Generalized Onsager algebras

Let $\mathfrak{g}(A)$ be the Kac-Moody algebra with respect to a symmetrizable generalized Cartan matrix $A$. We give an explicit presentation of the fix-point Lie subalgebra $\mathfrak{k}(A)$ of $\mathfrak{g}(A)$ with respect to the Chevalley involution. It is a presentation of $\mathfrak{k}(A)$ involving inhomogeneous versions of the Serre relations, or, from a different perspective, a presentation generalizing the Dolan-Grady presentation of the Onsager algebra. In the finite and untwisted affine case we explicitly compute the structure constants of $\mathfrak{k}(A)$ in terms of a Chevalley type basis of $\mathfrak{k}(A)$. For the symplectic Lie algebra and its untwisted affine extension we explicitly describe the one-dimensional representations of $\mathfrak{k}(A)$.


Introduction
The Onsager algebra is the infinite dimensional complex Lie algebra with linear basis {A k , G m } k∈Z,m∈Z >0 and Lie bracket for k, ℓ ∈ Z and m, n ∈ Z >0 , where G −m := −G m (m > 0) and G 0 := 0. It first appeared in Onsager's [19] paper on the two-dimensional Ising model in a zero magnetic field. It is generated as Lie algebra by B 0 := −A −1 and B 1 := A 0 . The corresponding defining relations, known as the Dolan-Grady [8]  The Onsager algebra plays an important role in integrable systems, representation theory and special functions, see, e.g., [19,20,8,6,7,22,14,10] and references therein. An important reason for its appearance in many different contexts is the fact that the Onsager algebra is isomorphic to the fix-point Lie subalgebra of the affine Lie algebra sl 2 with respect to the Chevalley involution, a result which is due to Roan [22,Prop. 1]. In this paper we define a generalization L(A) of the Onsager algebra in terms of explicit generators and relations depending on a symmetrizable generalized Cartan matrix A. It reduces to the Onsager algebra in Dolan-Grady form when A is of type A (1) 1 . We show that L(A) is isomorphic to the fix-point Lie subalgebra of the Kac-Moody algebra g(A) associated to A with respect to its Chevalley involution. For A of type A (1) 1 this is the earlier mentioned result of Roan [22,Prop. 1]. For A of type A (1) n (resp. D (1) n ) we recover the results of Uglov and Ivanov [23] (resp. Date and Usami [5]). Their techniques partly rely on the loop presentation of the associated affine Lie algebra, which is specific to the affine case.
Much work has recently been done on quantum group analogues of generalized Onsager algebras (see, e.g., [2,15] and references therein). Baseilhac and Belliard [2] introduced generalized q-Onsager algebras for A of affine type and provided algebra homomorphisms mapping the generalized q-Onsager algebras to the associated quantum affine algebra U q (g(A)). Kolb [15] puts these results in the more general framework of quantum symmetric Kac-Moody pairs, which concern quantum analogues of fix-point Kac-Moody subalgebras with respect to involutive automorphisms of the second kind. In this general context Kolb [15] gave a detailed study of the algebraic structures of the associated fix-point Kac-Moody Lie subalgebras and their quantum analogs, leading in particular cases to explicit identifications with Baseilhac's and Belliard's [2] generalized q-Onsager algebras. The presentations [15, §7] and [1, §3] of Kolb's coideal subalgebras in the general context are rather intricate and only explicit when the off-diagonal Cartan integers are ≥ −3 (for A of finite type, it goes back to the work of Letzter [18, §7]). I hope that the present paper will be a useful step towards a complete and explicit algebraic presentation of Kolb's [15] coideal subalgebras.
The content of the paper is as follows. In Section 2 we introduce the generalized Onsager algebra L(A) associated to a symmetrizable generalized Cartan matrix A. We prove that it is isomorphic to the fix-point Lie subalgebra k(A) of g(A) with respect to the Chevalley involution. We define a filtration on L(A) such that the associated graded Lie algebra is isomorphic to the graded nilpotent Lie subalgebra of g(A) generated by the positive root vectors. Finally, we describe the one-dimensional representations of L(A).
