Related papers: Cartan subgroups and regular points of o-minimal g…

If G is a locally essential subgroup of a compact abelian group K, then: (i) t(G)=w(G)=w(K), where t(G) is the tightness of G; (ii) if G is radial, then K must be metrizable; (iii) G contains a super-sequence S converging to 0 such that…

The spectrum of an admissible subalgebra $\mathscr{A}(G)$ of $\mathscr{LUC}(G)$, the algebra of right uniformly continuous functions on a locally compact group $G$, constitutes a semigroup compactification $G^\mathscr{A}$ of $G$. In this…

Let $G$ be a reductive algebraic group and $H$ its reductive subgroup. Fix a Borel subgroup $B\subset G$ and a maximal torus $T\subset B$. The Cartan space $\a_{G,G/H}$ is, by definition, the subspace of $\Lie(T)^*$ generated by the weights…

In this paper, we study definably compact semigroups in o-minimal structures, aiming to extend the theory of definable groups to a broader algebraic setting. We show that any definably compact semigroup contains idempotents and admits a…

We characterise the strictly closed left invariant C*-subalgebras of the C*-algebra C_b(G) of bounded continuous functions on a locally compact group G. On the dual side, we characterise the strictly closed invariant C*-subalgebras of the…

We say that an indecomposable Cartan matrix A with entries in the ground field of characteristic 0 is almost affine if the Lie sub(super)algebra determined by it is not finite dimensional or affine but the Lie (super)algebra determined by…

We give explicit formulas proving restrictedness of the following Lie (super)algebras: known exceptional simple vectorial Lie (super)algebras in characteristic 3, deformed Lie (super)algebras with indecomposable Cartan matrix, and (under…

Let G be a regular Lie group which is a directed union of regular Lie groups G_i (all modelled on possibly infinite-dimensional, locally convex spaces). We show that G is the direct limit of the G_i as a regular Lie group whenever G admits…

We establish the conjugacy of Cartan subalgebras for generic Lie tori "of type A". This is the only conjugacy problem of Lie tori related to Extended Affine Lie Algebras that remained open.

We establish the first global results for groups definable in tame expansions of o-minimal structures. Let $\mathcal N$ be an expansion of an o-minimal structure $\mathcal M$ that admits a good dimension theory. The setting includes dense…

We show that if $H \leq G$ is a closed amenable and cocompact subgroup of a unimodular locally compact group, then the reduced group C*-algebra of $G$ is not simple. Equivalently, there are unitary representations of $G$ that are weakly…

This is the lecture 3 of a mini-course of 4 lectures. Our purpose of this mini-curse is to explain some ideas of E. Cartan and S. Lie when we study differential geometry, particularly we will to explain the Cartan reduction method. The…

Let $L$ be a restricted Cartan type Lie algebra over an algebraically closed field $k$ of characteristic $p>3$, and let $G$ denote the automorphism group of $L$. We prove that there are no nontrivial invariants of $L^*$ under the coadjoint…

We explain by elementary means why the existence of a discrete series representation of a real reductive group $G$ implies the existence of a compact Cartan subgroup of $G$. The presented approach has the potential to generalize to real…

We provide a complete classification for regular subalgebras $B \subset M$ of injective factors satisfying a natural relative commutant condition. We show that such subalgebras are classified by their associated amenable discrete measured…

The paper is devoted to representation theoretic and algebraic geometric aspects of the theory of orthogonal decompositions of Lie algebra sl(n) into Cartan subalgebras orthogonal with respect to Killing form and the relevant theory of…

We show that an infinite group $G$ definable in a $1$-h-minimal field admits a strictly $K$-differentiable structure with respect to which $G$ is a (weak) Lie group, and show that definable local subgroups sharing the same Lie algebra have…

Let $\mathfrak{g}$ be a Lie algebra over an algebraically closed field $\Bbbk$ of characteristic zero. Define the universal grading group $\mathcal{C}(\mathfrak{g})$ as having one generator $g_{\rho}$ for each irreducible…

An algorithm for embedding finite dimensional Lie algebras into Lie algebras of vector fields (and Lie superalgebras into Lie superalgebras of vector fields) is offered in a way applicable over ground fields of any characteristic. The…

We show that a separable, nuclear C*-algebra satisfies the UCT if it has a Cartan subalgebra. Furthermore, we prove that the UCT is closed under crossed products by group actions which respect Cartan subalgebras. This observation allows us…