Generalised Onsager Algebra in Quantum Lattice Models

1. new integrable models related with generalized Onsager algebras
2. new examples of Yang-Baxter algebras

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A. Overview

The paper contains interesting results about generalizations of the Onsager algebra, Yang-Baxter algebras and lattice models recently introduced and studied by Fendley, Alcaraz-Pimenta, Minami, Miao,.... The paper also opens interesting perspectives. Before publication in SciPost, the content should be however revised according to the comments below.

B. Details

1. About the quotient of GTL$(\sqrt(2),r,N)$ mentionned around (2.5). This part should be improved.

a. The author is considering a quotient of GTL$(\sqrt(2),r,N)$, by some relations (for instance (2.6)). Let's denote this quotient by GTL' and its generators by $e'_i$. Instead of (2.5) a map $GTL' \rightarrow GC(r,N)$ should be defined (equality in (2.5) doesn't hold):
$e'_j \rightarrow 1/\sqrt(2)(id + h_j)$

b. The author says that additionnal relations such as (2.6) hold in GTL' compared with GTL. Is it possible to characterize the quotient by identifying all additional relations? A comment is welcome around (2.6).

2. Below (2.6) and top of page 5, a more precise reference where GO(r+1) is welcome (Definition number? Theorem number?)

3. In (2.11), (2.12), equalities should not be used since the maps are not isomorphisms. Arrows should be used. For instance a map $GO(r+1) \rightarrow GC(r,N)$ is given by $A^{(s)}\rightarrow ...$ instead of (2.11). Similar for (2.12). The conclusion below (2.12) should be improved accordingly.

4. About (2.13)-(2.15). The author explains that (2.14) holds thanks to (2.1). Details of the proof are not given, but I understand it is straightforward. A corollary is that (2.15) commutes with $A_0,A_1$. Then it is said 'Details are demonstrated in Appendix A'. This sentence is not suitable, as details are not given. In Appendix A, it is only recalled that the Onsager algebra admits two presentations, either (A.1) or the one with generators $A_0,A_1$ satisfying (2.9). So, any operator commuting with $A_0,A_1$, commutes with $A_n,G_m$. So, the sentence 'Details...Appendix A' should be modified.

5. Begining of section 3, it would be helpful to add a sentence explaining why reviewing known cases is useful and/or necessary for the rest of the paper.

6. Section 3.1. The sentence 'we make the following homomorphism to a vector space...' has to be improved. I guess the author means that he gives a homomorphism from $GC(1,2L)$ to an algebra of matrices in $End((C^2)^{\otimes L})$. It is simpler to write above (3.1):

In this case, we have the map: $GC(1,2L) \rightarrow End({C^2}^{\otimes L})$, then giving (3.1)

7. Above (3.2), it is written $GC(2,2L)$. Is this not a misprint? $GC(2,2L)\rightarrow GC(1,2L)$?

8. Below (3.6). The specific terminology 'spin current' is chosen. Please explain.

9. Below (3.6) 'presence of Onsager algebra' $\rightarrow$ 'presence of Onsager algebra's symmetry'.

10. About (3.7)-(3.9) and footnote 2. The presentation should be improved, simply giving the homomorphism (by analogy with section 3.1)

$ GC(1,L)\rightarrow End((C^2)^{\otimes L})$

Footnote 2 is then useless.

11. Top of page 8: constructing $\rightarrow$ studying.

12. Eq (3.11): is it not a misprint $L \rightarrow L/2$ in the first sum of $A_8V^(1)$?

13. Around equation (3.15), the author claims that Fendley's model with periodic boundary conditions is interacting, contrary to the case with free boundary conditions, which is free-fermionic. This seems to be a surprising result. In fact, the case $r=1$ with p.b.c. (Ising chain) is also free-fermionic. The author should clarify what he means by interacting' model, and at least to present some numerical evidence that this is indeed the case.

14. Around equation (3.16), the author seems to imply that the Hamiltonian (3.15) is only integrable for homogeneous couplings. The model should be integrable for any couplings according to Ref. [27]. Please verify.

15. In the sentence before (3.23), it seems that one should have $GTL(L,r)$ instead of $GTC(L,r)$.

16. Around equation (3.26), could the author expand on the Kramers-Wannier duality? What is a `generalization of the phase transition in the TIFM'?

17. In Section 3.3., same remark as above. Please indicate the homomorphism precisely: map $GC(r,L) \rightarrow End((C^2)^{\otimes L})$.

18. In begining of section 4, it is written 'As we have shown in the previous sections...'. However, saying 'As we have recalled in the previous sections..' is more appropriate, since it seems all results are taken from the literature. If new results are proven, it should be pointed out which ones.

19. In the third line of begining of section 4, it should be written
$GL(r,L), GL(L,r)$ (because of duality) and $GO(r+1)$.

20. Last sentence of 1st paragraph of begining of Section 4: 'generic case' $\rightarrow$ 'generic integer values of r'

21. The procedure of building a R-matrix from a Temperley-Lieb type generator is generally known as ``Baxterization''. Maybe the author can add this information and the reference {\it V. F. R. Jones, “Baxterization,” Int. J. Mod. Phys. A6 (1991) 2035-2043} around eq. (4.3).

22. The R-matrix (B.1) is intriguing. Does it fit into previous classifications of 8-vertex type R-matrices? See for example arXiv:1311.4994, arXiv:2010.11231 and references therein.

23. It seems that the algebra (5.1) was introduced in [28]. If so, the reference should appear.

24. Correct misprints in eqs. (5.8) and (5.9).

25. About Introduction:
The author gives a review on the subject of Onsager algebras and extensions, in relation with Yang-Baxter algebras. For clarity of the exposal of the subject, the introduction should be improved taking into account of the following informations.

a. About the Onsager algebra (a fixed point subalgebra of affine $sl(2)$). The main topic of the paper is the q=1 case (Onsager algebra), so it is natural to add the q=1 version of the presentation of the Onsager algebra studied in [23] (called alternating, different from the original one [5]). It has been introduced in arXiv:1806.07232, and its central extension has been studied in arXiv:2104.08106.
The author also cites important references on the Onsager algebra and recall how it relates with some Yang-Baxter algebras. Presentations of Onsager algebras using Yang-Baxter algebras have previously appear in the literature, and that should be mentionned. See arxiv:1709.08555 for the Onsager algebra using non-standard classical Yang-Baxter algebras.

b. About the q-extensions of the Onsager algebra (a coideal subalgebra of affine $U_q(\widehat{sl(2)}$). The q-deformed extension of the Onsager algebra in its original presentation [5] is introduced and studied in arxiv:1706.08747. The q-deformed extension of the Onsager algebra in its alternating presentation [arXiv:1806.07232, arXiv:2104.08106] is introduced in arXiv:0906.1482. In [23], proofs of existing conjectures in the literature are given.

c. About generalized q-Onsager algebras
As can be checked from the introduction of [26], the generalized q-Onsager algebras were first introduced in arXiv:0906.1215.

d. About generalized Onsager algebras. The paper is about generalized Onsager algebras and Yang-Baxter algebras. It is worth pointing out that the same year as [1], the Yang-Baxter algebra presentation of the generalized Onsager algebras was given in arxiv:1709.08555. For the $sl(N)$ case, it is discussed in details in arXiv:1811.02763.

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