Related papers: Braid group action and root vectors for the $q$-On…

It is a classical result in representation theory that the braid group $\mathscr{B}_\mathfrak{g}$ of a simple Lie algebra $\mathfrak{g}$ acts on any integrable representation of $\mathfrak{g}$ via triple products of exponentials in its…

Recently Pascal Baseilhac and Stefan Kolb obtained a PBW basis for the $q$-Onsager algebra $\mathcal O_q$. They defined the PBW basis elements recursively, and it is obscure how to express them in closed form. To mitigate the difficulty, we…

In this paper, we develop the PBW theory for the bosonic extension $\qbA{\g}$ of a quantum group $\mathcal{U}_q(\g)$ of \emph{any} finite type. When $\g$ belongs to the class of \emph{simply-laced type}, the algebra $\qbA{\g}$ arises from…

We describe the groups of automorphisms of two generated free braided associative algebras with involutive diagonal braidings over a field of characteristic $\neq 2$. Depending on the form of the diagonal involutive braiding, five different…

We establish isomorphisms between certain specializations of Birman-Murakami-Wenzl algebras and the symmetric squares of Temperley-Lieb algebras. These isomorphisms imply a link-polynomial identity due to W. B. R. Lickorish. As an…

Let $B'_n$ be the commutator subgroup of the braid group $B_n$. We prove that $Aut(B'_n)=Aut(B_n)$ for $n\ge4$. This answers a question asked by Vladimir Lin.

For a braided vector space $(V,\sigma)$ with braiding $\sigma$ of Hecke type, we introduce three associative algebra structures on the space $\oplus_{p=0}^{M}\mathrm{End}S_\sigma^p(V)$ of graded endomorphisms of the quantum symmetric…

In this thesis, the connection between recently introduced algebraic structures (tridiagonal algebra, $q$-Onsager algebra, generalized $q-$Onsager algebras), related representation theory (tridiagonal pair, Leonard pair, orthogonal…

Let V be the 7-dimensional irreducible representation of the quantum group U_q(g_2). For each n, there is a map from the braid group B_n to the endomorphism algebra of the n-th tensor power of V, given by R-matrices. We can extend this…

We construct a braiding operator in terms of the quantum dilogarithm function based on the quantum cluster algebra. We show that it is a q-deformation of the R-operator for which hyperbolic octrahedron is assigned. Also shown is that, by…

The present paper develops a general theory of quantum group analogs of symmetric pairs for involutive automorphism of the second kind of symmetrizable Kac-Moody algebras. The resulting quantum symmetric pairs are right coideal subalgebras…

We classify up to isomorphism the quantum generalized Weyl algebras and determine their automorphism groups in all cases in a uniform way, including those where the parameter q is a root of unity, thereby completing the results obtained by…

We define an action of the braid group of a simple Lie algebra on the space of imaginary roots in the corresponding quantum affine algebra. We then use this action to determine an explicit condition for a tensor product of arbitrary…

We establish PBW type bases for $\imath$quantum groups of arbitrary finite type, using the relative braid group symmetries. Explicit formulas for root vectors are provided for $\imath$quantum groups of each rank 1 type. We show that our PBW…

Let g be a simple complex finite dimensional Lie algebra and let U_q^+(g) be the positive part of the quantum enveloping algebra of g. If g is of type A_2, the group of algebra automorphisms of U_q^+(g) is a semidirect product of a…

We define a notion of quantum automorphism group of Graph C*-algebras for finite, connected graphs. Under the assumption that the underlying graph does not have any multiple edge or loop, the quantum automorphism group of underlying…

Let $\Gamma$ be a finite group acting faithfully and linearly on a vector space $V$. Let $T(V)$ ($S(V)$) be the tensor (symmetric) algebra associated to $V$ which has a natural $\Gamma$ action. We study generalized quadratic relations on…

This is a systematic introduction for physicists to the theory of algebras and groups with braid statistics, as developed over the last three years by the author. There are braided lines, braided planes, braided matrices and braided groups…

Suppose that $(G,T)$ is a second countable locally compact transformation group given by a homomorphism $\ell:G\to\Homeo(T)$, and that $A$ is a separable continuous-trace \cs-algebra with spectrum $T$. An action $\alpha:G\to\Aut(A)$ is said…

Inspired by the Basilica group $\mathcal B$, we describe a general construction which allows us to associate to any group of automorphisms $G \leq \operatorname{Aut}(T)$ of a rooted tree $T$ a family of Basilica groups…