By Michel Broué

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Proposition. Whenever we have three morphisms β : NX → NX , α : X1 → X , α : X → X1 , we have M α ◦ TrM N (β) ◦ α = TrN (N (α ) ◦ β ◦ N (α)) . In particular, the image if TrM N is a two-sided ideal in HomA ( , ). D2. 55. , M N (α) ◦ ηN,M (X1 ) = ηN,M (X) ◦ α . Similarly, we get εM,N (X1 ) ◦ M N (α ) = α ◦ εM,N (X ) . Using these equations, we obtain /X α X1 ηN,M (X1 ) ηN,M (X) M N X1 MNα / MNX εM,N (X1 ) εM,N (X ) Mβ /X1 O α XO / MNX MNα / MNX 1 α ◦ TrM N (β) ◦ α = α ◦ εM,N (X ) ◦ M (β) ◦ ηN,M (X) ◦ α = εM,N (X1 ) ◦ M (N (α ) ◦ β ◦ N (α)) ◦ ηN,M (X1 ) = TrM N (N (α ) ◦ β ◦ N (α)) .

Choose B := R, M :=A AR , N :=R AA and a, a := t(aa ). 1. The following conditions are equivalent. (i) A is strongly symmetric. (ii) R is A–split. 2. The following conditions are equivalent. (i) A is separable. (ii) A is R–split. Example : Induction–Restriction with a parabolic subalgebra. Let B be a parabolic subalgebra for A. Choose M :=A AB , N :=B AA , a, a := t(aa ) . Then B is always A–split, while A is B–split if and only if A is a summand of A ⊗B A in A ModA . If cA B = i ei ⊗B ei is the relative Casimir element, the “double relative trace” is  B   (A ⊗B A) → ZA TrA B: xj ⊗ yj → ei ( xj yj )ei   j i j Notice that the element 1 ⊗B 1 belongs to (A ⊗B A)B .

60. Lemma. A morphism X → X in A is M -split if and only if it factorizes through an M –split object of A. 61. Definition. Let A be an abelian category. The category Stab(A), is defined as follows: 1. the objects of Stab(A) are the objects of A, 2. , HomstA,M (X, X ) := HomA (X, X )/HomM A (X, X ) . Let A be an R-algebra. In the situation where A = A Mod, B = R Mod and A the biadjoint pair of functors is given by (IndA R , ResR ), we denote the corresponding stable category by A Stab. 36 SYMMETRIC ALGEBRAS Remarks.

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