On n-generalized commutators and Lie ideals of rings

Let [Formula: see text] be an associative ring. Given a positive integer [Formula: see text], for [Formula: see text] we define [Formula: see text], the [Formula: see text]-generalized commutator of [Formula: see text]. By an [Formula: see text]-generalized Lie ideal of [Formula: see text] (at the [Formula: see text]th position with [Formula: see text]) we mean an additive subgroup [Formula: see text] of [Formula: see text] satisfying [Formula: see text] for all [Formula: see text] and all [Formula: see text], where [Formula: see text]. In the paper, we study [Formula: see text]-generalized commutators of rings and prove that if [Formula: see text] is a noncommutative prime ring and [Formula: see text], then every nonzero [Formula: see text]-generalized Lie ideal of [Formula: see text] contains a nonzero ideal. Therefore, if [Formula: see text] is a noncommutative simple ring, then [Formula: see text]. This extends a classical result due to Herstein [Generalized commutators in rings, Portugal. Math. 13 (1954) 137–139]. Some generalizations and related questions on [Formula: see text]-generalized commutators and their relationship with noncommutative polynomials are also discussed.

On Multimatrix Models Motivated by Random Noncommutative Geometry I: The Functional Renormalization Group as a Flow in the Free Algebra

Random noncommutative geometry can be seen as a Euclidean path-integral quantization approach to the theory defined by the Spectral Action in noncommutative geometry (NCG). With the aim of investigating phase transitions in random NCG of arbitrary dimension, we study the nonperturbative Functional Renormalization Group for multimatrix models whose action consists of noncommutative polynomials in Hermitian and anti-Hermitian matrices. Such structure is dictated by the Spectral Action for the Dirac operator in Barrett’s spectral triple formulation of fuzzy spaces. The present mathematically rigorous treatment puts forward “coordinate-free” language that might be useful also elsewhere, all the more so because our approach holds for general multimatrix models. The toolkit is a noncommutative calculus on the free algebra that allows to describe the generator of the renormalization group flow—a noncommutative Laplacian introduced here—in terms of Voiculescu’s cyclic gradient and Rota–Sagan–Stein noncommutative derivative. We explore the algebraic structure of the Functional Renormalization Group equation and, as an application of this formalism, we find the $$\beta $$ β -functions, identify the fixed points in the large-N limit and obtain the critical exponents of two-dimensional geometries in two different signatures.

Boolean cumulants and subordination in free probability

Subordination is the basis of the analytic approach to free additive and multiplicative convolution. We extend this approach to a more general setting and prove that the conditional expectation [Formula: see text] for free random variables [Formula: see text] and a Borel function [Formula: see text] is a resolvent again. This result allows the explicit calculation of the distribution of noncommutative polynomials of the form [Formula: see text]. The main tool is a new combinatorial formula for conditional expectations in terms of Boolean cumulants and a corresponding analytic formula for conditional expectations of resolvents, generalizing subordination formulas for both additive and multiplicative free convolutions. In the final section, we illustrate the results with step by step explicit computations and an exposition of all necessary ingredients.