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On the Rmatrix realization of quantum loop algebras
by A. Liashyk, S. Z. Pakuliak
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Submission summary
Authors (as registered SciPost users):  Stanislav Pakuliak 
Submission information  

Preprint Link:  https://arxiv.org/abs/2106.10666v1 (pdf) 
Date submitted:  20210623 07:41 
Submitted by:  Pakuliak, Stanislav 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Mathematics 
Specialties: 

Abstract
We consider $\rm R$matrix realization of the quantum deformations of the loop algebras $\tilde{\mathfrak{g}}$ corresponding to nonexceptional affine Lie algebras of type $\widehat{\mathfrak{g}}=A^{(1)}_{N1}$, $B^{(1)}_n$, $C^{(1)}_n$, $D^{(1)}_n$, $A^{(2)}_{N1}$. For each $U_q(\tilde{\mathfrak{g}})$ we investigate the commutation relations between Gauss coordinates of the fundamental $\mathbb{L}$operators using embedding of the smaller algebra into bigger one. The new realization of these algebras in terms of the currents is given. The relations between all offdiagonal Gauss coordinates and certain projections from the ordered products of the currents are presented. These relations are important in applications to the quantum integrable models.
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Reports on this Submission
Report #2 by Anonymous (Referee 2) on 202219 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2106.10666v1, delivered 20220109, doi: 10.21468/SciPost.Report.4162
Report
The paper studies the Rmatrix realizations of quantum loop algebras of types $A_{N1}^{(1)}$, $B_{n}^{(1)}$, $C_{n}^{(1)}$, $D_{n}^{(1)}$, $A_{N1}^{(2)}$ in a uniform way. Using the Gauss decomposition of the fundamental $L$operator, the authors constructed the current generators of the corresponding quantum loop algebras and obtained the relations between the current generators in terms of generating series. The results for cases $A_{N1}^{(1)}$, $B_{n}^{(1)}$, $C_{n}^{(1)}$, $D_{n}^{(1)}$ were discussed in references [68]. The interesting case for this paper is the type of $A_{N1}^{(2)}$. I think it is good to show an isomorphism between these two realizations (physicists may be less interested in such statements).
The paper discussed
 the properties of Rmatrices,
 the central elements in the quantum groups in terms of $L$operator,
 Gauss decompositions and current generators,
 embedding theorem from smaller algebras into bigger algebras,
 commutator relations for current generators,
 certain projections which could possibly used to construct offshell Bethe vectors.
I think the paper is interesting and should also be important for studying quantum integrable systems like the associated XXZ models and representations of quantum loop algebras such as Rmatrix constructions of finitedimensional irreducible modules (fusion procedure). Therefore, I recommend publishing it.
Requested changes
Articles are missing or used incorrectly many times. There are also typos related to singular and plural nouns. Let me list a few of them.
 Page 0 Paragraph 3 Sentence 1: Let $q\in\mathbb C$ be arbitrary complex $\rightarrow$ Let $q\in\mathbb C$ be an arbitrary complex
 Page 1 Sec. 2 Sentence 2: Let $e_{ij}$ be a $N\times N$ $\rightarrow$ Let $e_{ij}$ be an $N\times N$
 Page 3 the line above the displayed formulas defining $P$ and $Q$: onto onedimensional subspace $\rightarrow$ onto a onedimensional subspace
 Page 4 bottom: has simple pole at $\rightarrow$ has a simple pole at
 Page 8 Prop. 3.2: means that product of the matrices $\rightarrow$ means that products of the matrices
 Page 9 proof of Prop. 3.2: chain of equalities $\rightarrow$ a chain of equalities
 Page 9 bottom: given by (3.12) satisfies $\rightarrow$ given by (3.12) satisfy
 Page 13 above eq. (5.5): and (5.4) implies that $\rightarrow$ and (5.4) imply that
 Page 16 after the first displayed equation: which is consequence of $\rightarrow$ which is a consequence of
 Page 16 beginning of Sec. 6: New realization of $\rightarrow$ A new realization of
 Pages 19 – 22: Many “which includes” should be changed to “which include” since “relations” are used before “which”
 Page 22 Sec 7 Paragraph 1 Sentence 2: Rigorous definitions of these projections depends on $\rightarrow$ Rigorous definitions of these projections depend on
Here are some other comments/suggestions.
 Since the title is about quantum loop algebras, I suggest add "which we call the quantum loop algebra" after "$U_q(\widetilde {\mathfrak g})$ [2]" in Paragraph 3 Sentence 2 of the introduction
 Page 4 the last sentence about Twist symmetry of Rmatrix: This sentence does not seem to be fine. Please check it
 Page 6 footnote: write simple $\rightarrow$ write simply
 Page 10 the end of the first sentence in Sec 4: "$U_q(\widetilde {\mathfrak g})$ quantum affine algebras $U_q(\widehat {\mathfrak g})$" does not seem to be fine. Please check it
 Page 11 Sec 4.1 Paragraph 2 Sentence 1: Please add ", respectively"
 Page 12 Line 3 of the last paragraph of Sec 4.1: different type Borel subalgebras $\rightarrow$ different types of Borel subalgebras
 Page 13 after Lem 5.2: them invertible operator $\mathrm{L}_{1,1}(v)$ $\rightarrow$ the invertible operator $\mathrm{L}_{1,1}(v)$?
 Page 14 Eq. (5.8): Dot at the end of (5.8) is missing.
 Page 21 before "and there are additional Serre relations": The dot should be changed to ","
 Page 22 at the end of Sec 6.5 after "as in the case of $U_q(C_n^{(1)})$ (6.6)": The dot should be changed to ","
Report #1 by Anonymous (Referee 1) on 20211214 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2106.10666v1, delivered 20211214, doi: 10.21468/SciPost.Report.4042
Report
This paper is about the representation of quantum group algebras using Rmatrix and $RLL$ relation. Such algbras are known to have two presentations, in terms of an infinite set of Cartan generators gathered in the LMatrix or in terms of a finite set of Chevaley generators. It is the purpose of this paper to provide the relation between the two presentations for most of the Lie algebras ($D^{(2)}_n$ is left for a future work). Starting from the Gauss decomposition of the Lmatrix, the authors obtain the presentation of the qdeformed Lie algebras in terms of currents generalizing the known result for $U_q(SL_N)$ and more recent results of refs [78].
The paper gathers many results and proofs such as the description of the central elements of $U_q(G)$ in terms of Lmatrix elements, the Gauss decomposition of the Lmatrix, various embeddings of smaller rank algebras. As such it is a good review article and will be useful to people working in intergrable systems and interested in the quantum group representations of Lie algebras. I recommend its publication.
Author: Stanislav Pakuliak on 20220112 [id 2091]
(in reply to Report 2 on 20220109)We thank referee for the report on our paper and accept all requested changes he proposed to include in the text of the paper. We will prepare a new version of the manuscript according to these changes and send it to the SciPost when the paper will be accepted for publication.