Clifford algebra of multivectors

CLIFFORD ALGEBRAOF MULTIVECTORS

Minimum polynomials of tensor product of theDirac matrices

and opposite Clifford algebra

Zbigniew Oziewicz∗

Universidad Nacional Autonoma de MexicoFacultad de Estudios Superiores

C.P. 54714 Cuautitlan Izcalli, Estado de Mexicooziewicz.zbigniew@gmail.com, oziewicz@unam.mx

Received April 16, 1996

Advances in Applied Clifford AlgebrasVolume 7 S (1997) pages 467–486

Abstract

I consider realization of an universal Clifford algebra and of anopposite Clifford algebra as a Chevalley deformation of an exterioralgebra. I discuss the realizations of a universal tensor algebra, arealization of a universal factor algebra as a deformation versus arealization by quantization and a generalization of a Clifford algebrafor a braided category. The technical aim is the determination ofthe minimum polynomial of the tensor product of the Dirac matricesγµ ⊗ γµ. This minimum polynomial is useful for understanding in the

∗A member of Sistema Nacional de Investigadores, Mexico. Supported by KomitetBadan Naukowych, Poland, KBN grant # 2 P302 023 07.

468 Zbigniew Oziewicz

framework of the Clifford algebra the particles with integer spin 0, 1and systems of particles with spin 1

Keywords: realizations of universal algebra, deformation of algebra versusthe quantization of algebra, the Chevalley deformation, braided monoidalcategory, Clifford algebra and opposite Clifford algebra

Physics and Astronomy Classification Scheme (PACS) 1999/2000.02.10.Tq;

2000 Mathematics Subject Classification. 15A66; 16S30; 17B35.

Contents

1 Aim of the present paper 469

2 Notation 472

3 Quantization Versus Deformation.The Poincare-Birkhoff-Witt Theoremas Corollary 474

4 Clifford Multiplication of Multivectors 476

5 Realizations of Universal Tensor Algebra 479

6 Two Examples of Deforming Map in a Braided Category 481

7 Involutive Braid 485

8 Minimum Polynomial of the Tensor Product of the DiracMatrices 486

9 Opposite Exterior Algebra 488

9.1 Deformed Exterior Algebra . . . . . . . . . . . . . . . . . . . . 489

10 Opposite Clifford Algebra 490

Clifford algebra of multivectors 469

1 Aim of the present paper

This paper is an introduction to realization of a universal Clifford algebraas a deformation of exterior algebra. This deformation goes back to the workby Hermann Graßmann [1877] and to monograph by Chevalley [1954, pp.38-42].

Throughout the paper k is a field with 12∈ k, M is a k-space of vectors,

M∗ ≡ hom(M, k) is a dual k-space of covectors and η = ηT ∈ hom(M,M∗) 'M∗⊗2 is a symmetric scalar product on vectors from M. If a scalar productη is variable then we have η-dependent family of Clifford algebras C`(M, η).This family of Clifford algebras is passing throughout the point η = 0 whichis an exterior algebra

∧M ≡ C`(M, 0).

A k-algebra is a pair: a k-space with a multiplication. In general anη-dependent family of realizations of the Clifford algebras C`(M, η) meansthat a k-space of algebra and his multiplication both depend on variable η.For example for mostly used realization for η = 0 a k-space of a k-algebra∧M ≡ C`(M, 0) is Z-graded and for η 6= 0 a k-space of an algebra C`(M, η)

is not Z-graded.An endomorphism σ ∈ End(M⊗2) is said to be a braid if σ is a solution

of the Artin braid equation

(σ ⊗ idM) ◦ (idM ⊗σ) ◦ (σ ⊗ idM) = (idM ⊗σ) ◦ (σ ⊗ idM) ◦ (idM ⊗σ), (1.1)

Figure 1: The Artin braid equation.

I wish to introduce a (η, σ)-dependent family of realizations C`(M, η, σ) ofa universal Clifford algebra for which a k-space of algebra is η-independent !In general this k-space depends on a braid σ, however in Section 7 a braidσ is supposed to be an involutive braid σ2 = id and the main result, theminimum polynomials of the tensor product of the Dirac matrices γµ ⊗ γµ,are calculated for a braid to be minus the switch,

∀ v, w ∈M, σ(v ⊗ w) ≡ −w ⊗ v. (1.2)

To distinguish between conceptually different realizations of a universalClifford algebra I will use the following notation and terminology.

