A pr 2 00 2 The Lawrence - Krammer representation
نویسندگان
چکیده
The Lawrence-Krammer representation of the braid groups recently came to prominence when it was shown to be faithful by myself and Krammer. It is an action of the braid group on a certain homology module H2(C̃) over the ring of Laurent polynomials in q and t . In this paper we describe some surfaces in C̃ representing elements of homology. We use these to give a new proof that H2(C̃) is a free module. We also show that the (n− 2, 2) representation of the Temperley-Lieb algebra is the image of a map to relative homology at t = −q , clarifying work of Lawrence. AMS Classification 20F36; 20C08
منابع مشابه
3 0 O ct 2 00 8 Irreducibility of the Lawrence - Krammer representation of the BMW algebra of type A n − 1
The Lawrence-Krammer representation introduced by Lawrence and Krammer in order to show the linearity of the braid group is generically irreducible. We show this fact and show further that for some values of its two parameters, when these are specialized to complex numbers, the representation becomes reducible. We describe what these values are and give a complete description of the dimensions ...
متن کاملar X iv : 0 90 1 . 38 56 v 1 [ m at h . R T ] 2 4 Ja n 20 09 Parameters for which the Lawrence - Krammer representation is reducible
We show that the representation, introduced by Lawrence and Krammer to show the linearity of Braid groups, is generically irreducible, but for that for some values of its two parameters when these are specialized to complex numbers , it becomes reducible. To do so, we construct a representation of degree n(n−1) 2 of the BMW algebra of type A n−1 inside the Lawrence-Krammer space. As a represent...
متن کاملFaithfulness of the Lawrence representation of braid groups
The Lawrence representation Ln,m is a family of homological representation of the braid group Bn, which specializes to the reduced Burau and the Lawrence-Krammer representation when m is 1 and 2. In this article we show that the Lawrence representation is faithful for m ≥ 2.
متن کاملInvariant Subspaces of the Lawrence–krammer Representation
The Lawrence–Krammer representation was used in 2000 to show the linearity of the braid group. The problem had remained open for many years. The fact that the Lawrence–Krammer representation of the braid group is reducible for some complex values of its two parameters is now known, as well as the complete description of these values. It is also known that when the representation is reducible, t...
متن کاملThe Lawrence–krammer Representation Is Unitary
We show that the Lawrence–Krammer representation is unitary. We explicitly present the non-singular matrix representing the sesquilinear pairing invariant under the action. We show that reversing the orientation of a braid is equivalent to the transposition of its Lawrence–Krammer matrix followed by a certain conjugation. As corollaries it is shown that the characteristic polynomial of the Lawr...
متن کامل