## 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

 (bd1) (bd2) ${X}^{{\epsilon }_{1}}{T}_{1}{X}^{{\epsilon }_{1}}{T}_{1}={T}_{1}{X}^{{\epsilon }_{1}}{T}_{1}{X}^{{\epsilon }_{1}},$ (bd3) (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)