Graph H\"ormander Systems

TL;DR

This paper generalizes the Bakry-Émery theorem to matrix-valued settings on Lie groups and introduces combinatorial methods to compute lower bounds for matrix-valued log-Sobolev inequalities in graph Hörmander systems.

Contribution

It extends classical curvature and inequality results to matrix-valued functions on Lie groups and develops combinatorial techniques for bounds in graph Hörmander systems.

Findings

Matrix-valued log-Sobolev inequalities are equivalent to uniform bounds across representations.

New computable lower bounds for these inequalities in graph Hörmander systems.

Application of noncommutative geometry tools to differential operators on Lie groups.

Abstract

This paper extends the Bakry-\'{E}mery theorem connecting the Ricci curvature and log-Sobolev inequalities to the matrix-valued setting. Using tools from noncommuative geometry, it is shown that for a right invariant second order differential operator on a compact Lie group, a lower bound for a matrix-valued modified log-Sobolev inequality is equivalent to a uniform lower bound for all finite dimensional representations. Using combinatorial tools, we obtain computable lower bounds for matrix-valued log-Sobolev inequalities of graph-H\"ormander systems using combinatorial methods.