Related papers: The $q$-Onsager algebra and its alternating centra…
The $q$-Onsager algebra $O_q$ is presented by two generators $W_0$, $W_1$ and two relations, called the $q$-Dolan/Grady relations. Recently Baseilhac and Koizumi introduced a current algebra $\mathcal A_q$ for $O_q$. Soon afterwards,…
The $q$-Onsager algebra $O_q$ is defined by two generators and two relations, called the $q$-Dolan/Grady relations. We investigate the alternating central extension $\mathcal O_q$ of $O_q$. The algebra $\mathcal O_q$ was introduced by…
The $q$-Onsager algebra $\mathcal O_q$ is defined by two generators $W_0, W_1$ and two relations called the $q$-Dolan/Grady relations. Recently Baseilhac and Kolb obtained a PBW basis for $\mathcal O_q$ with elements denoted $\lbrace B_{n…
The $q$-Onsager algebra, denoted $O_q$, is defined by two generators $W_0, W_1$ and two relations called the $q$-Dolan-Grady relations. Recently, Terwilliger introduced some elements of $O_q$, said to be alternating. These elements are…
The $q$-Onsager algebra $O_q$ is defined by two generators and two relations, called the $q$-Dolan/Grady relations. In 2019, Baseilhac and Kolb introduced two automorphisms of $O_q$, now called the Lusztig automorphisms. Recently, we…
The positive part $U^+_q$ of $U_q({\widehat{\mathfrak{sl}}}_2)$ has a presentation by two generators $X,Y$ that satisfy the $q$-Serre relations. The $q$-Onsager algebra $\mathcal O_q$ has a presentation by two generators $A,B$ that satisfy…
The $q$-Onsager algebra, denoted by $O_q$, is defined by generators $W_0, W_1$ and two relations called the $q$-Dolan-Grady relations. In 2017, Baseilhac and Kolb gave some elements of $O_q$ that form a Poincar\'e-Birkhoff-Witt basis. The…
Let $U^+_q$ denote the positive part of the quantized enveloping algebra $U_q(\widehat{\mathfrak{sl}}_2)$. The algebra $U^+_q$ has a presentation involving two generators $W_0$, $W_1$ and two relations, called the $q$-Serre relations. In…
The Onsager Lie algebra $O$ is often used to study integrable lattice models. The universal enveloping algebra of $O$ admits a $q$-deformation $O_q$ called the $q$-Onsager algebra. Recently, an algebra $\mathcal O_q$ was introduced called…
The $q$-Onsager algebra $\mathcal O_q$ is defined by two generators $A$, $A^*$ and two relations, called the $q$-Dolan/Grady relations. Recently P. Baseilhac and S. Kolb found an automorphism $L$ of $\mathcal O_q$, that fixes $A$ and sends…
An explicit homomorphism that relates the elements of the infinite dimensional non-Abelian algebra generating $O_q(\hat{sl_2})$ currents and the standard generators of the $q-$Onsager algebra is proposed. Two straightforward applications of…
Let $\textsf{A},\textsf{A}^*$ be the fundamental generators of the $q-$Onsager algebra. A linear basis for the $q-$Onsager algebra is known as the `zig-zag' basis [IT09]. In this letter, an attractive basis for the $q-$Onsager algebra is…
Let $\{A_j|j=0,1,...,rank(g)\}$ be the fundamental generators of the generalized $q-$Onsager algebra $\cal O_{q}(\widehat{g})$ introduced in \cite{BB1}, where $\widehat{g}$ is a simply-laced affine Lie algebra. New relations between certain…
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…
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…
This paper concerns the positive part $U^+_q$ of the quantum group $U_q({\widehat{\mathfrak{sl}}}_2)$. The algebra $U^+_q$ has a presentation involving two generators that satisfy the cubic $q$-Serre relations. We recently introduced an…
Let $\mathbb F$ denote a field, and fix a nonzero $q\in\mathbb F$ such that $q^4\not=1$. The universal Askey-Wilson algebra $\Delta_q$ is the associative $\mathbb F$-algebra defined by generators and relations in the following way. The…
Two sets of mutually commuting $q-$difference operators $x_i$ and $y_j$, $i,j=1, ...,N$ such that $x_i$ and $y_i$ generate a homomorphic image of the $q-$Onsager algebra for each $i$ are introduced. The common polynomial eigenfunctions of…
We construct W-algebra generalizations of the ^sl(2) algebra -- W-algebras W^{(2)}_n generated by two currents E and F with the highest pole of order n in their OPE. The n=3 term in this series is the Bershadsky--Polyakov algebra. We define…
A $q$-analogue of the Higgs algebra, which describes the symmetry properties of the harmonic oscillator on the $2$-sphere, is obtained as the commutant of the $\mathfrak{o}_{q^{1/2}}(2) \oplus \mathfrak{o}_{q^{1/2}}(2)$ subalgebra of…