RECORD
Intertwining operators for Sklyanin algebra and elliptic hypergeometric series
Mathematical Physics · Submitted 6 Dec 2010
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The intertwining operators obtained (W-operators) serve as building blocks for the elliptic R-matrix which intertwines tensor product of two L-operators taken in infinite-dimensional representations of the Sklyanin algebra with arbitrary spin.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
Intertwining operators for infinite-dimensional representations of the Sklyanin algebra with spins l and -l-1 are constructed using the technique of intertwining vectors for elliptic L-operator. They are expressed in terms of elliptic hypergeometric series with operator argument. The intertwining operators obtained (W-operators) serve as building blocks for the elliptic R-matrix which intertwines tensor product of two L-operators taken in infinite-dimensional representations of the Sklyanin algebra with arbitrary spin. The Yang-Baxter equation for this R-matrix follows from simpler equations of the star-triangle type for the W-operators. A natural graphic representation of the objects and equations involved in the construction is used.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Intertwining operators for infinite-dimensional representations of the Sklyanin algebra with spins l and -l-1 are constructed using the technique of intertwining vectors for elliptic L-operator. | contribution 0.82 | 0.90 | 0.20 | asserted 1.00 |
| s1 | They are expressed in terms of elliptic hypergeometric series with operator argument. | result 0.73 | 0.87 | 0.12 | asserted 1.00 |
| s2 | The intertwining operators obtained (W-operators) serve as building blocks for the elliptic R-matrix which intertwines tensor product of two L-operators taken in infinite-dimensional representations of the Sklyanin algebra with arbitrary spin. | result 0.67 | 0.89 | 0.53 | asserted 1.00 |
| s3 | The Yang-Baxter equation for this R-matrix follows from simpler equations of the star-triangle type for the W-operators. | result 0.93 | 0.85 | 0.25 | asserted 1.00 |
| s4 | A natural graphic representation of the objects and equations involved in the construction is used. | method 0.96 | 0.76 | 0.03 | asserted 0.99 |
What was examined, and what was not
- arXiv listing abstract v2
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
Intertwining operators for infinite-dimensional representations of the Sklyanin algebra with spins l and -l-1 are constructed using the technique of intertwining vectors for elliptic L-operator. They are expressed in terms of elliptic hypergeometric series with operator argument. The intertwining operators obtained (W-operators) serve as building blocks for the elliptic R-matrix which intertwines tensor product of two L-operators taken in infinite-dimensional representations of the Sklyanin algebra with arbitrary spin. The Yang-Baxter equation for this R-matrix follows from simpler equations of the star-triangle type for the W-operators. A natural graphic representation of the objects and equations involved in the construction is used.
Human review
References
How the literature uses it
5 arXiv papers cite this in 7 sentences
Papers citing it · 5 sentences without a confident label
Uses its method or tool 1 paper
-
Choosing this operator representation of the algebra generators one can obtain the factorized form for the $L$-operator [ref] [equation] where we denote matrices involving theta functions as follows [equation] and specify the relation between two sets of spectral parameters $(u_1,u_2)$ and $(u,ℓ)$ according to ([ref]) as [equation] All notations and properties of theta functions relevant for us are collected in Appendix [ref].
Baxter operators with deformed symmetry · Elliptic $L$-operator and parameter permutation operators · 0.85
Cites it as background 1 paper
-
The relevant algebra for the $r=1$ case is the Sklyanin algebra [ref] and their generalizations [ref] (see e.g. [ref] ), which are 2-parameter deformations of the universal enveloping algebra $U(𝔤𝔩_N)$.
New Integrable Models from the Gauge/YBE Correspondence · Further Comments · 0.81
What it cites · 3 sentences without a confident label
Compares with 1 paper
-
In particular, the elliptic $R$-operator has been found [ref] in terms of operators which implement elementary permutations of parameters entering the $RLL=LLR$ relation.
Yang-Baxter R operators and parameter permutations · Introduction · 0.91
Cites as background 4 papers
-
Integrable spin chains of XXX-type with infinite-dimensional representations of symmetry algebra at the sites were first studied in the seminal papers [ref] in the context of high energy QCD, see also [ref] .
math.QA/0507252 · Introduction · 0.98
-
Some closely related matters are discussed in the recent paper [ref] .
Elliptic beta integrals and solvable models of statistical mechanics · Introduction · 0.97
-
At last, one should keep in mind that the Sklyanin algebra is just a very particular representative of a wide family of elliptic algebras [ref] and, moreover, integrable systems associated to algebras from this class can be constructed [ref] .
Integrable systems associated with elliptic algebras · Concluding remarks · 0.98
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.