In Section 3 we consider the special case of finite and untwisted affine indecomposable Cartan matrices, corresponding to g(A) being a simple Lie algebra and an untwisted affine Lie algebra respectively. We give in these cases an integral form k Z (A) of the fix-point Lie subalgebra k(A) and explicitly describe the structure constants in terms of a Chevalley-type basis of k Z (A). This leads to a generalization of Onsager's original presentation [19] of the Onsager algebra. In Section 4 we further restrict attention to A of type C r and type C (1) r (r ≥ 1). For type C r the fix-point Lie subalgebra k(A) is isomorphic to gl r (C). In this case we make a detailed comparison of the Dolan-Grady type presentation of k(A) and its Serre presentation. Both in the finite and affine case we explicitly describe the one-dimensional representations of k(A). In the finite case this will play a role in the upcoming paper of the author and Reshetikhin [21] on vector-valued Harish-Chandra series. Reshetikhin for useful discussions on the topic of the paper. Shortly after the appearance on the arXiv of this paper Xinhong Chen, Ming Lu and Weiqiang Wang informed me that they have obtained an explicit presentation of Kolb's coideal subalgebra associated to a quasi-split Kac-Moody quantum symmetric pair using different techniques. Their results have appeared now in the preprint [3].

The generalized Onsager algebra
Let A = (a ij ) n i,j=1 be a symmetrizable generalized Cartan matrix and a realization of A. So h(A) is a complex vector space of dimension 2n − rk(A) and Π ⊂ h * , Π ∨ ⊂ h are linear independent subsets such that α j (h i ) = a ij . The Kac-Moody algebra g(A) associated to A is the complex Lie algebra generated by h(A) and the Chevalley generators e i , f i (i = 1, . . . , n), with defining relations Here we use the definition of the Kac-Moody algebra involving the Serre relations which has been shown in [4] to be equivalent to the usual definition [12] when A is symmetrizable.
We recall here some basic properties of Kac-Moody algebras, see [12] for further details. Let ω be the Chevalley involution of g(A). It is the involutive automorphism of g(A) be the Lie subalgebra of g(A) generated by e 1 , . . . , e n . It is a graded nilpotent Lie algebra with the generators e i having degree one. The defining relations of n(A) in terms of the Chevalley generators e i (i = 1, . . . , n) are the Serre relations (ad e i ) 1−a ij e j = 0 (i = j), see [4]. The triangular decomposition of g(A) is The g α (A) are finite dimensional and g 0 (A) = h(A). Write m(α) for the dimension of g α (A). Note that m(α i ) = 1 for i = 1, . . . , n.
The root system of g(A) is It decomposes in positive and negative roots, where Q + := n i=1 Z ≥0 α i . The height of a positive root α ∈ Φ + is defined by with r i (α) the nonnegative integers such that α = n i=1 r i (α)α i . The nilpotent Lie algebras n(A) and ω(n(A)) decompose as j=1 is a linear basis of (g α (A) ⊕ g −α (A)) ω and Consider the elements Proof. For completeness we recall the proof. Note that for α = α i ∈ Π a simple root we have m(α i ) = 1 and y (1) α i is a nonzero constant multiple of Y i . We now prove by induction to the height of α ∈ Φ + that y Suppose α ∈ Φ + has height k > 1 and fix 1 ≤ j ≤ m(α). Since n(A) is generated by e 1 , . . . , e n , we can write for certain constants c i ∈ C, with the sum over k-tuples i = (i 1 , . . . , i k ) such that α = α i 1 + · · · + α i k . Then By the induction hypothesis we conclude that y The defining relations of k(A) in terms of the generators Y 1 , . . . , Y n take the form of inhomogeneous Serre relations involving the following integers c ij s [r] for 1 ≤ i = j ≤ n and r ≥ s ≥ 0.
for r ≥ 0, where the right hand side is read as zero for r = 0. Applying the Chevalley involution ω shows that the formula with e i ↔ f i and e j ↔ f j also holds true.
We now proceed to prove (2.7) by induction to r, the statement being trivial for r = 0 and r = 1. If (2.7) holds true up to and including r ∈ Z >0 then acting by ad Y i = ad(e i −f i ) on both sides of (2.7) gives, in view of the previous paragraph, Applying the induction hypothesis to the second line and moving the resulting term to the other side of the equation establishes the induction step, since . We will show that these are the defining relations of k(A) in terms of the generators y 1 , . . . , y n of k(A). In other words, we will show that k(A) is isomorphic to the generalized Onsager algebra, which we define as follows: be a symmetrizable generalized Cartan matrix. The generalized Onsager algebra L(A) is the complex Lie algebra with generators B 1 , . . . , B n and defining relations the inhomogeneous Serre relations By (2.6), the inhomogeneous Serre relations (2.9) for a ij ≥ −4 are given by the following concrete list: (2.10) Remark 2.6. For the generalized Cartan matrix 1 , L(A) is the Onsager [19] algebra and the inhomogeneous Serre relations (2.9) are the Dolan-Grady [8] relations. For A of affine type A (1) n (respectively D (1) n ), the generalized Onsager algebra was introduced by Uglov and Ivanov [23] (respectively Date and Usami [5]). For A of arbitrary affine type, generalized Onsager algebras and their quantum analogs have been introduced by Baseilhac and Belliard [2].