C`(M, η) denotes a realization of universal Clifford k-algebras as η-dependentfamily of k-spaces and η-dependent family of multiplications. Such re-alization we call ‘a realization by quantization’. Compare with e.g.[Chevalley 1954, p. 39; Lawson and Michelsohn 1989, p. 8; Crumey-rolle 1990, p. 37].

C`(M, η, σ) denotes a realization of universal Clifford k-algebras as a familyof η-dependent multiplications on the same η-independent Z-graded k-space of multivectors. Such realization is said to be ‘a realization bydeformation’ and this family is said to be a family of Clifford algebrasof multivectors.

I shall explain a conceptual difference between realization of a universalClifford algebra ‘by quantization’ and realization as the Clifford algebra ofmultivectors.

An associative k-algebra A is a k-space FA with an associative multipli-cation m : FA⊗ FA→ FA.

Definition 1.1 (Presentation). Let TM beM -universal tensor k-algebra andlet A be an associative unital k-algebra. A following algebra epimorphism issaid to be an M -presentation of a k-algebra A,

π ∈ epi (TM,A).

A presentation is not unique. For example a following η-dependent alge-bra epimorphism is said to be a presentation of a Clifford algebra C`(M, η),

π(η) ∈ epi [TM, C`(M, η)]. (1.3)

Then kerπ(η)�TM is two-sided ideal in TM and C`(M, η) ' TM/ ker π(η).In (1.3) we need to distinguish between a k-space M imbeded in an algebraTM versus an imbeding in C`(M, η). Also an image π(η)M in C`(M, η) needsnot to coincide with M.

Definition 1.2 (Quantization). Let M be a k-space. A family of M -pre-sentations or equivalently a family of two-sided ideals in TM is said to be aquantization or a perturbation.

A family of η-dependent M -presentations π(η) which include a presen-tation with η = 0 is said to be a quantization of an exterior factor algebra∧M ≡ TM/ kerπ(0). In another words a quantization is a family of η-

dependent two-sided ideals

ker π(η) � TM.

For family of universal factor algebras C`(M, η) ' (TM)/ ker π(η) thek-spaces and the multiplications both depend on a choice of a scalar productη.

Definition 1.3 (Deformation). A family of multiplications on a fixed k-spaceis said to be an algebra deformation.

For example a family of η-dependent Clifford multiplications ∧η on ak-space M∧ of multivectors is said to be a η-deformation of an exterior k-algebra

An M -universal tensor algebra TM is unique up to isomorphism. Oneaim of this paper is to demonstrate the utility of realizations of M -universaltensor algebra TM. Different realizations are isomorphic algebras. LetM⊗ beZ-graded k-space of all M -words in ‘an alphabet’ M, i.e.M⊗ is k-vocabulary.

Let λ, µ ∈ aut(M⊗) be k-linear bijections such that

µ ◦ λ = idM⊗ = λ ◦ µ.

Our main tool is λ-dependent family of realizations of M -universal ten-sor algebra TM ' λTM ≡ {M⊗,⊗λ} with an associative λ-dependentmultiplication ⊗λ as a trivial deformation of concatenation ⊗ ≡ ⊗λ=id. Aλ-dependent realization of TM extends to λ-dependent realization of bi-universal bi-associative bi-unital tensor Hopf-gebra but this is beyond theaims of the present paper.

As an example we construct an algebra isomorphism µ ∈ iso(λTM, TM)as a polynomial in a scalar product η ∈ lin (M⊗2,k) and in a pre-braidσ ∈ End(M⊗2), such that µ(η = 0, σ) = id, see e.g. formulas (6.5-6.8).

Let I < M⊗ be a k-sub-space, M∧ ≡ M⊗/I be a factor k-space ofmultivectors (not an algebra) and let I be a two-sided ideal in λTM ≡{M⊗,⊗λ} what is denoted by I � λTM.

