Related papers: Crystal bases and categorifications
This is a survey of noncommutative generalizations of the spectrum of a ring, written for the Notices of the American Mathematical Society.
In this note we explain how to obtain cluster algebras from triangulations of (punctured) discs following the approach of S. Fomin, M. Shapiro and D. Thurston. Furthermore, we give a description of m-cluster categories via diagonals (arcs)…
In this research, a general theoretical framework for clustering is proposed over specific partial algebraic systems by the present author. Her theory helps in isolating minimal assumptions necessary for different concepts of clustering…
We propose a skein model for the quantum cluster algebras of surface type with coefficients. We introduce a skein algebra $\mathscr{S}_{\Sigma,\mathbb{W}}^{A}$ of a walled surface $(\Sigma,\mathbb{W})$, and prove that it has a quantum…
This is a survey paper on Alegbraic Geometry over Lie Algebras
This is a survey article devoted to the study of real structures on complex algebraic varieties endowed with a reductive group action.
This paper is partly a report on current knowledge concerning the structure of (generic) quantized coordinate rings and their prime spectra, and partly propaganda in support of the conjecture that since these algebras share many common…
We show that the coordinate ring of the Vinberg monoid of a simply connected semisimple complex group is an upper cluster algebra. As an application, we construct cluster structures on a large class of flat reductive monoids. After…
A cluster algebra is unistructural if the set of its cluster variables determines its clusters and seeds. It is conjectured that all cluster algebras are unistructural. In this paper, we show that any cluster algebra arising from a…
The main results of this paper are generalizations some classical theorems about transversals for families of finite sets to some cases of families of infinite sets.
We give several explicit examples of quantum cluster algebra structures, as introduced by Berenstein and Zelevinsky, on quantized coordinate rings of partial flag varieties and their associated unipotent radicals. These structures are shown…
In this survey we discuss some combinatorial aspects of higher cluster categories.
There has been a great deal of research on graphs defined on algebraic structures in the last two decades. In this paper we begin an exploration of hypergraphs defined on algebraic structures, especially groups, to investigate whether this…
We give a modern account of the construction and structure of the space of generalized connections, an extension of the space of connections that plays a central role in loop quantum gravity.
We review several aspects of clusters of galaxies and their application to cosmology. We present first results of numerical simulations of the dynamics of the intra-cluster gas and of different interaction processes between cluster galaxies…
We introduce and develop a structure theory of a new class of noncommutative rings - Galois orders, that generalize classical orders in noncommutative rings. Galois orders realized as certain subrings of invariants in skew semigroup rings.…
This paper describes the centers of the universal enveloping algebras and the invariant rings of the standard filiform Lie algebras over fields of characteristic zero and also over large enough prime characteristic. We determine explicit…
We give an overview on recent results concerning additive unit representations. Furthermore the solutions of some open questions are included. The central problem is whether and how certain rings are (additively) generated by their units.…
We first study a new family of graded quiver varieties together with a new $t$-deformation of the associated Grothendieck rings. This provides the geometric foundations for a joint paper by Yoshiyuki Kimura and the author. We further…
The basic methods of constructing the sets of mutually unbiased bases in the Hilbert space of an arbitrary finite dimension are discussed and an emerging link between them is outlined. It is shown that these methods employ a wide range of…