27 research outputs found
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