We turn L(A) into a filtered Lie algebra with filtration Elements in Gr j (L(A)) will be denoted by Note that the inhomogeneous Serre relations (2.9) in L(A) turn into the usual Serre relations Proof. By the earlier remarks it is clear that ψ and ϕ are well defined and surjective. a. We show that ψ is injective. Let y ∈ Ker(ψ) ⊆ L(A) and let ℓ ≥ 1 such that y ∈ L ℓ (A). Write with coefficients d i 1 ,...,i k ∈ C. We prove that y = 0 by induction to ℓ. If ℓ = 1 then where g(0) := h(A) and, for k ∈ Z >0 , g(k) := α∈Φ + :ht(α)=k g α (A) and g(−k) := ω(g(k)).
Choosing the g(ℓ)-component of the identity (2.16) along the direct sum decomposition (2.17) we obtain Returning to (2.15) we conclude that y ∈ L ℓ−1 (A), hence y = 0 by the induction hypothesis. This show that ψ : L(A) → k(A) is an isomorphism of Lie algebras. b. We show that the surjective graded Lie algebra homomorphism ϕ : n(A) → Gr(L(A)) is injective. Fix ℓ ≥ 1. It suffices to show that the set Identify L(A) ≃ k(A) using the Lie algebra isomorphism ψ. Then we have for k ≥ 1. Hence we have a well defined linear map π ℓ : Gr ℓ (L(A)) → g ≤ℓ /g ≤ℓ−1 defined by π ℓ ([x] ℓ ) := x + g ≤ℓ−1 . For α ∈ Φ + with ht(α) = ℓ and j ∈ {1, . . . , m(α)} one shows, in a similar manner as in the proof of part a, that which is a linear independent set in g ≤ℓ /g ≤ℓ−1 . We conclude that S ℓ is a linear independent set in Gr ℓ (L). This completes the proof of b.
In view of Remark 2.6 and the previous theorem, we introduce the following terminology.  Define Proposition 2.9. We have a linear isomorphism By the defining inhomogeneous Serre relations (2.9) for L(A), a one-dimensional representation χ ∈ ch(L(A)) can thus take any value at B j if j ∈ E A , while it must map B j to zero if j ∈ E A . The result now immediately follows from Theorem 2.7 a.

Onsager type presentation in the finite and untwisted affine case
In this section we describe some additional properties of k(A) in case the symmetrizable generalized Cartan matrix A is of finite or untwisted affine type. In these two cases we discuss integral forms of k(A) and explicitly give the structure constants of k(A) with respect to a suitable integral basis of k(A). We show that this leads to Onsager type presentations of k(A) if A is of untwisted affine type. .
3.1. The finite case. Let A be an indecomposable generalized Cartan matrix of finite type (which is automatically symmetrizable by [12,Lem. 4.6]). By Serre's Theorem [11,Thm. 18.2] the corresponding Kac-Moody Lie algebra g(A) is a complex simple Lie algebra and the pair (g(A), k(A)) is the complexification of an irreducible real split symmetric pair. Let us discuss this case now from this perspective.
Let g be a simple Lie algebra over C of rank r and fix a Cartan subalgebra h ⊂ g. Write Φ ⊂ h * for its root system, Φ + for a choice of positive roots, and Π := {α 1 , . . . , α r } for the corresponding simple roots. Let (·, ·) be a nondegenerate invariant symmetric bilinear form on g. It is unique up to a nonzero scalar multiple. It is non-degenerate when restricted to h × h. For λ ∈ h * let t λ ∈ h be the Cartan element such that (t λ , h) = λ(h) for all h ∈ h. Write (λ, µ) := µ(t λ ) (λ, µ ∈ h * ) for the induced bilinear form on h * . For later purposes (see Subsection 3.2) it is convenient to normalize the form (·, ·) on g such that (α, α) = 2 for long roots α. Define and write h i := h α i for i = 1, . . . , r. Set Π ∨ := {h i } r i=1 ⊂ h. Then A = (a ij ) r i,j=1 := (α j (h i )) r i,j=1 is the Cartan matrix of g, and (h, Π, Π ∨ ) is a realization of A. Fix e i ∈ g α i and f i ∈ g −α i (1 ≤ i ≤ r) such that [e i , f i ] = h i . Then Serre's Theorem [11,Thm. 18.2] shows that g ≃ g(A) by identifying e 1 , . . . , e r , f 1 , . . . , f r with the Chevalley generators of g(A). We will freely use the resulting notations and results on g(A) from the previous section, only dropping the dependence on A. In particular, we write g α for g α (A), k for k(A), etc. Note that the natural number n from the previous section equals the rank r of g.