It is shown that a condition I � λTM generalize the definition of theDirac gamma matrices [to (λ, I)-dependent case γ = γ(λ, I)] identified witha Clifford (µI)-map [see (2.4)],

γ ∈ lin [M,F End(M∧)], γEnd ∈ alg [TM,End(M∧)]. (1.4)

A factor algebra λTM/I is a λ-deformation of TM/I. A λ-deformed factormultiplication ∧λ of multivectors, ∧λ : M∧ ⊗M∧ −→ M∧, is expressed interms of (λ, I)-dependent Dirac matrix γ (1.4) and of an algebra isomorphismµ ∈ iso(λTM, TM),

π ∈ epi (λTM, λTM/I), ∧λ ◦ π = γEnd ◦ µ. (1.5)

The formula (1.5) allows to calculate the minimum polynomials of

Γ ≡ γµ ⊗ γµ ∈ End(M∧ ⊗M∧). (1.6)

The Casimir operators of the Lorentz Lie algebra L represented in two-particle space M∧ ⊗ M∧ can be expressed in terms of γµ ⊗ γµ. Thereforethe minimum polynomial of Γ (1.6) allows to decompose the Dirac operatorfor a systems of two particles - antiparticles with spin 1

2into the Dirac op-

erators for composite (anti)particles - with spin zero, spin 1 and for a vectorfield (1

2, 12).

Acknowledgements

The author wishes to thank to Adan Ruben Rodrıguez Domınguez forproving that Conjecture 8.2 holds true for dimM = 4 and to Bertfried Fauserand Bernard Jancewicz for helpful reading this text. The author is greatlyindebted to Jaime Keller for the invitation and hospitality in Mexico.

2 Notation

For a k-alphabet M ∈ k-space, M⊗ ∈ k-space denotes Z-graded k-vocabulary (not an algebra) i.e. a totality of all finite sentences in M.

By definition functors T and F are adjoint [Kan 1958]: bifunctors lin k(·, F ·)and alg k(T ·, ·) are naturally equivalent. This means that a natural set bi-jection holds,

∀M × A ∈ k-space×k-alg,

lin k(M,FA) 3 `←→ `A ∈ alg k(TM,A), `A|M ≡ `. (2.1)

An example of realization of M -universal tensor k-al-gebra is TM '{M⊗,⊗} i.e. a Z-graded k-vocabulary M⊗ of all finite words in an alphabetM with a concatenation ⊗ as a multiplication of a grade ⊗ = 0.

Table 1: Notations

k is a field with 12∈ k

k-space a category of k-spaces (of k-alphabets)

k-alg a category of associative unital k-algebras

T : k-space → k-alg the tensor algebra functornot additive on morphisms;

F : k-alg → k-space the forgetful functor;

⊗ bifunctor of tensor product:k-space×k-space→ k-space.

⊗ means ⊗k if not otherwise stated;

lin ≡ lin k, End ≡ Endk are both sided k-linear bifunctors;

M ∈ k-space is a k-space (a k-alphabet);

M⊗ = FTM a Z-graded k-vocabulary in a k-alphabet M ;

fk(M⊗) ≡⊕

i≤kM⊗i a filtration of a graded k-space M⊗;

M∗ ≡ lin (M, k) a dual k-space of co-vectors;

η = ηT ∈ lin (M,M∗) 'M∗⊗2, is a symmetric scalar product;

id⊗n ≡ (idM)⊗n = idM⊗n .

An example of realization ofM -co-universal co-gebra is, ShM ' {M⊗, sh}.Diagram 1 shows the universal and co-universal lifts,

`m ≡ `A = m⊗ ◦ `⊗ ∈ alg (TM,A),

`4 ≡ `C = `⊗ ◦ 4⊗ ∈ cog(C, ShM).(2.2)

Minjection−−−−−−−−→ M⊗y` y`⊗

Am⊗←−−−−− A⊗

Mprojection←−−−−−−−−− M⊗x` x`⊗

C4⊗−−−−−→ C⊗

Diagram 1. The universal and co-universal lifts

A notation I � A means that I is two sided ideal in an algebra A. LetI � TM. An algebra map `A (2.1) factors to algebra map

`A/I ∈ alg (TM/I,A) if and only if `AI = 0. (2.3)

This suggest to define a set of linear I-maps

for I � TM, lin [(M, I), FA] ≡ {` ∈ lin (M,FA), `AI = 0 ∈ A}. (2.4)

Definition 2.1 (Universal algebra). Let I � TM. An (M, I)-universal k-algebra T (M, I) is an object in k-alg with an I-map (2.4)

ı ∈ lin [(M, I), FT (M, I)] such that ker ı = I ∩M,

having the universal property: for any A ∈ Obj-k-alg there is a set bijection

lin [(M, I), FA] ' alg [T (M, I), A].

