On Automorphisms and Universal R-Matrices at Roots of Unity

Daniel Arnaudon

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On Automorphisms and Universal R-Matrices at Roots of Unity

Daniel Arnaudon

High Energy Physics - Theory · Submitted 18 Mar 1994

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The authors' main claim We prove that this happens only when q^4=1, and we explicitly give the expressions for the automorphisms and for the transformed universal R-matrices in this case.

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.45 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.60 · compares with prior results 0.25 · mentions code or data availability 0.02.

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Invertible universal R-matrices of quantum Lie algebras do not exist at roots of unity. There exist however quotients for which intertwiners of tensor products of representations always exist, i.e. R-matrices exist in the representations. One of these quotients, which is finite dimensional, has a universal R-matrix. In this paper, we answer the following question: on which condition are the different quotients of U_q(sl(2)) (Hopf)-equivalent? In the case when they are equivalent, the universal R-matrix of one can be transformed into a universal R-matrix of the other. We prove that this happens only when q^4=1, and we explicitly give the expressions for the automorphisms and for the transformed universal R-matrices in this case.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Invertible universal R-matrices of quantum Lie algebras do not exist at roots of unity.	background 0.97	0.27	0.10	asserted 1.00
s1	There exist however quotients for which intertwiners of tensor products of representations always exist, i.e. R-matrices exist in the representations.	background 0.87	0.26	0.05	asserted 1.00
s2	One of these quotients, which is finite dimensional, has a universal R-matrix.	background 0.83	0.30	0.07	asserted 1.00
s3	In this paper, we answer the following question: on which condition are the different quotients of U_q(sl(2)) (Hopf)-equivalent?	contribution 0.52	0.36	0.04	unclear 0.85
s4	In the case when they are equivalent, the universal R-matrix of one can be transformed into a universal R-matrix of the other.	result 0.63	0.47	0.13	asserted 0.98
s5	We prove that this happens only when q^4=1, and we explicitly give the expressions for the automorphisms and for the transformed universal R-matrices in this case.	result 0.92	0.96	0.82	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Original abstract

Invertible universal R-matrices of quantum Lie algebras do not exist at roots of unity. There exist however quotients for which intertwiners of tensor products of representations always exist, i.e. R-matrices exist in the representations. One of these quotients, which is finite dimensional, has a universal R-matrix. In this paper, we answer the following question: on which condition are the different quotients of U_q(sl(2)) (Hopf)-equivalent? In the case when they are equivalent, the universal R-matrix of one can be transformed into a universal R-matrix of the other. We prove that this happens only when q^4=1, and we explicitly give the expressions for the automorphisms and for the transformed universal R-matrices in this case.

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2 arXiv papers cite this in 19 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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Cites it as background 2 papers

In the fusions indecomposable representation $ℐ^(ℛ)_{r,ℛ-r}$ arises from the "merging" of the representations $V_r$ and $V_ℛ-r$ at roots of unity, when $c_r=c_ℛ-r$ and $V_r⇒𝒰, V_ℛ-r⇒U$ (see for details [ref] ).

New solutions to the $s\ell_q(2)$-invariant Yang-Baxter equations at roots of unity · $sl_q(2)$ algebra and Jimbo's relations for composite $R$-matrices. · 0.81
Such matrices, as well as intertwiner matrices for affine extensions of the quantum algebras were constructed and there were observed the models possessing the quantum algebra symmetry at roots of unity [ref] .

New solutions to the $s\ell_q(2)$-invariant Yang-Baxter equations at roots of unity: cyclic representations · Introduction · 0.91

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