Recall that the Chevalley involution ω is given by ω| h = −Id h and ω(e i ) = −f i for i = 1, . . . , r. The corresponding generators . . , r). Theorem 2.7 a leads to the explicit presentation of k in terms of the generators Y i (i = 1, . . . , r) by identifying k with the generalized Onsager algebra L with respect to the Cartan matrix A.
Example 3.1. Let g = sl r+1 (C) be the special linear Lie algebra and E i,j (i, j = 1, . . . , r+1) the standard matrix units in gl r+1 (C). Take the diagonal matrices in g as the Cartan subalgebra h of g. Then are Chevalley generators of g. One has for the associated Chevalley involution, with X T is the transpose of the matrix X, and the associated fix-point Lie subalgebra is the orthogonal Lie algebra so r+1 (C).
We extend the subset {h i , e i , f i } r i=1 to a Chevalley basis {h i , e α } 1≤i≤r,α∈Φ of g as follows. We set e α i := e i and e −α i := f i for i = 1, . . . , r and choose for the remaining roots α root vectors e α ∈ g α such that [e α , e −α ] = h α and ω(e α ) = −e −α (see, e.g., [11, §25.2] for a detailed discussion on the existence of Chevalley bases of g). Then is an integral form of g, in the sense that the Lie bracket [·, ·] of g restricts to a Lie bracket [·, ·] : g Z × g Z → g Z on g Z and g ≃ C ⊗ Z g Z . Note that the Chevalley involution ω restricts to an involution on g Z .
Chevalley involutions of g are complex linear extensions of Cartan involutions of split real forms of g. In our present notations the split real form g 0 of g is defined as the real span of the Chevalley basis {h i , e α } 1≤i≤r,α∈Φ . It contains the real form h 0 := r i=1 Rh i of h as its Cartan subalgebra, and ω 0 := ω| g 0 is the Cartan involution of g 0 containing h 0 in its −1-eigenspace. The fix-point Lie subalgebra k 0 of g 0 with respect to ω 0 is generated as real Lie algebra by Y i := e i − f i (i = 1, . . . , r), and its complexification is isomorphic to k. The real version of Theorem 2.7 holds true, with k replaced by k 0 and L(A) (Definition 2.5) defined over the real numbers.
The fix-point Lie subalgebras k can now be described using the explicit description of k 0 , see [13,Appendix C] (note that in the list of properties of g 0 ≃ so(2p + 1, 2q + 1) in [13, Appendix C, page 528], the fact that so(2p + 1, 2p + 1) is a split real form is missing). It leads to the following table (the type refers to the type of the Cartan matrix A, i.e. the type of the simple Lie algebra g).
From the list it is clear that k is reductive, and that k is semisimple unless the Cartan matrix A is of type C r (r ≥ 1). Remark 3.3. Dolan-Grady type presentations of the quantum analogue of k for any symmetric pair (g, k) were considered in [16,15]. They play an important role in quantum harmonic analysis because of the intrinsic rigidity of quantum symmetric pairs. The quantized universal enveloping algebra U q (g), defined by quantizing a Serre presentation of U(g), depends on a distinguished choice of Cartan subalgebra h. In addition, the quantum analogue of a symmetric pair requires fixing a representative of the isomorphism class of the involution that stabilizes h and that has the additional property that its −1-eigenspace in h is of maximal dimension (the special case under consideration in this paper corresponds to the most extreme case that the whole Cartan subalgebra h is contained in the −1-eigenspace of the involution). It is in this setup that the Serre type presentation of g (resp. U q (g)) does not induce a Serre type presentation of (the quantum analogue of) k, but instead leads to Dolan-Grady type presentations. Note the recent paper [18] in which an important first step is made to reveal the reductive nature of k in the quantum context. So on the one hand k is reductive, hence admits a Serre type presentation, while on the other hand k admits an Dolan-Grady type presentation (Theorem 2.7 a). The interplay between these two presentations of k is worked out for type C r (r ≥ 1) in Section 4.

3.2.
