An aide-memoire concocted for personal use, which might also be useful to anyone else with a memory as treacherous as mine. 7.5 Atiyah-Bott-Shapiro and all that ___________________________________ The algebraic structure of the general Clifford algebra was surveyed by these fellows in 1964. Nobody actually reads that, of course --- see Gallier, Porteous, or the following viciously compressed crib-sheet.
From the point of view of practical geometric computation, more to the point would be tabulation of the structure of the versor semigroups Vl(p,q) --- not to mention Vl(p,q,r) --- rather than the full algebras.
Notation: with \P denoting any of \R,\C,\H, etc, \P (x) \P' == tensor product of \P,\P'; \P(n) == n x n matrices over \P; 2 \P == \P (+) \P == double ring, defined by (a,b)(c,d) = (ac,bd) for a,b,c,d in \P. Bott structure of Cl(p,q-1) as offset symmetric matrix (below): for p = 0,...,7, Cl(0,p-1) ~= Cl(p,0-1) ~= 2 \R(1/2), \R, \C, \H, 2 \H, \H(2), \C(4), \R(8); for min(p,q) >= 0, Cl(p+8,q-1) ~= Cl(p,q-1+8) ~= Cl(p,q-1)(16); ================================================================================ p\q = -1 0 1 2 3 4 5 6 7 8 0 0 R C H 2H H(2) C(4) R(8) 2R(8) R(16) 1 R 2R R(2) C(2) H(2) 2H(2) H(4) C(8) R(16) 2 C R(2) 2R(2) R(4) C(4) H(4) 2H(4) H(8) 3 H C(2) R(4) 2R(4) R(8) C(8) H(8) 4 2H H(2) C(4) R(8) 2R(8) R(16) 5 H(2) 2H(2) H(4) C(8) R(16) 6 C(4) H(4) 2H(4) H(8) 7 R(8) C(8) H(8) 8 2R(8) R(16) 9 R(16) [view without proportional spacing] ================================================================================ Vahlen: Cl(p+1,q-1+1) ~= Cl(q+1,p-1+1) ~= Cl(p,q-1)(2). Even subalgebra: Cl^0(p,q) ~= Cl(p,q-1) ~= Cl(q,p-1) ~= Cl^0(q,p) [versors & grades not conserved]. Splitting: == Cl(p,q) simple iff p-q <> 1 (mod 4); else Cl(p,q) ~= Cl(p,q)(1+J)/2 (+) Cl(p,q)(1-J)/2, where J denotes unit pseudar (product of generators). Complex scalars: with m = p+q, Cl(m;\C) == Cl(p,q) (x) \C; for m even, Cl(m;\C) ~= Cl(m/2, m/2+1) ~= Cl(m+1,0), Cl(0,m+1) as m/2 odd, even [versors & grades not conserved]; Cl(m;\C) ~= \C(2^{m/2}) for m even, ~= 2\C(2^[m/2]) for m odd. Degenerate algebra: Cl(p,q,r) has p squares +1, q squares -1, r squares 0; Cl(p,q,r) ~= Cl(p,q) (x) Cl(0,0,r) [Ablamovicz] Reduction isomorphisms: \R(l) (x) \R(m) ~= \R(lm); \C (x) \C ~= 2\C; \H (x) \H ~= \R(4); \H (x) \C ~= \C(2), via a + b \i + c \j + d \k \in \H <-> [[a - d \I, b \I + c], [b \I - c, a + d \I]] \in \C(2). Cl(1,0) ~= 2\R via \x <-> [[1,0],[0,-1]] (diagonal matrix)! WFL