The Other Classical Groups

Abstract

We consider only the algebraic groups, over a groundfield k, which, over the universal domain, are isogenous to products of simple groups of the three “classical” types : “special” linear, orthogonal and symplectic. Excluding the case of characteristic 2 (which has not been fully investigated) and certain “exceptional” forms of the orthogonal group in 8 variables (depending upon the principle of triality), such groups, up to isogeny, can be reduced to the following types, which will be called “classical” (the letter indicates the type over the universal domain, and K denotes any separably algebraic extension of k): L1. Special linear group (or projective group) over a division algebra DK over K. L2. (a) Hermitian (i.e., “special” unitary) group for a hermitian form over a quadratic extension K’ of K. (b) Id. for a non-commutative central division algebra DK’ over K’, with an involution inducing on K’ the non-trivial automorphism of K’ over K. O1. Orthogonal group for a quadratic form over K. O2. Antihermitian group for an antihermitian (or “skewhermitian”) form over a quaternion algebra over K, with its usual involution. S1. Symplectic group over K. S2. Hermitian group for a hermitian form over a quaternion algebra over K, with its usual involution.