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On the R-matrix realization of quantum loop algebras

by A. Liashyk, S. Z. Pakuliak

Submission summary

As Contributors: Stanislav Pakuliak
Arxiv Link: (pdf)
Date submitted: 2021-06-23 07:41
Submitted by: Pakuliak, Stanislav
Submitted to: SciPost Mathematics
Academic field: Mathematics
  • Mathematical Physics


We consider $\rm R$-matrix realization of the quantum deformations of the loop algebras $\tilde{\mathfrak{g}}$ corresponding to non-exceptional affine Lie algebras of type $\widehat{\mathfrak{g}}=A^{(1)}_{N-1}$, $B^{(1)}_n$, $C^{(1)}_n$, $D^{(1)}_n$, $A^{(2)}_{N-1}$. 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 off-diagonal 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.

Current status:
Editor-in-charge assigned

Reports on this Submission

Report 2 by Kang Lu on 2022-1-9 (Invited Report)


The paper studies the R-matrix realizations of quantum loop algebras of types $A_{N-1}^{(1)}$, $B_{n}^{(1)}$, $C_{n}^{(1)}$, $D_{n}^{(1)}$, $A_{N-1}^{(2)}$ 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 $A_{N-1}^{(1)}$, $B_{n}^{(1)}$, $C_{n}^{(1)}$, $D_{n}^{(1)}$ were discussed in references [6-8]. The interesting case for this paper is the type of $A_{N-1}^{(2)}$. 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 R-matrices,
- 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 off-shell 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 R-matrix constructions of finite-dimensional 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 one-dimensional subspace $\rightarrow$ onto a one-dimensional 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 R-matrix: 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 ","

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Author:  Stanislav Pakuliak  on 2022-01-12  [id 2091]

(in reply to Report 2 by Kang Lu on 2022-01-09)

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.

Anonymous Report 1 on 2021-12-14 (Invited Report)


This paper is about the representation of quantum group algebras using R-matrix and $RLL$ relation. Such algbras are known to have two presentations, in terms of an infinite set of Cartan generators gathered in the L-Matrix 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 L-matrix, the authors obtain the presentation of the q-deformed Lie algebras in terms of currents generalizing the known result for $U_q(SL_N)$ and more recent results of refs [7-8].

The paper gathers many results and proofs such as the description of the central elements of $U_q(G)$ in terms of L-matrix elements, the Gauss decomposition of the L-matrix, 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.

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