dorsal/arxiv
View SchemaOn the Structure of Monodromy Algebras and Drinfeld Doubles
| Authors | Florian Nill |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9609020 |
| URL | https://arxiv.org/abs/q-alg/9609020 |
| DOI | 10.1142/S0129055X97000142 |
| Journal | Rev.Math.Phys. 9 (1997) 371-395 |
Abstract
We give a review and some new relations on the structure of the monodromy algebra (also called loop algebra) associated with a quasitriangular Hopf algebra H. It is shown that as an algebra it coincides with the so-called braided group constructed by S. Majid on the dual of H. Gauge transformations act on monodromy algebras via the coadjoint action. Applying a result of Majid, the resulting crossed product is isomorphic to the Drinfeld double D(H). Hence, under the so-called factorizability condition given by N. Reshetikhin and M. Semenov-Tian- Shansky, both algebras are isomorphic to the algbraic tensor product H\otimes H. It is indicated that in this way the results of Alekseev et al. on lattice current algebras are consistent with the theory of more general Hopf spin chains given by K. Szlach\'anyi and the author. In the Appendix the multi-loop algebras L_m of Alekseev and Schomerus [AS] are identified with braided tensor products of monodromy algebras in the sense of Majid, which leads to an explanation of the ``bosonization formula'' of [AS] representing L_m as H\otimes\dots\otimes H.
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"abstract": "We give a review and some new relations on the structure of the monodromy\nalgebra (also called loop algebra) associated with a quasitriangular Hopf\nalgebra H. It is shown that as an algebra it coincides with the so-called\nbraided group constructed by S. Majid on the dual of H. Gauge transformations\nact on monodromy algebras via the coadjoint action. Applying a result of Majid,\nthe resulting crossed product is isomorphic to the Drinfeld double D(H). Hence,\nunder the so-called factorizability condition given by N. Reshetikhin and M.\nSemenov-Tian- Shansky, both algebras are isomorphic to the algbraic tensor\nproduct H\\otimes H. It is indicated that in this way the results of Alekseev et\nal. on lattice current algebras are consistent with the theory of more general\nHopf spin chains given by K. Szlach\\\u0027anyi and the author. In the Appendix the\nmulti-loop algebras L_m of Alekseev and Schomerus [AS] are identified with\nbraided tensor products of monodromy algebras in the sense of Majid, which\nleads to an explanation of the ``bosonization formula\u0027\u0027 of [AS] representing\nL_m as H\\otimes\\dots\\otimes H.",
"arxiv_id": "q-alg/9609020",
"authors": [
"Florian Nill"
],
"categories": [
"q-alg",
"hep-lat",
"math.QA"
],
"doi": "10.1142/S0129055X97000142",
"journal_ref": "Rev.Math.Phys. 9 (1997) 371-395",
"title": "On the Structure of Monodromy Algebras and Drinfeld Doubles",
"url": "https://arxiv.org/abs/q-alg/9609020"
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