The group algebra of the affine braid group

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 13 April 2011

The group algebra of the affine braid group

The affine braid group $Bk$ is the group given by generators $T_1,T_2,\dots ,T_{k-1}$ and $X^{{\epsilon }_1}$, with relations

$$T_i{T}_j=T_j{T}_i,\text{if }j\ne i\pm 1,$$	(bd1)
$T_i{T}_{i+1}{T}_i=T_{i+1}{T}_i{T}_{i+1},\text{for }1\le i\le k-2,$	(bd2)
$X^{{\epsilon }_1}{T}_1{X}^{{\epsilon }_1}{T}_1=T_1{X}^{{\epsilon }_1}{T}_1{X}^{{\epsilon }_1},$	(bd3)
$$X^{{\epsilon }_1}{T}_i=X^{{\epsilon }_1}{T}_i,\text{for }2\le i\le k-1.$$	(bd4)

The affine braid group is isomorphic to the group of braids in the thickened annulus (see, for example [GH2]), where the generators $T_i$ and $X^{{\epsilon }_1}$ are identified with the diagrams

$T_i=$    

and

For $i=1,\dots ,k$ define

The pictorial computation PUTTHISPICTUREIN shows that the $X^{{\epsilon }_i}$ all commute with each other.

Notes and References

This section is based on forthcoming joint work with Z. Daugherty and R. Virk [DRV]. See [GH2] or [OR] for pictures of braids in an annulus, or in a cylinder.

References

[AMR] S. Ariki, A. Mathas, and H. Rui, Cyclotomic Nazarov-Wenzl algebras, Nagoya Math. J. 182 (2006), 47-134. MR2235339 (2007d:20005)

[BB] A. Beliakova and C. Blanchet, Skein construction of idempotents in Birman-Murakami-Wenzl algebras, Math. Ann. 321 (2001), 347-373. MR1866492 (2002h:57018)

[Bou] N. Bourbaki, Groupes et Algèbres de Lie, Masson, Paris, 1990.

[DRV] Z. Daugherty, A. Ram, and R. Virk, Affine and graded BMW algebras, in preparation.

[GH1] F. Goodman and H. Hauschild Mosley, Cyclotomic Birman-Wenzl-Murakami algebras. I. Freeness and realization as tangle algebras, J. Knot Theory Ramifications 18 (2009), 1089-1127. MR2554337 (2010j:57014)

[Naz] M. Nazarov, Young's orthogonal form for Brauer's centralizer algebra, J. Algebra 182 (1996), no. 3, 664-693. MR1398116 (97m:20057)

[OR] R. Orellana and A. Ram, Affine braids, Markov traces and the category $𝒪$, Algebraic groups and homogeneous spaces, 423-473, Tata Inst. Fund. Res. Stud. Math., Tata Inst. Fund. Res., Mumbai, 2007. MR2348913 (2008m:17034)

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