An action of the free product $\mathbb Z_2 \star \mathbb Z_2 \star \mathbb Z_2$ on the $q$-Onsager algebra and its current algebra

Recently Pascal Baseilhac and Stefan Kolb introduced some automorphisms $T_0$, $T_1$ of the $q$-Onsager algebra $\mathcal O_q$, that are roughly analogous to the Lusztig automorphisms of $U_q(\widehat{\mathfrak{sl}}_2)$. We use $T_0, T_1$ and a certain antiautomorphism of $\mathcal O_q$ to obtain an action of the free product $\mathbb Z_2 \star \mathbb Z_2 \star \mathbb Z_2$ on $\mathcal O_q$ as a group of (auto/antiauto)-morphisms. The action forms a pattern much more symmetric than expected. We show that a similar phenomenon occurs for the associated current algebra $\mathcal A_q$. We give some conjectures and problems concerning $\mathcal O_q$ and $\mathcal A_q$.


Introduction
We will be discussing the q-Onsager algebra O q [2,20]. This algebra is infinite-dimensional and noncommutative, with a presentation involving two generators and two relations called the q-Dolan/Grady relations. The algebra appears in a number of contexts which we now summarize. The algebra O q is a q-deformation of the Onsager algebra from mathematical physics [17], [21,Remark 9.1] and is currently being used to investigate statistical mechanical models such as the XXZ open spin chain [1,2,4,[6][7][8]. The algebra O q appears in the theory of tridiagonal pairs; this is a pair of diagonalizable linear transformations on a finitedimensional vector space, each acting on the eigenspaces of the other in a block-tridiagonal fashion [13,19]. A tridiagonal pair of q-Racah type [14] is essentially the same thing as a finitedimensional irreducible O q -module [20,Theorem 3.10]. See [12][13][14][15]19,21,26] for work relating O q and tridiagonal pairs. The algebra O q comes up in algebraic combinatorics, in connection with the subconstituent algebra of a Q-polynomial distance-regular graph [13,18]. This topic is where O q originated; to our knowledge the q-Dolan/Grady relations first appeared in [18,Lemma 5.4]. The algebra O q appears in the theory of quantum groups, as a coideal subalgebra of U q ( sl 2 ) [3,11,16]. There exists an injective algebra homomorphism from O q into the algebra q [23,Proposition 5.6], and a noninjective algebra homomorphism from O q into the universal Askey-Wilson algebra [22,Sections 9,10], [25].
We will be discussing some automorphisms and antiautomorphisms of O q . In [9] Pascal Baseilhac and Stefan Kolb introduced two automorphisms T 0 , T 1 of O q that are roughly analogous to the Lusztig automorphisms of U q ( sl 2 ). More information about T 0 , T 1 is given in [24]. Using T 0 , T 1 and a certain antiautomorphism of O q , we will obtain an action of the free product Z 2 ⋆ Z 2 ⋆ Z 2 on O q as a group of (auto/antiauto)-morphisms. The action seems remarkable because it forms a pattern much more symmetric than expected. We show that a similar phenomenon occurs for the current algebra A q of O q . Our main results are Theorem 2.11 and Theorem 3.9. At the end of the paper we give some conjectures and problems concerning O q and A q .
2 The q-Onsager algebra O q We will define the q-Onsager algebra after a few comments. Let F denote a field. All vector spaces discussed in this paper are over F. All algebras discussed in this paper are associative, over F, and have a multiplicative identity. A subalgebra has the same multiplicative identity as the parent algebra. For an algebra A, an automorphism of A is an algebra isomorphism If A is commutative, then there is no difference between an automorphism and antiautomorphism of A. If A is noncommutative, then no map is both an automorphism and antiautomorphism of A. Recall the natural numbers N = {0, 1, 2, . . .}. Fix 0 = q ∈ F that is not a root of unity. We will use the notation We call O q the q-Onsager algebra. The relations (1), (2) are called the q-Dolan/Grady relations.
We now consider some automorphisms of O q . By the form of the relations (1), (2) there exists an automorphism of O q that swaps A, B. The following automorphisms of O q are less obvious. In [9] Pascal Baseilhac and Stefan Kolb introduced some automorphisms T 0 , T 1 of O q that satisfy The inverse automorphisms satisfy In [9] the automorphisms T 0 , T 1 are used to construct a Poincaré-Birkhoff-Witt (or PBW) basis for O q . In that construction the following result is used.
and the map T 1 sends Proof. The map T 0 is an automorphism of O q that fixes A. Therefore, (3) and evaluate the result using (1). We have verified the assertion about T 0 . The assertion about T 1 is similarly verified.
The automorphism group Aut(O q ) consists of the automorphisms of the algebra O q ; the group operation is composition.
The group N is freely generated by T ±1 0 , T ±1 1 . We have been discussing automorphisms of O q . We now bring in antiautomorphisms of O q .

