Abstract
The Temperley–Lieb algebra may be thought of as a quotient of the Hecke algebra of type A, acting on tensor space as the commutant of the usual action of quantum \({\mathfrak{s}\mathfrak{l}}_{2}\) on \({(\mathbb{C}{(q)}^{2})}^{\otimes n}\). We define and study a quotient of the Birman–Wenzl–Murakami algebra, which plays an analogous role for the three-dimensional representation of quantum \({\mathfrak{s}\mathfrak{l}}_{2}\). In the course of the discussion, we prove some general results about the radical of a cellular algebra, which may be of independent interest.
To Toshiaki Shoji on his 60th birthday
Mathematics Subject Classifications (2000): 17B37, 20G42
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J. Birman and H. Wenzl, Braids, link polynomials and a new algebra, Trans. Amer. Math. Soc. 313 (1989), 249–273
R. Brauer, On algebras which are connected with the semisimple continuous groups, Ann. Math. (2) 38 (1937), 857–872
W. F. Doran IV, D. B. Wales and P.J. Hanlon, On the semisimplicity of the Brauer centralizer algebras, J. Algebra 211 (1999), 647–685
V. G. Drinfel’d, Quantum groups. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, CA, 1986), 798–820, Amer. Math. Soc., RI, 1987
J. Graham and G. I. Lehrer, Cellular algebras, Inventiones Math. 123 (1996), 1–34
J. J. Graham and G. I. Lehrer, Diagram algebras, Hecke algebras and decomposition numbers at roots of unity, Ann. Sci. École Norm. Sup. 36 (2003), 479–524
J. J. Graham and G. I. Lehrer, Cellular algebras and diagram algebras in representation theory, Representation theory of algebraic groups and quantum groups, Adv. Stud. Pure Math. 40, Math. Soc. Japan, Tokyo, (2004), 141–173
G. I. Lehrer and R. B. Zhang, Strongly multiplicity free modules for Lie algebras and quantum groups, J. Alg. 306 (2006), 138–174
G. Lusztig, Quantum deformations of certain simple modules over enveloping algebras. Adv. Math. 70 (1988), no. 2, 237–249
G. Lusztig, Introduction to quantum groups. Progress in Mathematics, 110. Birkhäuser, Boston, 1993
H. Rui and M. Si, A criterion on the semisimple Brauer algebras. II. J. Combin. Theor. Ser. A 113 (2006), 1199–1203
C. Xi, On the quasi-heredity of Birman–Wenzl algebras. Adv. Math. 154(2), (2000), 280–298
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Lehrer, G.I., Zhang, R.B. (2010). A Temperley–Lieb Analogue for the BMW Algebra. In: Gyoja, A., Nakajima, H., Shinoda, Ki., Shoji, T., Tanisaki, T. (eds) Representation Theory of Algebraic Groups and Quantum Groups. Progress in Mathematics, vol 284. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4697-4_7
Download citation
DOI: https://doi.org/10.1007/978-0-8176-4697-4_7
Published:
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-4696-7
Online ISBN: 978-0-8176-4697-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)