Brauer Algebras of Simply Laced Type

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 An−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 An−1, Dn, E6, E7, E8). 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. keywords: associative algebra, Brauer algebra, Brauer diagram, Coxeter group, partially ordered set, root system AMS 2000 Mathematics Subject Classification: 20M05, 16K20, 17Bxx, 20Fxx, 20F36