Related papers: Kac-Moody groups and cluster algebras

We present a variant of the Theory of Lorentzian (i. e. with a hyperbolic generalized Cartan matrix) Kac-Moody algebras recently developed by V. A. Gritsenko and the author. It is closely related with and strongly uses results of R.…

The cluster-tilted algebras have been introduced by Buan, Marsh and Reiten, they are the endomorphism rings of cluster-tilting objects $T$ in cluster categories; we call such an algebra cluster-concealed in case $T$ is obtained from a…

Let $G$ be a locally compact group and $P \subset G$ be a closed Ore semigroup containing the identity element. Let $V: P \to B(\clh)$ be a representation such that for every $a \in P$, $V_{a}$ is an isometry and the final projections of…

For the quiver Hecke algebra $R$ associated with a simple Lie algebra, let $R$-gmod be the category of finite-dimensional graded $R$-modules. It is well-known that it categorifies the unipotent quantum coordinate ring. The localization of…

Let $\mathscr{C}$ be the category of finite-dimensional modules over a simply-laced quantum affine algebra $U_q(\widehat{\mathfrak{g}})$. For any height function $\xi$ and $\ell\in \mathbb{Z}_{\geq 1}$, we introduce certain subcategories…

We prove that an invariant subalgebra A_n^W of the Weyl algebra A_n is a Galois order over an adequate commutative subalgebra \Gamma when W is a two-parameters irreducible unitary reflection group G(m,1,n), m\geq 1, n\geq 1, including the…

In earlier work, the author introduced a method for constructing a Frobenius categorification of a cluster algebra with frozen variables by starting from the data of an internally Calabi-Yau algebra, which becomes the endomorphism algebra…

The cluster algebra of any acyclic quiver can be realized as the coordinate ring of a subvariety of a Kac-Moody group -- the quiver is an orientation of its Dynkin diagram, defining a Coxeter element and thereby a double Bruhat cell. We use…

In the setting of von Neumann algebras, measurable quantum groupoids have successfully been axiomatized and studied by Enock, Vallin, and Lesieur, whereas in the setting of $C^{*}$-algebras, a similar theory of locally compact quantum…

Let $X= \{x_1, x_2, \cdots, x_n\}$ be a finite alphabet, and let $K$ be a field. We study classes $\mathfrak{C}(X, W)$ of graded $K$-algebras $A = K\langle X\rangle / I$, generated by $X$ and with a fixed set of obstructions $W$. Initially…

We describe a ring whose category of Cohen-Macaulay modules provides an additive categorification of the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of k-planes in n-space. More precisely, there is a…

Let $A$ be an arbitrary symmetrizable Cartan matrix of rank $r$, and ${\bf n}={\bf n_+}$ be the standard maximal nilpotent subalgebra in the Kac-Moody algebra associated with $A$ (thus, ${\bf n}$ is generated by $E_1,\ldots,E_r$ subject to…

A localized quantum unipotent coordinate category $\widetilde{\mathscr{C}_w}$ associated with a Weyl group element $w$ is a rigid monoidal category which is obtained by applying the localization process to a subcategory of the category of…

Let G be a split, simple, simply connected, algebraic group over Q. The degree 4, weight 2 motivic cohomology group of the classifying space BG of G is identified with Z. We construct cocycles representing the generator of this group, known…

Recently, we have constructed a non{linear (polynomial) extension of the 1-mode Heisenberg group and the corresponding Fock and Weyl representations. The transition from the 1-mode case to the current algebra level, in which the operators…

Given a tagged triangulation of a once-punctured polygon $P^*$ with $n$ vertices, we associate an ice quiver with potential such that the frozen part of the associated frozen Jacobian algebra has the structure of a Gorenstein $K[X]$-order…

W-algebras are a class of non-commutative algebras related to the classical universal enveloping algebras. They can be defined as a subquotient of U(g) related to a choice of nilpotent element e and compatible nilpotent subalgebra m. The…

We construct countable groups $G$ with the following new degree of W*-superrigidity: if $L(G)$ is virtually isomorphic, in the sense of admitting a bifinite bimodule, with any other group von Neumann algebra $L(\Lambda)$, then the groups…

This article presents a proof of the Feldman-Moore theorem on Cartan subalgebras in W*-algebras based on the non-commutative Stone equivalence between Boolean inverse semigroups and Boolean groupoids. The proof is decomposed into two parts.…

We prove a spectral theorem for bimodules in the context of graph C*-algebras. A bimodule over a suitable abelian algebra is determined by its spectrum (i.e., its groupoid partial order) iff it is generated by the Cuntz-Krieger partial…