Semisimplicity of Even Brauer Algebras

In 1937, Richard Brauer introduced certain diagram algebras corresponding to the centralizer algebra of transformations commuting with the action of the complex special orthogonal group SO(2n). This algebra, denoted by D_r (2n), is called the even Brauer algebra. The even Brauer algebra plays the same role for the special orthogonal group that the symmetric group algebra does for the representation theory of the general linear group in Schur-Weyl duality. Studying the semisimplicity of the even Brauer algebra is useful in studying the representations of the special orthogonal groups. Since the even Brauer algebra D_r (2n) is not associative, we study the semisimplicity of the largest associative quotient of D_r (2n), denoted by D_r (2n). In this paper, we study the even Brauer algebra D_r (2) and find a chain of its two-sided ideals. Finally we prove that D₁(2), D₂(2) and D₃(2) are semisimple algebras over C.