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 Baseilhac and Koizumi, who called it the current algebra of $O_q$. Recently Baseilhac and Shigechi gave a presentation of $\mathcal O_q$ by generators and relations. The presentation is attractive, but the multitude of generators and relations makes the presentation unwieldy. In this paper we obtain a presentation of $\mathcal O_q$ that involves a subset of the original set of generators and a very manageable set of relations. We call this presentation the compact presentation of $\mathcal O_q$. This presentation resembles the compact presentation of the alternating central extension for the positive part of $U_q(\widehat{\mathfrak{sl}}_2)$.
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, Baseilhac and Shigechi gave a presentation of $\mathcal A_q$ by generators and relations. We show that these generators give a PBW basis for $\mathcal A_q$. Using this PBW basis, we show that the algebra $\mathcal A_q$ is isomorphic to $O_q \otimes \mathbb F \lbrack z_1, z_2, \ldots \rbrack$, where $\mathbb F$ is the ground field and $\lbrace z_n \rbrace_{n=1}^\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 alternating central extension of $O_q$. In this paper we introduce a Lie algebra $\mathcal O$ that is roughly described by the following two analogies: (i) $\mathcal O$ is to $O$ as $\mathcal O_q$ is to $O_q$; (ii) $O_q$ is to $O$ as $\mathcal O_q$ is to $\mathcal O$. We call $\mathcal O$ the alternating central extension of $O$. This paper contains a com…
The $q$-Onsager algebra $O_q$ has a presentation involving two generators $W_0$, $W_1$ and two relations, called the $q$-Dolan/Grady relations. The alternating central extension $\mathcal O_q$ has a presentation involving the alternating generators $\lbrace \mathcal W_{-k}\rbrace_{k=0}^\infty$, $\lbrace \mathcal W_{k+1}\rbrace_{k=0}^\infty$, $ \lbrace \mathcal G_{k+1}\rbrace_{k=0}^\infty$, $\lbrace \mathcal {\tilde G}_{k+1}\rbrace_{k=0}^\infty$ and a large number of relations. Let $\langle \mathcal W_0, \mathcal W_1 \rangle$ denote the subalgebra of $\mathcal O_q$ generated by $\mathcal W_0$, …
Automorphisms of the infinite dimensional Onsager algebra are introduced. Certain quotients of the Onsager algebra are formulated using a polynomial in these automorphisms. In the simplest case, the quotient coincides with the classical analog of the Askey-Wilson algebra. In the general case, generalizations of the classical Askey-Wilson algebra are obtained. The corresponding class of solutions of the non-standard classical Yang-Baxter algebra are constructed, from which a generating function of elements in the commutative subalgebra is derived. We provide also another presentation of the Ons…
The $sl_N$-Onsager algebra has been introduced by Uglov and Ivanov in 1995. In this letter, a FRT presentation of the $sl_N$-Onsager algebra is given, its current algebra and commutative subalgebra are constructed. Certain quotients of the $sl_N$-Onsager algebra are then considered, which produce `classical' analogs of higher rank extensions of the Askey-Wilson algebra. As examples, the cases $N=3$ and $N=4$ are described in details.
For a Coxeter system (G,S) the multi-parametric alternating subalgebra H^+(G) of the Hecke algebra and the alternating subgroup B^+(G) of the braid group are defined. Two presentations for H^+(G) and B^+(G) are given; one generalizes the Bourbaki presentation for the alternating subgroups of Coxeter groups, another one uses generators related to edges of the Coxeter graph.
An infinite dimensional algebra denoted $\bar{\cal A}_q$ that is isomorphic to a central extension of $U_q^+$ - the positive part of $U_q(\widehat{sl_2})$ - has been recently proposed by Paul Terwilliger. It provides an `alternating' Poincaré-Birkhoff-Witt (PBW) basis besides the known Damiani's PBW basis built from positive root vectors. In this paper, a presentation of $\bar{\cal A}_q$ in terms of a Freidel-Maillet type algebra is obtained. Using this presentation: (a) finite dimensional tensor product representations for $\bar{\cal A}_q$ are constructed; (b) explicit isomorphisms from $\bar…
This paper is about the positive part $U^+_q$ of the quantum group $U_q(\widehat{\mathfrak{sl}}_2)$. The algebra $U^+_q$ has a presentation with two generators $A,B$ that satisfy the cubic $q$-Serre relations. Recently we introduced a type of element in $U^+_q$, said to be alternating. Each alternating element commutes with exactly one of $A$, $B$, $qBA-q^{-1}AB$, $qAB-q^{-1}BA$; this gives four types of alternating elements. There are infinitely many alternating elements of each type, and these mutually commute. In the present paper we use the alternating elements to obtain a central extensio…
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 $q$-Dolan/Grady relations. We give two results that describe how $U^+_q$ and $\mathcal O_q$ are related. First, we consider the filtration of $\mathcal O_q$ whose $n$th component is spanned by the products of at most $n$ generators. We show that the associated graded algebra is isomorphic to $U^+_q$. Second, we introduce an algebra $\square_q$ and show h…
The main result of this paper shows that, over large enough fields of characteristic different from $2$, the alternating Hecke algebras are $\mathbb{Z}$-graded algebras that are isomorphic to fixed-point subalgebras of the quiver Hecke algebra of the symmetric group $\mathfrak{S}_n$. As a special case, this shows that the group algebra of the alternating group, over large enough fields of characteristic different from $2$, is a $\mathbb{Z}$-graded algebra. We give a homogeneous presentation for these algebras, compute their graded dimension and show that the blocks of the quiver Hecke algebras…
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 1993 I. Damiani obtained a PBW basis for $U^+_q$, consisting of some elements $\lbrace E_{n δ+ α_0} \rbrace_{n=0}^\infty$, $\lbrace E_{n δ+ α_1} \rbrace_{n=0}^\infty$, $\lbrace E_{n δ} \rbrace_{n=1}^\infty$. In 2019 we introduced the alternating central extension $\mathcal U^+_q$ of $U^+_q$. We defined $\mathcal U^+_q$ by generators and relations. The generato…
Classical symmetric pairs consist of a symmetrizable Kac-Moody algebra $\mathfrak{g}$, together with its subalgebra of fixed points under an involutive automorphism of the second kind. Quantum group analogs of this construction, known as quantum symmetric pairs, replace the fixed point Lie subalgebras by one-sided coideal subalgebras of the quantized enveloping algebra $U_q(\mathfrak{g})$. We provide a complete presentation by generators and relations for these quantum symmetric pair coideal subalgebras. These relations are of inhomogeneous $q$-Serre type and are valid without restrictions on …