Lemma 2.5. There exists an antiautomorphism S of O q that fixes A and B. Moreover
Proof. By the form of the q-Dolan/Grady relations.
The antiautomorphism S is related to the automorphisms T 0 , T 1 in the following way. Lemma 2.6. For the algebra O q , Proof. We verify the equation on the left in (9). In that equation, each side is an automorphism of O q . These automorphisms agree at A and B; this is checked using (3) and (5). These automorphisms are equal since A, B generate O q . We have verified the equation on the left in (9). The equation on the right in (9)  (i) The group H has order 2 and is not contained in N.
(ii) The group N is a normal subgroup of G with index 2.
Proof. (i) The group H has order 2 by the last assertion of Lemma 2.5. The group H is not contained in N, since Aut(O q ) contains N but not S.
(ii) By Lemma 2.6 and part (i) above.
(iv) The group G is the union of cosets N and NS. The elements of N are in Aut(O q ), and the elements of NS are not in Aut(O q ).
We now consider G from another point of view. Let Z 2 denote the group with two elements. The free product Z 2 ⋆ Z 2 ⋆ Z 2 has a presentation by generators a, b, c and relations a 2 = b 2 = c 2 = 1. Shortly we will display a group isomorphism To motivate this isomorphism we give a second presentation of Z 2 ⋆ Z 2 ⋆ Z 2 by generators and relations.
An isomorphism sends The inverse isomorphism sends Proof. One checks that each map is a group homomorphism and the maps are inverses. Consequently each map is a group isomorphism.
The inverse isomorphism sends Proof. For notational convenience we identify the group Z 2 ⋆ Z 2 ⋆ Z 2 with the group defined in Lemma 2.9, via the isomorphism in Lemma 2.9. Comparing (11) with the relations in Lemmas 2.5, 2.6 we obtain a surjective group homomorphism γ : Using the identification (12) we find that γ acts as in (14). We show that γ is an isomorphism. Let N denote the subgroup of Z 2 ⋆ Z 2 ⋆ Z 2 generated by t ±1 0 , t ±1 1 . From the relations (11) we see that Z 2 ⋆ Z 2 ⋆ Z 2 is the union of N and N s. We have γ(N ) = N and γ(N s) = NS. The cosets N, NS are disjoint and N contains the identity, so the kernel of γ is contained in N . This kernel is the identity by Lemma 2.4. Therefore γ is injective and hence an isomorphism. Line (15) follows from (13).
We now give our first main result. For notational convenience define Theorem 2.11. The free product Z 2 ⋆Z 2 ⋆Z 2 acts on the algebra O q as a group of (auto/antiauto)morphisms in the following way.
(i) The generator a acts as an antiautomorphism that sends (ii) The generator b acts as an antiautomorphism that sends (iii) The generator c acts as an antiautomorphism that sends The above Z 2 ⋆ Z 2 ⋆ Z 2 action is faithful.

