, with a Lie-type bracket. We establish a one-to-one correspondence between the structural constants for it and bi-differential Christoffel's symbols of flat affine connections in the triads of two Lie algebras and a morphism. Thence we recognize the Riemannian geometry, which corresponds to the degenerate case of jet bundles over a point, as the zero-order term in all expansions of (bi-)differential operators.

Finally, we demonstrate that the operators with the commutation closure of images are not the anchors of any Lie algebroids over the infinite jet spaces. However, we describe the Chevalley-Eilenberg complexes that arise in this setting.

Die Universität der Informationsgesellschaft