Related papers: Theta bases are atomic

With a view toward studying the homotopy type of spaces of Boolean formulae, we introduce a simplicial complex, called the theta complex, associated to any hypergraph, which is the Alexander dual of the more well-known independence complex.…

We relate the canonical basis of the Fock space representation of the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_{n})$, as defined by Leclerc and Thibon, to the canonical basis of its restriction to $U_q(\mathfrak{sl}_{n})$,…

In recent years, a large class of nuclear $C^\ast$-algebras have been classified, modulo an assumption on the Universal Coefficient Theorem (UCT). We think this assumption is redundant and propose a strategy for proving it. Indeed,…

A cluster variety of Fock and Goncharov is a scheme constructed from the data related to the cluster algebras of Fomin and Zelevinsky. A seed is a combinatorial data which can be encoded as an $n\times n$ matrix with integer entries, or as…

This is a survey paper of the theory of crystal bases, global bases and the cluster algebra structure on the quantum coordinate rings.

In 2006, Fock and Goncharov constructed a nice basis of the ring of regular functions on the moduli space of framed ${\rm PGL}_2$-local systems on a punctured surface $S$. The moduli space is birational to a cluster $\mathcal{X}$-variety,…

We propose a generalization of the classical theta function to higher cohomology of the polarization line bundle on a family of complex tori with positive index. The constructed cocycles vary horizontally with respect to the (projective)…

This paper concerns a class of orbital integrals in Lie algebras over p-adic fields. The values of these orbital integrals at the unit element in the Hecke algebra count points on varieties over finite fields. The construction, which is…

Let $Y$ be a (partial) minimal model of a scheme $V$ with a cluster structure. Under natural assumptions, for every choice of seed we associate a Newton--Okounkov body to every divisor on $Y$ supported on $Y \setminus V$ and show that these…

For every algebraically closed field $\boldsymbol k$ of characteristic different from $2$, we prove the following: (1) Generic finite dimensional (not necessarily associative) $\boldsymbol k$-algebras of a fixed dimension, considered up to…

We obtain some simple relations between decomposition numbers of quantized Schur algebras at an n-th root of unity (over a field of characteristic 0). These relations imply that every decomposition number for such an algebra occurs as a…

The quantum Grothendieck ring of a certain category of finite-dimensional modules over a quantum loop algebra associated with a complex finite-dimensional simple Lie algebra $\mathfrak{g}$ has a quantum cluster algebra structure of…

The Temperley-Lieb algebra \tln(\beta) can be defined as the set of rectangular diagrams with n points on each of their vertical sides, with all points joined pairwise by non-intersecting strings. The multiplication is then the…

A topological space $X$ is defined to have an $\omega^\omega$-base if at each point $x\in X$ the space $X$ has a neighborhood base $(U_\alpha[x])_{\alpha\in\omega^\omega}$ such that $U_\beta[x]\subset U_\alpha[x]$ for all $\alpha\le\beta$…

V. Nestoridis conjectured that if $\Omega$ is a simply connected subset of $\mathbb{C}$ that does not contain $0$ and $S(\Omega)$ is the set of all functions $f\in \mathcal{H}(\Omega)$ with the property that the set…

Let $G$ be a metrizable compact group, $A$ a separable C*-algebra and $\alpha$ a strongly continuous action of $G$ on $A$. Provided that $\alpha$ satisfies the continuous Rokhlin property, we show that the property of satisfying the UCT in…

This is a non-standard exposition of the main notions of quantum mechanics and quantum field theory including some recent results. It is based on the algebraic approach where the starting point is a star-algebra and on the geometric…

For a field $F$ and an integer $d\geq 1$, we consider the universal associative $F$-algebra $A$ generated by two sets of $d+1$ mutually orthogonal idempotents. We display four bases for the $F$-vector space $A$ that we find attractive. We…

Let Q be a finite quiver without oriented cycles, and let $\Lambda$ be the corresponding preprojective algebra. Let g be the Kac-Moody Lie algebra with Cartan datum given by Q, and let W be its Weyl group. With w in W is associated a…

The definitions of scattering matrix and inclusive scattering matrix in the framework of formulation of quantum field theory in terms of associative algebras with involution are presented. The scattering matrix is expressed in terms of…