The current algebra A q
In the previous section we obtained an action of Z 2 ⋆ Z 2 ⋆ Z 2 on the q-Onsager algebra O q . In this section we do something similar for the corresponding current algebra A q . In [6] Baseilhac and Koizumi introduce A q in order to solve boundary integrable systems with hidden symmetries related to a coideal subalgebra of U q ( sl 2 ). In [ and relations In the above equations ℓ ∈ N and ρ = −(q 2 − q −2 ) 2 . We are using the notation [X There is a redundancy among the generators (22), since we could use (23) to eliminate {G k+1 } k∈N or {G k+1 } k∈N in (24)-(33). These eliminations yield the equations in the next lemma.
Lemma 3.2. The following equations hold in A q . For k ∈ N, For k, ℓ ∈ N, We now consider some automorphisms of A q .
Definition 3.5. Define T 1 = ΩT 0 Ω, where Ω is from Lemma 3.3 and T 0 is from Lemma 3.4. By construction T 1 is an automorphism of the algebra A q .
We have been discussing automorphisms of A q . We now consider antiautomorphisms of A q .
Lemma 3.7. There exists an antiautomorphism S of A q that sends For k ∈ N. Moreover S fixes ∆ k+1 for k ∈ N. We have S 2 = 1.
Proof. The antiautomorphism S exists by the form of the defining relations (23)-(33) for A q . The map S 2 is an automorphism of A q that fixes W −k , W k+1 , G k+1 ,G k+1 for k ∈ N. These elements generate A q , so S 2 = 1. For k ∈ N the map S fixes ∆ k+1 by the form of ∆ k+1 given in [5, Lemma 2.1].
Lemma 3.8. For the algebra A q , Proof. Similar to the proof of Lemma 2.6.
We now obtain our second main result. Recall the free product Z 2 ⋆ Z 2 ⋆ Z 2 from above Lemma 2.9. For k ∈ N define Note by (23), (29), (31) that (42) Theorem 3.9. The free product Z 2 ⋆Z 2 ⋆Z 2 acts on the algebra A q as a group of (auto/antiauto)morphisms in the following way.
(i) The generator a acts as an antiautomorphism that sends (ii) The generator b acts as an antiautomorphism that sends (iii) The generator c acts as an antiautomorphism that sends Proof. For notational convenience we identify the group Z 2 ⋆ Z 2 ⋆ Z 2 with the group defined in Lemma 2.9, via the isomorphism in Lemma 2.9. Comparing (11) with the relations in Lemmas 3.7, 3.8, we obtain a group homomorphism This group homomorphism gives an action of Z 2 ⋆ Z 2 ⋆ Z 2 on the algebra A q as a group of (auto/antiauto)-morphisms such that s, t ±1 0 , t ±1 1 act as S, T ±1 0 , T ±1 1 , respectively. Using the identifications (12), (13) we find that this action satisfies condition (iv) in the theorem statement. By (43) each of a, b, c acts on A q as an antiautomorphism. For these elements the action on W −k , W ′ −k , W ′′ −k , ∆ k+1 is routinely obtained using Lemmas 3.4, 3.6, 3.7 along with Lemma 3.2 and (42).

Suggestions for future research
In this section we give some conjectures and problems concerning O q and A q .
Earlier in this paper we gave a Z 2 ⋆ Z 2 ⋆ Z 2 action on O q and A q . It is natural to ask whether these algebras are characterized by this sort of Z 2 ⋆Z 2 ⋆Z 2 action. As we pursue this question, let us begin with the simpler case of O q . The following concept is motivated by Theorem 2.11. (ii) there exists an antiautomorphism of A that sends (iii) there exists an antiautomorphism of A that sends (iv) the algebra A is generated by A, B, C.   We define some notation. Let λ 1 , λ 2 , . . . denote mutually commuting indeterminates. Let F[λ 1 , λ 2 , . . .] denote the algebra of polynomials in λ 1 , λ 2 , . . . that have all coefficients in F. For a subset Y ⊆ A q let Y denote the subalgebra of A q generated by Y . Shortly we will encounter some tensor products. All tensor products in this paper are understood to be over F.

Acknowledgment
The author thanks Pascal Baseilhac and Samuel Belliard for many discussions about the q-Onsager algebra and its current algebra.