
Journal of Lie Theory 26 (2016), No. 2, 383438 Copyright Heldermann Verlag 2016 Isomorphy Classes of Involutions of SO(n, k, β), n>2 Robert W. Benim Dept. of Mathematics and Computer Science, Pacific University, 2043 College Way, Forest Grove, OR 97116, U.S.A. rbenim@gmail.com Christopher E. Dometrius Math, Science and Technologies Division, Forsyth Technical Community College, 100 Silas Creek Parkway, WinstonSalem, NC 27103, U.S.A. cdometrius@forsythtech.edu Aloysius G. Helminck Dept. of Mathematics, North Carolina State University, 2108 SAS Hall, Box 8205, Raleigh, NC 27695, U.S.A. loek@ncsu.edu Ling Wu Dept. of Mathematics, North Carolina State University, 2108 SAS Hall, Box 8205, Raleigh, NC 27695, U.S.A. ling_wu@hotmail.com A first characterization of the isomorphism classes of kinvolutions for any reductive algebraic group defined over a perfect field was given by A. G. Helminck [On the classification of kinvolutions I, Adv. in Math. 153 (2000) 1117] using $3$ invariants. In another paper by A. G. Helminck, L. Wu and C. Dometrius [Involutions of Sl(n, k), (n > 2), Acta Appl. Math. 90 (2006) 91119] a full classification of all kinvolutions on SL(n,k) for k algebraically closed, the real numbers, the padic numbers or a finite field was provided. In a paper by R. W. Benim, A. G. Helminck and F. Jackson Ward [Isomorphy classes of involutions of Sp(2n,k), n>2, J. of Lie Theory 25 (2015) 903947] a similar classification was given for all kinvolutions of SP(2n,k). In this paper, we find analogous results to develop a detailed characterization of the kinvolutions of SO(n,k,β), where β is any nondegenerate symmetric bilinear form and k is any field not of characteristic 2. We use these results to characterize the isomorphy classes of kinvolutions of SO(n,k,β) for all bilinear forms, β when char(k) is not equal to 2 or 3, and for some bilinear forms when char(k) = 3. When n unequal 3, 4, 6, 8, then the characterization considers all involutions. If n = 3, 4, 6, 8, then the characterization only considers inner involutions. Keywords: Orthogonal Group, symmetric spaces, involutions, inner automophisms. MSC: 14M15, 20G05, 20G15, 20K30 [ Fulltextpdf (522 KB)] for subscribers only. 