An algebra T (M, I) if exists is unique up to isomorphism. The existence isproved by setting T (M, I) = TM/I. The composition of injection M ↪→ TMwith projection TM → TM/I is an I-map which kernel is equal to I∩M. Weneed to check the universal property of a factor algebra TM/I. According to(2.3) each I-map ` determine unique algebra map (`A)/I ∈ alg (TM/I,A)such that (`A/I)|M ≡ `.

3 Quantization Versus Deformation.

The Poincare-Birkhoff-Witt Theorem

as Corollary

Let λ, µ ∈ aut(M⊗) be k-linear bijections such that

µ ◦ λ = idM⊗ = λ ◦ µ. (3.1)

Let TM denote a realization with concatenation, TM = {M⊗,⊗}. ThenλTM is a k-space M⊗ with λ-dependent multiplication ⊗λ,

λTM = {M⊗,⊗λ ≡ λ ◦ ⊗ ◦ (µ⊗ µ)}.

Therefore the bijections λ and µ are isomorphisms of associative algebras

λ ∈ iso(TM, λTM), µ ∈ iso(λTM, TM). (3.2)

Throughout of this paper we assume that a k-sub-space I < M⊗ is atwo-sided ideal in both isomorphic tensor algebras,

I � TM and I � λTM. (3.3)

Then λI � λTM and µI � TM are two-sided ideals.

Definition 3.1. We have the factor k-algebras∧M ≡ TM/I an exterior algebra of multivectors,

λTM/λI an exterior algebra by quantization,

C`(M,µI) ≡ TM/µI a Clifford algebra as quantization,

C`(M,λ, I) ≡ λTM/I a Clifford algebra of multivectors as deformation.

A name ‘Clifford algebra’ could be misleading, however it is convenientbecause, as we are going to show in the next sections, the names ‘Cliffordalgebra as quantization’ and ‘a Clifford algebra of multi-vectors’ reflects thegeneralization of the known cases for the particular algebra isomorphismλ ∈ iso(TM, λTM) [in terms of a filtered map j ∈ End(M⊗) of grade j =−2 (Definition 5.1) constructed later on in terms of a scalar product η ∈lin (M⊗2,k) (6.4)] and for a particular ideal I�TM [generated by a quadraticform v ⊗ v −Q(v)].

I will use the following notation for the factor k-spaces and for the factorproducts with the algebra epimorphism π,

M∧ ≡M⊗/I, ∧ ≡ ⊗/I, ∧λ ≡ ⊗λ/I,

TM/I = {M∧,∧}, λTM/I = {M∧,∧λ}, (3.4)

π ∈ epi (TM, TM/I), π ∈ epi [λTM, λTM/I].

By definition a factor k-algebra TM/µI ≡ {M⊗/µI,⊗/µI} is a µ-quantizationof a factor algebra TM/I.

A factor algebra (λTM)/I is a k-space M∧ ≡M⊗/I with a λ-dependentmultiplication ∧λ (3.4) and is said to be the λ-deformation of a factor algebra∧M ≡ TM/I ≡ {M∧,∧}, ∧λ=id = ∧,

TM/Iλ-deformation−−−−−−−−−−−→ (λTM)/I.

Corollary 3.2 (Poincare, Birkhoff 1937, Witt 1937). Let µ and λ be thealgebra isomorphisms (3.1-3.2) and let a k-sub-space I < M⊗ be a two-sidedideal in λ-deformed tensor algebra, I � λTM. Then a factor algebra TM/µIis isomorphic with an algebra of multivectors (λTM)/I.

In another words under conditions (3.3) the µ-quantization is equivalentto λ-deformation of a factor algebra TM/I.

Proof. An algebra isomorphisms µ ∈ iso(λTM, TM) and λ (3.2) factor toisomorphisms of factor algebras

I � λTM =⇒ (λTM)/I ' TM/µI,

I � TM =⇒ TM/I ' (λTM)/λI.

The factor algebras (λTM)/I and TM/I are isomorphic if and only ifλI = I.

