Related papers: Motzkin Algebras
A Gelfand model for a semisimple algebra A over C is a complex linear representation that contains each irreducible representation of A with multiplicity exactly one. We give a method of constructing these models that works uniformly for a…
In linear algebra applications, elementary matrices hold a significant role. This paper presents a diagrammatic representation of all $2^m\times 2^n$-sized elementary matrices in algebraic ZX-calculus, showcasing their properties on…
We refine a construction of Choi, Farah and Ozawa to build a nonseparable amenable operator algebra $\mathcal A\subseteq\ell_\infty(M_2)$ whose nonseparable subalgebras, including $\mathcal A$, are not isomorphic to a $C^*$-algebra. This is…
This paper introduces an algebra structure on the part of the skein module of an arbitrary $3$-manifold $M$ spanned by links that represent $0$ in $H_1(M;\mathbb{Z}_2)$ when the value of the parameter used in the Kauffman bracket skein…
We introduce a type affine $C$ analogue of the nil Temperley--Lieb algebra, in terms of generators and relations. We show that this algebra $T(n)$, which is a quotient of the positive part of a Kac--Moody algebra of type $D_{n+1}^{(2)}$,…
We study a mixed tensor product $\mathbf{3}^{\otimes m} \otimes \mathbf{\overline{3}}^{\otimes n}$ of the three-dimensional fundamental representations of the Hopf algebra $U_{q} s\ell(2|1)$, whenever $q$ is not a root of unity. Formulas…
Two different types of centrally extended quantum reflection algebras are introduced. Realizations in terms of the elements of the central extension of the Yang-Baxter algebra are exhibited. A coaction map is identified. For the special…
A presentation of the centralizer of the three-fold tensor product of the spin $s$ representation of the quantum group $U_q(\mathfrak{sl}_2)$ is provided. It is expressed as a quotient of the Askey-Wilson braid algebra. This newly defined…
We provide a necessary and sufficient condition for a type D Temperley-Lieb algebra ${\rm TLD}_n(\delta)$ being semi-simple by studying branching rule for cell modules. As a byproduct, our result is used to study the so-called forked…
Let $n$ be a maximal nilpotent subalgebra of a complex symmetric Kac-Moody Lie algebra. Lusztig has introduced a basis of U(n) called the semicanonical basis, whose elements can be seen as certain constructible functions on varieties of…
For a finite-dimensional Hopf algebra $A$, the McKay matrix $M_V$ of an $A$-module $V$ encodes the relations for tensoring the simple $A$-modules with $V$. We prove results about the eigenvalues and the right and left (generalized)…
Unitary representations of the Temperley-Lieb algebra $TL_N(Q)$ on the tensor space $({\mathbb C^n})^{\otimes N}$ are considered. Two criteria are given for determining when an orthogonal projection matrix $P$ of a rank $r$ gives rise to…
A new basis of the $q$-Brauer algebra is introduced, which is a lift of Murphy bases of Hecke algebras of symmetric groups. This basis is a cellular basis in the sense of Graham and Lehrer. Subsequently, using combinatorial language we…
We investigate pairwise quasi-orthogonal subalgebras in $M_{p^{kn}}$ which are isomorphic to $M_{p^{k}}$ for $k \ge 1$, $n \ge 2$ and a prime number $p$ with $p \ge 3$. We prove there exist $p^{2kn}-1/p^{2k}-1$ such subalgebras and they…
Given a minuscule representation of a simple Lie algebra, we find an algebraic model for the action of a regular element and show that these models can be glued together over the adjoint quotient, viewed as the set of all regular conjugacy…
In this paper we prove that the subalgebras of cocommutative elements in the quantized coordinate rings of $M_{n}$, $GL_{n}$ and $SL_{n}$ are the centralizers of the trace $x_{1,1}+\dots+x_{n,n}$ in each algebra, for…
We develop the basic constructions of homological algebra in the (appropriately defined) unbounded derived categories of modules over algebras over coalgebras over noncommutative rings (which we call semialgebras over corings). We define…
We refine a conjecture by Lehrer and Solomon on the structure of the Orlik-Solomon algebra of a finite Coxeter group $W$ and relate it to the descent algebra of $W$. As a result, we claim that both the group algebra of $W$, as well as the…
We prove that a countable dimensional associative algebra (resp. a countable semigroup) of locally subexponential growth is $M_\infty$-embeddable as a left ideal in a finitely generated algebra (resp. semigroup) of subexponential growth.…
Algebraic basics on Temperley-Lieb algebras are proved in an elementary and straightforward way with the help of tensor categories behind them.