Involutions on composition algebras (Q1433048)

In this paper involutions on composition algebras are studied. First, let \(R\) be a unital commutative associative ring where 2 is invertible in \(R\), and let \(C\) be a composition algebra over \(R\). Then \(C\) has a standard involution \(\overline{}\) given by \(\overline x= t(x)l_C- x\), where \(t: C\to R\) is the trace map. An involution \(\tau: C\to C\) is an anti-automorphism of period 2, i.e. \(\tau(x+ y)= \tau(x)+ \tau(y)\), \(\tau(xy)=\tau(y)\tau(x)\), \(\tau^2= \text{id}\), for all \(x,y\in C\). If \(\tau|_R= \text{id}\) then the involution \(\tau\) is called of the first kind. The author shows that there is a one-one correspondence between non-standard involutions of the first kind, and composition algebras of half \(\text{rank\,}{r\over 2}\). Moreover, every non-standard involution \(\tau\) of the first kind is isomorphic to the natural generalization of Lewis's hat involution: \(C\cong \text{Cay}(B, P, N)\) and \(\tau((u, v))= (\overline u, v)\) is the hat involution on \(C\). Now let \(S\) be a quadratic étale \(R\)-algebra with standard involution \(s_0\), let \(C\) be a composition algebra over \(S\) of \(\text{rank\,}\geq 4\), and assume \(A= \text{Cay}(c,\mu_1,\dots, \mu_m)\) is a generalized Cayley-Dickson algebra of \(\text{rank\,}2^m\cdot\text{rank\,}C\geq 16\). An involution on C whose restriction to \(S\) is the standard involution is called of the second kind. Now let \(\sigma={}^-\) denote the scalar involution on \(A\), and let \(\tau\) be any involution on \(A\) of the second kind. Then \(\tau\) can be written as the tensor product of the standard involution of a unique \(R\)-composition subalgebra of \(C\) and the standard involution of \(S/R\).