The isomorphic factor algebras (λTM)/I and TM/µI for j-dependentalgebra isomorphism µ (see Sections 5 & 6) include the realizations

for grade j = −2 of a universal Clifford and Weyl algebras,

for grade j = −1 of a universal enveloping algebra of an algebra

m ∈ lin (M⊗2,M).

A Clifford multiplication of multivectors ∧λ ≡ ⊗λ/I must not be confusedwith the Clifford multiplication ⊗/µI.

Let a k-sub-space I < M⊗ be Z-graded. Then a factor k-space M⊗/I isZ-graded. If an algebra isomorphism µ and λ (3.2) for λ 6= id do not preservegradation then k-space M⊗/µI is not Z-graded. Contrary to realization ofthe (M,µI)-universal algebra as a factor algebra TM/µI, a factor algebraof multivectors λTM/I is a Z-graded space M⊗/I with a multiplicationwhich do not need to be Z-graded. A priori ‘a bivector’ is meaningless in arealization as a factor k-algebra TM/µI.

4 Clifford Multiplication of Multivectors

Let λ, µ be given algebra isomorphisms (3.1-3.2) and let a k-sub-spaceI < M⊗ be a two-sided ideal in a deformed tensor algebra λTM,

I � λTM (4.1)

The first aim of this section is to give, under the above condition (4.1),an abstract definition of the ‘factored deformed creator’ γ = γ(λ, I) whichgeneralize the matrices introduced by Dirac in 1928.

Let e ∈ lin k[M,F End(M⊗)] be a concatenation of one letter from theleft (creating one letter from the left). Then ⊗ ≡ eEnd ∈ alg [TM,End(M⊗)]is a multiplication by concatenation.

A λ-deformed creator ` ∈ lin (M ⊗M⊗,M⊗) ' lin [M,F End(M⊗)] isdefined in terms of the given algebra isomorphism λ (3.1-3.2) as follows

` ≡ λ ◦ e ◦ (idM ⊗µ), `End = λ ◦ eEnd ◦ (idM⊗ ⊗µ). (4.2)

Definition 4.1 (The Dirac matrix). Let π ∈ epi (M⊗,M∧) be k-linear epi-morphism such that ker π ≡ I. The Dirac matrix γ = γ(λ, I) ∈ lin [M,F End(M∧)]is defined as factored deformed creator by the following diagram,

M ⊗M⊗ `−−−−→ M⊗yidM ⊗ πyπ

M ⊗M∧ γ−−−−→ M∧

Theorem 4.2. Let the boxed condition (4.1) holds. Then

(i) The exists the Dirac matrix γ(λ, I).

(ii) The Dirac matrix is (µI)-map.

Proof. We shall prove that the condition (4.1) imply the existence of γ. Thecondition for I to be two-sided ideal in λTM is equivalent to two conditions,

π ◦ ⊗λ ◦ (idM⊗ ⊗I) = 0 left ideal, (4.4)

π ◦ ⊗λ ◦ (I ⊗ idM⊗) = 0 right ideal, (4.5)

⊗λ = `End ◦ (µ⊗ idM⊗). (4.6)

The condition (4.4) imply that exists the Dirac matrix γ (4.3). To show thisnote that (4.4) gives

(π ◦ `End)(idM⊗ ⊗ I) = 0. (4.7)

Then exists the Dirac matrix γ (4.3) such that

`End ∈ alg [TM,End(M⊗)],

γEnd ∈ alg [TM,End(M∧)],

π ◦ `End = γEnd ◦ (idM⊗ ⊗π). (4.8)

The factored deformed creator γ (4.8) is a (µI)-map iff

γEnd(µI) ≡ (γEnd ◦ µ)I = 0 the Dirac condition, (4.9)

γEnd ◦ (µI ⊗ idM∧) = 0,

and then γEnd/µI ∈ alg [TM/µI,End(M∧)] . (4.10)

Due to (4.8) the Dirac condition (4.9) is equivalent to (4.5).

