Related papers: $q$-Coxeter matrix and $q$-Cartan matrix for a hom…
The $(q,t)$-Cartan matrix specialized at $t=1$, usually called the quantum Cartan matrix, has deep connections with (i) the representation theory of its untwisted quantum affine algebra, and (ii) quantum unipotent coordinate algebra, root…
We first review some invariant theoretic results about the finite subgroups of SU(2) in a quick algebraic way by using the McKay correspondence and quantum affine Cartan matrices. By the way it turns out that some parameters (a,b,h;p,q,r)…
As every simple module of a quiver Hecke algebra appears as the image of the R-matrix defined on the convolution product of certain cuspidal modules, knowing the $\mathbb{Z}$-invariants of the R-matrices between cuspidal modules is quite…
Let Q be a finite quiver without oriented cycles, and let k be an algebraically closed field. The main result in this paper is that there is a natural bijection between the elements in the associated Coxeter group W_Q and the cofinite…
Let C be a coalgebra and consider the Grothendieck groups of the categories of the socle-finite injective right and left C-comodules. One of the main aims of the paper is to study Coxeter transformation, and its dual, of a pointed sharp…
We associate to a bound quiver (Q,I) a CW-complex which we denote by B(Q,I), and call the classifying space of (Q,I). We show that the fundamental group of B(Q,I) is isomorphic to the fundamental group of (Q,I). Moreover, we show that this…
We introduce and study a class of Iwanaga-Gorenstein algebras defined via quivers with relations associated with symmetrizable Cartan matrices. These algebras generalize the path algebras of quivers associated with symmetric Cartan…
We find the R matrix for the inhomogeneous quantum groups whose homogeneous part is $GL_q(n)$, or its restrictions to $SL_q(n)$,$U_q(n)$ and $SU_q(n)$. The quantum Yang-Baxter equation for R holds because of the Hecke relation for the…
The aim of this paper is first to introduce and study Rota-Baxter cosystems and bisystems as generalization of Rota-Baxter coalgebras and bialgebras, respectively, with various examples. The second purpose is to provide an alternative…
We provide a formalism using the $q$-Cartan matrix to compute the instanton partition function of quiver gauge theory on various manifolds. Applying this formalism to eight dimensional setups, we introduce the notion of double quiver gauge…
Based on properties of the universal R-matrix, we derive universal Baxter TQ-relations for quantum integrable systems with (diagonal) open boundaries associated with $U_{q}(\widehat{sl_{2}})$. The Baxter TQ-relations for the open XXZ-spin…
The aim of the present study is to establish some properties for q-Bessel matrix polynomials such as several q-differential matrix equation, q-differential matrix relations and q-recurrence matrix relations, and integral representation,…
We provide conditions on the defining graph of a right-angled Coxeter group presentation that guarantees the boundary of any CAT(0) space on which the group acts geometrically will be locally connected. This is a revised version of a…
A simple relation between inhomogeneous transfer matrices and boundary quantum KZ equations is exhibited for quantum integrable systems with reflecting boundary conditions, analogous to an observation by Gaudin for periodic systems. Thus…
Let $M$ be a closed surface, $q\geq 2$ and $n\geq 2$. In this paper, we analyze the Coxeter-type quotient group $B_n(M)(q)$ of the surface braid group $B_{n}(M)$ by the normal closure of the element $\sigma_1^q$, where $\sigma_1$ is the…
To each symmetrizable Cartan matrix, we associate a finite free EI category. We prove that the corresponding category algebra is isomorphic to the algebra defined in [C. Geiss, B. Leclerc, and J. Schr\"{o}er, Quivers with relations for…
As part of our study of the $q$-tetrahedron algebra $\boxtimes_q$ we introduce the notion of a $q$-inverting pair. Roughly speaking, this is a pair of invertible semisimple linear transformations on a finite-dimensional vector space, each…
In a recent paper by K.-H. Lee, K. Lee and M. Mills, a mutation of reflections in the universal Coxeter group is defined in association with a mutation of a quiver. A matrix representation of these reflections is determined by a linear…
The main objective of this paper is to construct a symmetric monoidal closed model category of coherently commutative monoidal quasi-categories. We construct another model category structure whose fibrant objects are (essentially) those…
In the previous article we introduced the new concept of mixed representations of quivers and described the generators of their algebras of invariants. In this article we describe the defining relations of these algebras. Some applications…