We determine which finite dimensional central simple superalgebras posses a superinvolution of the second kind and put these results in the context of the AlbertReihm Theorem on the existence of involutions of the second kind. | Turk J Math 35 (2011) , 11 – 21. ¨ ITAK ˙ c TUB doi: Central simple superalgebras with superantiautomorphism of order two of the second kind Ameer Jaber Abstract Our main purpose is to develop the theory of existence of superantiautomorphisms of order two of the second kind (which are caled superinvolutions of the second kind) on finite dimensional central simple superalgebras A = Mn (D) , where D is a finite dimensional division superalgebra with nontrivial grading over K , where K is a field of any characteristic. We determine which finite dimensional central simple superalgebras posses a superinvolution of the second kind and put these results in the context of the AlbertReihm Theorem on the existence of involutions of the second kind. Key word and phrases: Central simple superalgebras, Superantiautomorphisms, Superinvolutions, BrauerWall Groups. 1. Introduction An associative super-ring R = R0 + R1 is nothing but a Z2 -graded associative ring. A Z2 -graded ideal I = I0 + I1 of an associative super-ring R is called a superideal of R . An associative super-ring R is simple if it has no non-trivial superideals. Let R be an associative super-ring with 1 ∈ R0 then R is said to be a division super-ring if all nonzero homogeneous elements are invertible, ., every 0 = rα ∈ Rα has an inverse rα−1 , necessarily in Rα . An associative Z2 -graded K -algebra A = A0 + A1 is a finite dimensional central simple superalgebra over a field K , if Z(A) ∩ A0 = K , where Z(A) = {a ∈ A | ab = ba ∀ b ∈ A} is the center of A, and the only superideals of A are A and (0). Finite dimensional central simple associative superalgebras over a field K are isomorphic to EndV ∼ = Mn (D), where D = D0 + D1 is a finite dimensional associative division superalgebra over K , ., all nonzero elements of Dα , α = 0, 1 , are invertible, and V = V0 + V1 is an n-dimensional D -superspace. If D1 = {0} , the grading of Mn (D) is induced by that of V = V0 + V1 , A = Mp+q (D), p = .