The untwisted affine case. Let A be an indecomposable generalized Cartan matrix of untwisted affine type X We recall the loop presentation of the untwisted affine Lie algebras following [12,Chpt. 7]. The starting point is a simple Lie algebra g of type X r (we will freely use the notations from the previous subsection regarding the structure theory of g). The loop algebra of g is . We extend ·, · : g × g → C to a nondegenerate invariant symmetric form on g × g by The abelian Lie subalgebras h = h ⊕ Cc of g and h = h ⊕ Cd of g play the role of Cartan subalgebras. Define Λ 0 , δ ∈ h * by Λ 0 (c) = 1, Λ 0 (h) = 0 = Λ 0 (d), We identify h * with the subspace of h * vanishing at Cc ⊕ Cd. Then Since (·, ·)| h× h is nondegenerate there exists for λ ∈ h * a unique t λ ∈ h such that λ(h) = (h, t λ ) for all h ∈ h. For λ ∈ h * we have t λ ∈ h, which coincides with the element t λ as defined in the previous subsection. Note furthermore that The bilinear form on h * obtained from the nondegenerate symmetric bilinear form (·, ·)| h× h by dualizing is again denoted by (·, ·). It satisfies Note that We take as associated set of positive affine roots Φ + := Φ re The corresponding simple roots are Π := {α 0 , α 1 , . . . , α r } with the additional affine simple root given by α 0 := −θ + δ with θ ∈ Φ + the highest root of Φ. Note that θ is a long root, hence (θ, θ) = 2 by our convention on the normalisation of (·, ·).
The root space decomposition of g is The root spaces are concretely given by In particular, m(γ) = 1 if γ ∈ Φ re and m(γ) = r if γ ∈ Φ im . The description of g as a Kac-Moody algebra is obtained as follows. Define , γ ∈ Φ re and write Π ∨ := {h 0 , h 1 , . . . , h r } with h j := h α j for j = 0, . . . , r. Note that for γ = α ∈ Φ and j = i ∈ {1, . . . , r} the elements h γ and h j are the elements h α ∈ h and h i ∈ h as defined in the previous subsection. In addition, Then ( h, Π, Π ∨ ) is a realization of the affine Cartan matrix A := α j (h i ) r i,j=0 of type X (1) r (it contains the Cartan matrix A = α j (h i ) r i,j=1 of type X r ). The affine Cartan matrix A is symmetrizable by [12,Lem. 4.6]. Furthermore, the rank of A is r and the dimension of h is r + 2 = 2n − r with n := r + 1.
Let e i ∈ g α i and f i ∈ g −α i (i = 1, . . . , r) be the Chevalley generators of g and ω the corresponding Chevalley involution of g. Choose E 0 ∈ g −θ such that (E 0 , ω(E 0 )) = −1 and set Then (E 0 , F 0 ) = 1 and [E 0 , F 0 ] = −h θ . Define now e 0 ∈ g α 0 and f 0 ∈ g −α 0 by Note that [e 0 , f 0 ] = h 0 . For i = 1, . . . , r interpret e i and f i as elements in g by the canonical Lie algebra embedding g ֒→ g, x → x ⊗ 1. Then [12,Thm. 7.4] shows that g ≃ g( A) by identifying e 0 , . . . , e r , f 0 , . . . , f r with the Chevalley generators of g. We will freely use the identification g ≃ g( A) in this subsection. We write ω for the Chevalley involution of g. It is characterized by ω(e j ) = −f j and ω(h j ) = −h j for j = 0, . . . , r. Write k for the fix-point Lie subalgebra. The corresponding generators Y j (0 ≤ j ≤ r) of k are Y j = e j − f j for j = 0, . . . , r. Theorem 2.7a leads to the Dolan-Grady type presentation of k in terms of the generators Y j (j = 0, . . . , r) by the identification of k with the generalized Onsager algebra L := L( A).

The symplectic case
We start this section by explicitly comparing the Serre presentation and the Dolan-Grady type presentation of k when A is of type C r (r ≥ 1). We will explicitly evaluate its one-dimensional representations on the Chevalley basis of k. We end the section by considering the untwisted affine case.
Let g = sp r (C) be the symplectic Lie algebra consisting of 2r × 2r complex-valued matrices B C D −B T with B, C, D ∈ gl r (C) and C T = C, D T = D. Let h be the Cartan subalgebra of g consisting of the diagonal matrices in sp r (C).
Transporting the Dolan-Grady type presentation of k (see Theorem 2.7) through the Lie algebra isomorphism η gives the following presentation of the general linear Lie algebra gl r (C).
Then gl r (C) is generated by K 1 , . . . , K r . The defining relations of gl r (C) in terms of the generators K 1 , . . . , K r are [K j , K k ] = 0, |j − k| > 1, Remark 4.2. In a similar manner one can write down an explicit Dolan-Grady type presentation of so r+1 (C) using the fact that so r+1 (C) is the fix-point Lie subalgebra for the Chevalley involution of sl r+1 (C) (see Example 3.1).