Related papers: Interplay between Algebraic Groups, Lie Algebras a…

Let $A$ be a unital operator algebra. Let us assume that every {\it bounded\/} unital homomorphism $u\colon \ A\to B(H)$ is similar to a {\it contractive\/} one. Let $\text{\rm Sim}(u) = \inf\{\|S\|\, \|S^{-1}\|\}$ where the infimum runs…

We propose an operadic framework suitable for describing algebraic structures with operations being multilinear differential operators of varying orders or, more generally, formal series of such operators. The framework is built upon the…

Lie algebras are an important class of algebras which arise throughout mathematics and physics. We report on the formalisation of Lie algebras in Lean's Mathlib library. Although basic knowledge of Lie theory will benefit the reader, none…

In this paper we consider some Lie groups in complexified Clifford algebras. Using relations between operations of conjugation in Clifford algebras and matrix operations we prove isomorphisms between these groups and classical matrix groups…

We construct and study various dual pairs between finite dimensional classical Lie groups and infinite dimensional Lie algebras in some Fock representations. The infinite dimensional Lie algebras here can be either a completed infinite rank…

The main result is that the only non trivial closed ideal in the Banach algebra $L(L^p)$ of bounded linear operators on $L^p(0,1)$, $1\le p < \infty$, that has a left approximate identity is the ideal of compact operators. The algebra…

We study the relation between two special classes of Riemannian Lie groups $G$ with a left-invariant metric $g$: The Einstein Lie groups, defined by the condition $\operatorname{Ric}_g=cg$, and the geodesic orbit Lie groups, defined by the…

Let $G$ be a finite group. There is a standard theorem on the classification of $G$-equivariant finite dimensional simple commutative, associative, and Lie algebras (i.e., simple algebras of these types in the category of representations of…

A finite-dimensional Lie algebra $L$ over a field $F$ is called an $A$-algebra if all of its nilpotent subalgebras are abelian. This is analogous to the concept of an $A$-group: a finite group with the property that all of its Sylow…

We develop the Lie theory of Lie-admissible algebras whose product is enriched with higher operations modeled on directed graphs with a view to apply it to the deformation theories controlled by this kind of Lie algebras. We produce…

Let $G$ be a connected Lie group, $C$ be the maximal compact connected subgroup of the center of $G$, and let Aut$(G)$ denote the group of Lie automorphisms of $G$, viewed, canonically, also as a subgroup of GL$(\frak G)$, where $\frak G$…

A left ideal of any C*-algebra is an example of an operator algebra with a right contractive approximate identity (r.c.a.i.). Conversely, we show here and in a `pre-quel' to this paper [B], that operator algebras with r.c.a.i. should be…

We study the action of the inertia operator on the motivic Hall algebra, and prove that it is diagonalizable. This leads to a filtration of the Hall algebra, whose associated graded algebra is commutative. In particular, the degree 1…

The Leibniz bracket of an operator on a (graded) algebra is defined and some of its properties are studied. A basic theorem relating the Leibniz bracket of the commutator of two operators to the Leibniz bracket of them, is obtained. Under…

We present an axiomatic approach to finite- and infinite-dimensional differential calculus over arbitrary infinite fields (and, more generally, suitable rings). The corresponding basic theory of manifolds and Lie groups is developed.…

Operator systems are the unital self-adjoint subspaces of the bounded operators on a Hilbert space. Complex operator systems are an important category containing the C*-algebras and von Neumann algebras, which is increasingly of interest in…

In this paper, we study the structure of operators in a type $\mathrm{I}_{n}$ von Neumann algebra $\mathscr{A}$. Inspired by the Jordan canonical form theorem, our main motivation is to figure out the relation between the structure of an…

We show that finitely subgraded Lie algebras of compact operators have invariant subspaces when conditions of quasinilpotence are imposed on certain components of the subgrading. This allows us to obtain some useful information about the…

We discuss and develop some connections between analysis on singular spaces and operator algebras, as presented in my sequence of four lectures at the conference "Noncommutative geometry and applications," Frascati, Italy, June 16-21, 2014.…

Bilinear systems emerge in a wide variety of fields as natural models for dynamical systems ranging from robotics to quantum dots. Analyzing controllability of such systems is of fundamental and practical importance, for example, for the…