Recent results on the linearity of braid groups are extended in two ways. We generalize the Lawrence Krammer representation as well as Krammer's faithfulness proof for this linear representation to Artin groups of finite type.

It is known that the recently discovered representations of the Artin groups of type A n , the braid groups, can be constructed via BMW algebras. We introduce similar algebras of type D n and E n which also lead to the newly found faithful representations of the Artin groups of the corresponding types. We establish finite dimensionality of these algebras. Moreover, they have ideals I 1 and I 2 with I 2 ⊂ I 1 such that the quotient with respect to I 1 is the Hecke algebra and I 1 /I 2 is a module for the corresponding Artin group generalizing the Lawrence-Krammer representation. Finally we give conjectures on the structure, the dimension and parabolic subalgebras of the BMW algebra, as well as on a generalization of deformations to Brauer algebras for simply laced spherical type other than A n .

Abstract. The diagram algebra introduced by Brauer that describes the centralizer algebra of the n-fold tensor product of the natural representation of an orthogonal Lie group has a presentation by generators and relations that only depends on the path graph A n−1 on n − 1 nodes. Here we describe an algebra depending on an arbitrary graph M , called the Brauer algebra of type M , and study its structure in the cases where M is a Coxeter graph of simply laced spherical type (so its connected components are of type A n−1 , Dn, E 6 , E 7 , E 8 ). We determine the representations and find the dimension. The algebra is semisimple and contains the group algebra of the Coxeter group of type M as a subalgebra. It is a ring homomorphic image of the Birman-MurakamiWenzl algebra of type M ; this fact will be used in later work determining the structure of the Birman-Murakami-Wenzl algebras of simply laced spherical type.

Abstract. A generalization of the Kauffman tangle algebra is given for Coxeter type Dn. The tangles involve a pole of order 2. The algebra is shown to be isomorphic to the Birman-Murakami-Wenzl algebra of the same type. This result extends the isomorphism between the two algebras in the classical case, which, in our set-up, occurs when the Coxeter type is A n−1 . The proof involves a diagrammatic version of the Brauer algebra of type Dn of which the generalized Temperley-Lieb algebra of type Dn is a subalgebra.

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