Ablamowicz+Fauser: On Transposition Anti-Inv. in Real Cliff. Algs. II: Stab. Groups of Prim. Idemp.

On the Transposition Anti-Involution in Real Clifford Algebras II: Stabilizer Groups of Primitive Idempotents

by Rafal Ablamowicz, Department of Mathematics TTU,
and Bertfried Fauser, School of Computer Science, The University of Birmingham, Edgbaston-Birmingham,
Tech Report No. 2010-2,
Date: May 13, 2010.
Download: downloadable file, 340KB

In the first article of this work [1] we showed that real Clifford algebras Cl(V,Q) posses a unique transposition anti-involution T_eps˜. There it was shown that the map reduces to reversion (resp. conjugation) for any Euclidean (resp. anti-Euclidean) signature. When applied to a general element of the algebra, it results in transposition of the associated matrix of that element in the left regular representation of the algebra. In this paper we show that, depending on the value of (p − q) mod 8, where eps = (p, q) is the signature of Q, the anti-involution gives rise to transposition, Hermitian complex, and Hermitian quaternionic conjugation of representation matrices in spinor representations. We realize spinors in minimal left ideals S = Cl(p,q)f generated by a primitive idempotent f. The map T_eps˜ allows us to define a dual spinor space S*, and a new spinor norm on S, which is different, in general, from two spinor norms known to exist. We study a transitive action of generalized Salingaros’ multiplicative vee groups G_p,q on complete sets of mutually annihilating primitive idempotents. Using the normal stabilizer subgroup G_p,q(f) we construct left transversals, spinor bases, and maps between spinor spaces for different orthogonal idempotents fi summing up to 1. We classify the stabilizer groups according to the signature in simple and semisimple cases.

1991 Mathematics Subject Classification.
11E88, 15A66, 16S34, 20C05, 68W30.

Key words and phrases.
correlation, dual space, exterior algebra, grade involution, group action, group ring, indecomposable module, involution, left regular representation, minimal left ideal, monomial order, primitive idempotent, reversion, simple algebra, semisimple algebra, spinor, stabilizer, transversal, universal Clifford algebra.

Source: Email by R. Ablamowicz of 2010/06/07 23:39 (rablamowicz_at_tntech.edu)


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