Pertti Lounesto

Scalar products of spinors and an extension of Brauer-Wall groups

The automorphism groups of scalar products of spinors are determined. Spinors are considered as elements of minimal left ideals of Clifford algebras on quadratic modules, e.g., on orthogonal spaces. Orthogonal spaces of any dimension and arbitrary signature are discussed. For example, the automorphism groups of scalar products of Pauli spinors and Dirac spinors are, respectively, isomorphic to the matrix groups U(2) and U(2, 2). It is found that there are, in general, 32 different types or similarity classes of such automorphism groups, if one considers real orthogonal spaces of arbitrary dimension and arbitrary signature. On a more abstract level this means that the Brauer-Wall group of the real field R, consisting of graded central simple algebras over R, can be extended from the cyclic group of 8 elements to an abelian group with32 elements. This extended Brauer-Wall group is seen to be isomorphic with the group (Z8 × Z8)/Z2 and it consists of real Clifford algebras with antiinvolutions. Moreover, the subgroup determined by the antiinvolution is isomorphic to the automorphism group of scalar products of spinors.

Axially symmetric vector fields and their complex potentials

The main result of this paper is the following: if a is a fixed direction in a euclidean space and r is the variable vector, then the polynomials are regular in the sense of R. Delanghe, when the coefficients Pl,j are calculated by the following algorithm These regular polynomials are complex potentials of axial regular vector fields. The paper also refers the usual approach involving partial differential equations to axial vector fields.

Matrix representations of Clifford algebras

Abstract As is well known, Clifford algebras can be faithfully realized as certain matrix algebras, the matrix entries being real numbers, complex numbers, or quaternions, depending on the particular Clifford algebra. We show that the matrix representations of the basis elements of a Clifford algebra can be chosen to satisfy a certain additional trace condition; we then use this trace condition to establish optimal inequalities involving norms in Clifford algebras.

Clifford Algebras and Spinor Structures

Reading is a hobby to open the knowledge windows. Besides, it can provide the inspiration and spirit to face this life. By this way, concomitant with the technology development, many companies serve the e-book or book in soft file. The system of this book of course will be much easier. No worry to forget bringing the clifford algebras and spinor structures book. You can open the device and get the book by on-line.

On Clifford algebras of a bilinear form with an antisymmetric part

We explicitly demonstrate with a help of a computer that Clifford algebra Cl(B) of a bilinear form B with a non-trivial antisymmetric part A is isomorphic as an associative algebra to the Clifford algebra Cl(Q) of the quadratic form Q induced by the symmetric part of B [in characteristic ≠ 2], However, the multivector structure of Cl(B) depends on A and is therefore different than the one of Cl(Q). Operation of reversion is still an anti-automorphism of Cl(B). It preserves a new kind of gradation in ⋀ V determined by A but it does not preserve the gradation in ⋀ V. The demonstration is given for Clifford algebras in real and complex vector spaces of dimension ≤ 9 with a help of a Maple package ‘Clifford’. The package has been developed by one of the authors to facilitate computations in Clifford algebras of an arbitrary bilinear form B.

Idempotent structure of Clifford algebras

AbstractSpinor spaces can be represented as minimal left ideals of Clifford algebras and they are generated by primitive idempotents. Primitive idempotents of the Clifford algebras Rp, q are shown to be products of mutually nonannihilating commuting idempotent % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabaGaaiaacaqabeaadaqaaqGaaO% qaamaaleaaleaacaaIXaaabaGaaGOmaaaaaaa!3DBD!\[{\textstyle{1 \over 2}}\]2}}\](1+eT), where the k=q−rq−p basis elements eT satisfy eT2=1. The lattice generated by a set of mutually annihilating primitive idempotents is examined. The final result characterizes all Clifford algebras Rp, q with an anti-involution such that each symmetric elements is either a nilpotent or then some right multiple of it is a nonzero symmetric idempotent. This happens when p+q<-3 and (p, q)≠(2, 1).

Some remarks on clifford algebras

(1989). Some remarks on clifford algebras. Complex Variables, Theory and Application: An International Journal: Vol. 12, No. 1-4, pp. 201-209.

Walsh Functions, Clifford Algebras and Cayley-Dickson Process

The present paper scrutinizes how the sign of the product of two elements in the basis for the Clifford algebra of dimension 2n can be computed by the Walsh functions of degree less than 2n. In the multiplication formula the basis elements are labelled by the binary n-tuples, which form an abelian group Ω which in turn gives rise to the maximal grading of the Clifford algebra. The group of the binary n-tuples is also employed to the Cayley-Dickson process.

OCTONIONS AND TRIALITY

In this chapter, we explore another generalization of C and IE, a non-associative real algebra, the Cayley algebra of octonions, (D. Like complex numbers and quaternions, octonions f o r m a real division algebra, of the highest possible dimension, 8. As an extreme case, @ makes its presence felt in classifications, for instance, in conjunction with exceptional cases of simple Lie algebras. Like C and lE, @ has a geometric interpretation. The automorphism group of lE is SO(3), the rotation group of IR 3 in IHI = IR| 3. The automorphism group of 9 = IR @ i~7 is not all of SO(7), but only a subgroup, the exceptional LŸ group G2. The subgroup G2 fixes a 3-vector, in A s IR T, whose choice determines the product rule of (D. The Cayley algebra @ is a tool to handle an esoteric phenomenon in dimension 8, namely triality, an automorphism of the universal covering group Spin(8) of the rotation group SO(8) of the Euclidean space E s. In general, all automorphisms of SO(n) ate either inner or similarities by orthogonal matrices in O(n), and all automorphisms of Spin(n) ate restrictions of linear transformations Cgn --+ Cgn, and project down to automorphisms of SO(n). The only exception is the triality automorphism of Spin(8), which cannot be linear while it permutes cyclically the three non-identity elements -1 , e12...s,-e12...8 in the center of Spin(8).

Geometry of spinor regularization

The Kustaanheimo theory of spinor regularization is given a new formulation in terms of geometric algebra. The Kustaanheimo-Stiefel matrix and its subsidiary condition are put in a spinor form directly related to the geometry of the orbit in physical space. A physically significant alternative to the KS subsidiary condition is discussed. Derivations are carried out without using coordinates.