A faithful tensor space representation for the blob algebra

Representations of Toroidal general linear Superalgebra

We consider general linear superalgebra (type A) and tensor with Laurent polynomial ring in several variables. We then consider the universal central extension of this Lie superalgebra which we call toroidal superalgebra. We give a faithful representation of toroidal superalgebra using vertex operators and bosons.

The periplectic Brauer algebra III: The Deligne category

We construct a faithful categorical representation of an infinite Temperley-Lieb algebra on the periplectic analogue of Deligne's category. We use the corresponding combinatorics to classify thick tensor ideals in this periplectic Deligne category. This allows to determine the objects in the kernel of the monoidal functor going to the module category of the periplectic Lie supergroup. We use this to classify indecomposable direct summands in the tensor powers of the natural representation, determine which are projective and determine their simple top.

Schur-Weyl duality, Verma modules, and row quotients of Ariki-Koike algebras

We prove a Schur-Weyl duality between the quantum enveloping algebra of $\mathfrak{gl}_m$ and certain quotient algebras of Ariki-Koike algebras, which we give explicitly. The duality involves several algebraically independent parameters and is realized through the tensor product of a parabolic universal Verma module and a tensor power of the natural representation of $\mathfrak{gl}_m$. We also give a new presentation by generators and relations of the generalized blob algebras of Martin and Woodcock as well as an interpretation in terms of Schur-Weyl duality by showing they occur as a particul…

全忠实函子的张量三角几何

The tensor triangular geometry of fully faithful functors

该研究证明全忠实几何函子诱导的Balmer谱映射是纤维连通的商映射，引入tt-范畴的集中与幺元化概念，计算相关范畴的Balmer谱并研究比较映射性质，给出理解tt-范畴谱的分层方法。

On the Representation Theory of an Algebra of Braids and Ties

We consider the algebra ${\cal E}_n(u)$ introduced by F. Aicardi and J. Juyumaya as an abstraction of the Yokonuma-Hecke algebra. We construct a tensor space representation for ${\cal E}_n(u)$ and show that this is faithful. We use it to give a basis for ${\cal E}_n(u)$ and to classify its irreducible representations.

On the Burnside-Brauer-Steinberg theorem

A well-known theorem of Burnside says that if $ρ$ is a faithful representation of a finite group $G$ over a field of characteristic $0$, then every irreducible representation of $G$ appears as a constituent of a tensor power of $ρ$. In 1962, R. Steinberg gave a module theoretic proof that simultaneously removed the constraint on the characteristic, and allowed the group to be replaced by a monoid. Brauer subsequently simplified Burnside's proof and, moreover, showed that if the character of $ρ$ takes on $r$ distinct values, then the first $r$ tensor powers of $ρ$ already contain amongst them a…

The dihedral Dunkl--Dirac symmetry algebra with negative Clifford signature

The Dunkl--Dirac symmetry algebra is an associative subalgebra of the tensor product of a Clifford algebra and the faithful polynomial representation of a rational Cherednik algebra. In previous work, the finite-dimensional representations of the Dunkl--Dirac symmetry algebra in three dimensions linked with a dihedral group were given. We give here the necessary results to proceed to the same construction when the Clifford algebra in the tensor product has negative signature.

On certain representations of automorphism groups of an algebraically closed field

Let k be an algebraically closed field of characteristic zero, F its algebraically closed extension, and G be the group of k-automorphisms of F endowed with a natural topology. One of the purposes of this paper is to show that any non-faithful continuous representation of G factors through a discrete quotient of G. Properties of representation of G arising from geometry are studied. In some cases the groups of morphisms between geometric objects are identified with the groups of morphisms between corresponding G-modules, and the ${\rm Ext}^1$'s are related. In particular, the category of abeli…

A Framization of the Hecke algebra of Type B

In this article we introduce a framization of the Hecke algebra of type B. For this framization we construct a faithful tensorial representation and two linear bases. We finally construct a Markov trace on these algebras and from this trace we derive isotopy invariants for framed and classical knots and links in the solid torus.

Cell structures for the Yokonuma-Hecke algebra and the algebra of braids and ties

We construct a faithful tensor representation for the Yokonuma-Hecke algebra Y, and use it to give a concrete isomorphism between Y and Shoji's modified Ariki-Koike algebra. We give a cellular basis for Y and show that the Jucys-Murphy elements for Y are JM-elements in the abstract sense. Finally, we construct a cellular basis for the Aicardi-Juyumaya algebra of braids and ties.

Meromorphic tensor equivalence for Yangians and quantum loop algebras

Let ${\mathfrak g}$ be a complex semisimple Lie algebra, and $Y_h({\mathfrak g})$, $U_q(L{\mathfrak g})$ the corresponding Yangian and quantum loop algebra, with deformation parameters related by $q=\exp(πi h)$. When $h$ is not a rational number, we constructed in arXiv:1310.7318 a faithful functor $Γ$ from the category of finite-dimensional representations of $Y_h ({\mathfrak g})$ to those of $U_q(L{\mathfrak g})$. The functor $Γ$ is governed by the additive difference equations defined by the commuting fields of the Yangian, and restricts to an equivalence on a subcategory of $Y_h({\mathfrak…

Tensor Representations for the Drinfeld Double of the Taft Algebra

Over an algebraically closed field $\mathbb k$ of characteristic zero, the Drinfeld double $D_n$ of the Taft algebra that is defined using a primitive $n$th root of unity $q \in \mathbb k$ for $n \geq 2$ is a quasitriangular Hopf algebra. Kauffman and Radford have shown that $D_n$ has a ribbon element if and only if $n$ is odd, and the ribbon element is unique; however there has been no explicit description of this element. In this work, we determine the ribbon element of $D_n$ explicitly. For any $n \geq 2$, we use the R-matrix of $D_n$ to construct an action of the Temperley-Lieb algebra $\m…

$U_q\mathfrak{sl}_2$-invariant non-compact boundary conditions for the XXZ spin chain

We introduce new $U_q\mathfrak{sl}_2$-invariant boundary conditions for the open XXZ spin chain. For generic values of $q$ we couple the bulk Hamiltonian to an infinite-dimensional Verma module on one or both boundaries of the spin chain, and for $q=e^{\frac{iπ}{p}}$ a $2p$-th root of unity $ - $ to its $p$-dimensional analogue. Both cases are parametrised by a continuous "spin" $α\in\mathbb{C}$. To motivate our construction, we first specialise to $q=i$, where we obtain a modified XX Hamiltonian with unrolled quantum group symmetry, whose spectrum and scaling limit is computed explicitly usin…