Fundamental representations and algebraic properties of biquaternions or complexified quaternions

The fundamental properties of biquaternions (complexified quaternions) are presented including several different representations, some of them new, and definitions of fundamental operations such as the scalar and vector parts, conjugates, semi-norms, polar forms, and inner and outer products. The notation is consistent throughout, even between representations, providing a clear account of the many ways in which the component parts of a biquaternion may be manipulated algebraically

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

In the first article of this work [... I: The transposition map] we showed that real Clifford algebras CL(V,Q) posses a unique transposition anti-involution \tp. 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 \ve=(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 \tp allows us to define a dual spinor space S^\ast, 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 f_i summing up to 1. We classify the stabilizer groups according to the signature in simple and semisimple cases.Comment: 27 page

On the Transposition Anti-Involution in Real Clifford Algebras III: the Automorphism Group of the Transposition Scalar Product on Spinor Spaces

A signature epsilon=(p,q) dependent transposition anti-involution T of real Clifford algebras Cl_{p,q} for non-degenerate quadratic forms was introduced in [arXiv.1005.3554v1]. In [arXiv.1005.3558v1] we showed that, depending on the value of (p-q) mod 8, the map T gives rise to transposition, complex Hermitian, or quaternionic Hermitian conjugation of representation matrices in spinor representation. The resulting scalar product is in general different from the two known standard scalar products [Lounesto, Clifford algebras and Spinors 2001]. We provide a full signature (p,q) dependent classification of the invariance groups Gpq_{p,q} of this product for p+q <= 9. The map T is identified as the "star" map known [Passmann, The Algebraic Structure of Group Rings 1985] from the theory of (twisted) group algebras, where the Clifford algebra Cl_{p,q} is seen as a twisted group ring k^t[(Z_2)^n], n=p+q. We discuss and list important subgroups of stabilizer groups Gpq(f)_{p,q} and their transversals in relation to generators of spinor spaces.Comment: LaTeX - 24 page

On the Transposition Anti-Involution in Real Clifford Algebras I: the Transposition Map

A particular orthogonal map on a finite dimensional real quadratic vector space (V,Q) with a non-degenerate quadratic form Q of any signature (p,q) is considered. It can be viewed as a correlation of the vector space that leads to a dual Clifford algebra CL(V^*,Q) of linear functionals (multiforms) acting on the universal Clifford algebra CL(V,Q). The map results in a unique involutive automorphism and a unique involutive anti-automorphism of CL(V,Q). The anti-involution 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 element matrix in the left regular representation of CL(V,Q). We give also an example for real spinor spaces. The general setting for spinor representations will be treated in part II of this work [...II: Spabilizer groups of primitive idempotents].Comment: 28 page

On the transposition anti-involution in real Clifford algebras I: the transposition map

A particular orthogonal map on a finite dimensional real quadratic vector space (V,Q) with a non-degenerate quadratic form Q of any signature (p,q) is considered. It can be viewed as a correlation of the vector space that leads to a dual Clifford algebra CL(V^*,Q) of linear functionals (multiforms) acting on the universal Clifford algebra CL(V,Q). The map results in a unique involutive automorphism and a unique involutive anti-automorphism of CL(V,Q). The anti-involution 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 element matrix in the left regular representation of CL(V,Q). We give also an example for real spinor spaces. The general setting for spinor representations will be treated in part II of this work [...II: Spabilizer groups of primitive idempotents].Comment: 28 page