Integrality Properties of the Symmetries

Let e = ±1 and let i \( \in \) I. The symmetry T’ ie : U→ U (resp. T" I,e : U → U) induces for each λ’, λ" a linear isomorphism \(\lambda \prime{\rm{U}}\lambda \prime\prime \to s_i (\lambda \prime){\rm{U}}s_i (\lambda \prime\prime)\) (notation of 23.1.1; s i : X → X is as in 2.2.6). Taking direct sums, we obtain an algebra automorphism \(T\prime_{i,e} :{\rm{\dot U}} \to {\rm{\dot U}}\) (resp. \(T\prime\prime_{i,e} :{\rm{\dot U}} \to {\rm{\dot U}}\)) such that \(T\prime_{i,e} (1_\lambda ) = 1_{s_i (\lambda )} \) (resp. \(T\prime\prime_{i,e} (1_\lambda ) = 1_{s_i (\lambda )} \)) for all λ and \(T\prime_{i,e} (uxx\prime u\prime) = T\prime_{i,e} (u)T\prime_{i,e} (x)T\prime_{i,e} (x\prime)T\prime_{i,e} (u\prime)\) (resp. \(T\prime\prime_{i,e} (uxx\prime u\prime) = T\prime\prime_{i,e} (u)T\prime\prime_{i,e} (x)T\prime\prime_{i,e} (x\prime)T\prime\prime_{i,e} (u\prime)\)) for all \(u,u\prime \in {\rm{\dot U}}\) and \(x,x\prime \in {\rm{\dot U}}\). Then T’ ie is an automorphism of the algebra U with inverse T" i-e . These automorphisms satisfy braid group relations just like those of U.