Geodesic
On R-Matrix Representations of Birman–Murakami–Wenzl Algebras
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algebraic geometry: Methods, relations, and applications, Tome 246 (2004), pp. 147-153 
A. P. Isaev; O. V. Ogievetskii; P. N. Pyatov. On R-Matrix Representations of Birman–Murakami–Wenzl Algebras. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algebraic geometry: Methods, relations, and applications, Tome 246 (2004), pp. 147-153. http://geodesic.mathdoc.fr/item/TM_2004_246_a9/
@article{TM_2004_246_a9,
     author = {A. P. Isaev and O. V. Ogievetskii and P. N. Pyatov},
     title = {On {R-Matrix} {Representations} of {Birman{\textendash}Murakami{\textendash}Wenzl} {Algebras}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {147--153},
     year = {2004},
     volume = {246},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2004_246_a9/}
}
TY  - JOUR
AU  - A. P. Isaev
AU  - O. V. Ogievetskii
AU  - P. N. Pyatov
TI  - On R-Matrix Representations of Birman–Murakami–Wenzl Algebras
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2004
SP  - 147
EP  - 153
VL  - 246
UR  - http://geodesic.mathdoc.fr/item/TM_2004_246_a9/
LA  - ru
ID  - TM_2004_246_a9
ER  - 
%0 Journal Article
%A A. P. Isaev
%A O. V. Ogievetskii
%A P. N. Pyatov
%T On R-Matrix Representations of Birman–Murakami–Wenzl Algebras
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2004
%P 147-153
%V 246
%U http://geodesic.mathdoc.fr/item/TM_2004_246_a9/
%G ru
%F TM_2004_246_a9

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

It is shown that, to every local representation of the Birman–Murakami–Wenzl algebra defined by a skew invertible R-matrix $\hat R\in\mathrm{Aut}(V^{\otimes 2})$, one can associate pairings $V\otimes V\rightarrow\mathbb C$ and $V^*\otimes V^*\rightarrow\mathbb C$, where $V$ is the representation space. Further, conditions are investigated under which the corresponding quantum group is of SO or Sp type.

[1] Drinfel'd V. G., “Quantum groups”, Proc. Intern. Congr. Math. (Berkeley, 1986), 1, ed. A. M. Gleason, Amer. Math. Soc., Providence, RI, 1987, 798–820 | MR

[2] Reshetikhin N. Yu., Takhtadzhyan L. A., Faddeev L. D., “Kvantovanie grupp i algebr Li”, Algebra i analiz, 1:1 (1989), 178–206 | MR

[3] Brauer R., “On algebras which are connected with the semisimple continuous groups”, Ann. Math., 38 (1937), 854–872 | DOI | MR

[4] Murakami J., “The Kauffman polynomial of links and representation theory”, Osaka J. Math., 24 (1987), 745–758 | MR | Zbl

[5] Birman J. S., Wenzl H., “Braids, link polynomials and a new algebra”, Trans. Amer. Math. Soc., 313:1 (1989), 249–273 | DOI | MR | Zbl

[6] Chari V., Pressley A., A guide to quantum groups, Cambridge Univ. Press, Cambridge, 1994 | MR | Zbl

[7] Wenzl H., “Quantum groups and subfactors of type B, C and D”, Commun. Math. Phys., 133 (1990), 383–432 | DOI | MR | Zbl

[8] Reshetikhin N. Yu., “Kvazitreugolnye algebry Khopfa i invarianty svyazok”, Algebra i analiz, 1:2 (1989), 169–188 | MR

[9] Ogievetsky O., “Uses of quantum spaces”, Contemp. Math., 294 (2002), 161–232 | MR | Zbl

[10] Drinfeld V. G., “O pochti kokommutativnykh algebrakh Khopfa”, Algebra i analiz, 1:2 (1989), 30–46 | MR | Zbl

[11] Ewen H., Ogievetsky O., Wess J., “Quantum matrices in two dimensions”, Lett. Math. Phys., 22 (1991), 297–305 | DOI | MR | Zbl

[12] Isaev A. P., Ogievetsky O. V., Pyatov P. N., “On quantum matrix algebras satisfying the Cayley–Hamilton–Newton identities”, J. Phys. A: Math. and Gen., 32 (1999), L115–L121 | DOI | MR | Zbl

[13] Reshetikhin N. Yu., “Multiparameter quantum groups and twisted quasitriangular Hopf algebras”, Lett. Math. Phys., 20 (1990), 331–335 | DOI | MR | Zbl

[14] Isaev A. P., “Kvantovye gruppy i uravneniya Yanga–Bakstera”, EChAYa, 26:5 (1995), 1204–1263 | MR