Letters in Mathematical Physics

, Volume 86, Issue 2–3, pp 209–227 | Cite as

On Endomorphisms of Quantum Tensor Space



We give a presentation of the endomorphism algebra \({\rm End}_{\mathcal {U}_{q}(\mathfrak {sl}_{2})}(V^{\otimes r})\) , where V is the three-dimensional irreducible module for quantum \({\mathfrak {sl}_2}\) over the function field \({\mathbb {C}(q^{\frac{1}{2}})}\) . This will be as a quotient of the Birman–Wenzl–Murakami algebra BMWr(q) : =  BMWr(q−4, q2 − q−2) by an ideal generated by a single idempotent Φq. Our presentation is in analogy with the case where V is replaced by the two-dimensional irreducible \({\mathcal {U}_q(\mathfrak {sl}_{2})}\) -module, the BMW algebra is replaced by the Hecke algebra Hr(q) of type Ar-1, Φq is replaced by the quantum alternator in H3(q), and the endomorphism algebra is the classical realisation of the Temperley–Lieb algebra on tensor space. In particular, we show that all relations among the endomorphisms defined by the R-matrices on \({V^{\otimes r}}\) are consequences of relations among the three R-matrices acting on \({V^{\otimes 4}}\). The proof makes extensive use of the theory of cellular algebras. Potential applications include the decomposition of tensor powers when q is a root of unity.

Mathematics Subject Classification (2000)

Primary 17B10 17B37 Secondary 20C08 


cellular algebras Lie algebras quantum groups presentations of endomorphism algebras BMW algebra Brauer algebra 


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  1. 1.
    Birman J., Wenzl H.: Braids, link polynomials and a new algebra. Trans. Am. Math. Soc. 313, 249–273 (1989)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Brauer R.: On algebras which are connected with the semisimple continuous groups. Ann. Math. (2) 38, 857–872 (1937)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Doran W.F. IV, Wales D.B., Hanlon P.J.: On the semisimplicity of the Brauer centralizer algebras. J. Algebra 211, 647–685 (1999)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Drinfel’d, V.G.: Quantum groups. In: Proceedings of the International Congress of Mathematicians (Berkeley, Calif., 1986), vols. 1, 2, pp. 798–820. American Mathematical Society, Providence (1987)Google Scholar
  5. 5.
    Du J., Parshall B., Scott L.: Quantum Weyl reciprocity and tilting modules. Commun. Math. Phys. 195, 321–352 (1998)MATHCrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Graham J., Lehrer G.I.: Cellular algebras. Invent. Math. 123, 1–34 (1996)MATHCrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Graham J.J., Lehrer G.I.: Diagram algebras, Hecke algebras and decomposition numbers at roots of unity. Ann. Sci. École Norm. Sup. 36, 479–524 (2003)MATHMathSciNetGoogle Scholar
  8. 8.
    Graham, J.J., Lehrer, G.I.: Cellular algebras and diagram algebras in representation theory. Representation Theory of Algebraic Groups and Quantum Groups. Advanced Studies in Pure Mathematics, vol. 40, pp. 141–173. Mathematical Society of Japan, Tokyo (2004)Google Scholar
  9. 9.
    Green J.A.: Polynomial representations of GLn. Lecture Notes in Mathematics, vol. 830. Springer, Berlin (1980)Google Scholar
  10. 10.
    Hanlon P., Wales D.: On the decomposition of Brauer’s centralizer algebras. J. Algebra 121, 409–445 (1989)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Hilton P., Wu Y.C.: A course in modern algebra. Pure and Applied Mathematics. Wiley, New York (1974)Google Scholar
  12. 12.
    Lehrer G.I., Zhang R.B.: Strongly multiplicity free modules for Lie algebras and quantum groups. J. Algebra 306, 138–174 (2006)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Lehrer, G.I., Zhang, R.B.: A Temperley–Lieb analogue for the BMW algebra. Representation Theory of Algebraic Groups and Quantum Groups. Progress in Mathematics. Birkhäuser, Basel (2009, in press). arXiv:0806.0687v1[math.RT]Google Scholar
  14. 14.
    Lusztig G.: Quantum deformations of certain simple modules over enveloping algebras. Adv. Math. 70(2), 237–249 (1988)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Lusztig G.: Introduction to quantum groups. Progress in Mathematics, vol. 110. Birkhäuser, Boston (1993)Google Scholar
  16. 16.
    Martin P.: The structure of the partition algebras. J. Algebra 183, 319–358 (1996)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Rui H., Si M.: A criterion on the semisimple Brauer algebras. II. J. Comb. Theory Ser. A 113, 1199–1203 (2006)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Weyl H.: The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton (1939)Google Scholar
  19. 19.
    Xi C.: On the quasi-heredity of Birman–Wenzl algebras. Adv. Math. 154(2), 280–298 (2000)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer 2008

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of SydneySydneyAustralia

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