Millionschikov D.V. Growth of Lie algebras and integrability

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Speaker: Dmitry Millionschikov

Title: Growth of Lie algebras and integrability

Abstract:
We consider naturally graded Lie algebras   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathfrak g} = \oplus_{i = 1}^{n}{\mathfrak g}_i, \; [{\mathfrak g}_1, {\mathfrak g} _i]={\mathfrak g}_{i + 1}, \; i \ge 1.}

In the finite-dimensional case they are called Carnot algebras and play an important role in non-holonomic geometry and geometric control theory. A naturally graded Lie algebra Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathfrak g}} is generated by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathfrak g}_1} and one can define its natural growth function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_{\mathfrak g}^{gr}(n)=\sum_{i=1}^n\dim{{\mathfrak g}_i} } which is well-defined.

It turned out that the characteristic Lie algebras Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} of some nonlinear hyperbolic partial differential equations are precisely such positively graded Lie algebras. The integability of these equations in the sense of Darboux or higher symmetries leads to the slow growth of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} .

I will also try to discuss another geometric integrability, the integrability of complex structures on Carnot algebras. It turns out that in this case, on the contrary, Lie algebras must grow sufficiently fast.

Video


Event: One day workshop in honor of Maxim Pavlov's 60th birthday, 14 December 2022, Independent University of Moscow