A space of multivectors M∧ with (4.10) is a one-sided modul for an(M,µI)-universal algebra C`(M,µI). This C`-modulM∧ is faithful, ker

(γEnd/µI

0.A fact that an (M,µI)-universal algebra C`(M,µI) can be represented in

a factor k-space M∧ should not be confused that a factor space M∧ itself isan algebra with a factor multiplication ∧λ (3.4),

M∧ ⊗M∧ ∧λ−−−−−→M∧. (4.11)

Theorem 4.3 (The Clifford multiplication of multivectors). Let boxed con-dition (4.1) holds. Then a Clifford multiplication ∧λ of multivectors (3.4)on a factor k-space M∧ is a composition of an algebra isomorphism µ ∈iso(λTM, TM) with a product of (λ, I)-dependent Dirac matrices γEnd (4.8),i.e. the following commutative diagram of algebra maps holds,

λTM = {M⊗,⊗λ} µ−−−−→ TM = {M⊗,⊗}yπ= mod I

yγEnd

C`(M,λ, I) ≡ {M∧,∧λ} ∧λ−−−−−→ End(M∧)

(4.12)

Proof.

∧λ ◦ (π ⊗ π) = π ◦ ⊗λ

= π ◦ `End ◦ (µ⊗ idM⊗)

= γEnd ◦ (µ⊗ π),

∧λ ◦ π = γEnd ◦ µ.

The formula for ∧λ (4.12) with (4.2) and (5.8) can be related to the Dyson[1949] and the Wick [1950] expansion in a quantum field theory.

5 Realizations of Universal Tensor Algebra

In this section we shall construct an algebra isomorphism µ ∈ iso(λTM, TM)(3.2) in terms of a filtered map j ∈ End(M⊗) generalizing in his way aconstruction made by Bourbaki [1959]. This leads to a j-dependent familyof realizations of an M -universal tensor algebra TM ' λjTM ≡ {M⊗,⊗j}with j-deformed multiplication ⊗j which not need to be Z-graded.

Definition 5.1. A filtered map j ∈ End(M⊗) is said to be a deforming mapif

j|M⊗m ∈ lin [M⊗m, fm−1(M⊗)]. (5.1)

A deforming map j is said to be homogeneous of grade j = s < 0 if

j|M⊗m ∈ lin (M⊗m,M⊗(m+s)), j|(f−s−1M⊗) ≡ 0.

Let j ∈ End(M⊗) be deforming map. Consider j-deformed creator,

` ≡ e+ j ◦ e ∈ lin [M,F End(M⊗)],

` : M 3 v 7−→ `v ≡ ev + j ◦ ev ∈ End(M⊗), (5.2)

`End ∈ alg [TM,End(M⊗)].

Therefore {M⊗, `End} is a j-dependent family of left (TM)-modules.The Bourbaki lemma [Bourbaki 1959] (copied in [Crumeyrolle 1990, pp.

43-44, lemmas 3.1.2 and 3.1.4]) has the following generalization.

Lemma 5.2. Let j ∈ End(M⊗) be a deforming map (5.1) Then(i) exist unique bijective j-dependent maps µ and λ ∈ End(M⊗) such that,

µ = idM ⊗ µ− µ ◦ j, µ|k = id, (5.3)

λ = idM ⊗ λ+ j ◦ (idM ⊗ λ), λ|k = id . (5.4)

(ii) µ ◦ λ = idM⊗ = λ ◦ µ.

Proof. Because (idM ⊗µ) ◦ ev = ev ◦µ etc, then conditions (5.3-5.4) have theequivalent forms

eEnd ◦ (idM⊗ ⊗ µ) = µ ◦ `End,

λ ◦ eEnd = `End ◦ (idM⊗ ⊗ λ). (5.5)

The existence and uniqueness of µ and λ is proved by induction as in [Bour-baki 1959]. Moreover

µ ◦ λ ◦ eEnd = eEnd ◦ [idM⊗ ⊗ (µ ◦ λ)],

`End ◦ [idM⊗ ⊗ (λ ◦ µ)] = λ ◦ µ ◦ `End.Then µ ◦ λ = idM⊗ = λ ◦ µ follows by induction.

A following j-dependent composition is a trivial associative deformationof a tensor product,

⊗j ≡ λ ◦ eEnd ◦ (µ⊗ µ)

= `End ◦ (µ⊗ idM⊗) : M⊗ ⊗M⊗ −→ M⊗. (5.6)

Corollary 5.3. Let λTM ≡ {M⊗,⊗j} be an algebra with j-dependent pro-duct (5.6) on a k-space M⊗. Then the bijections of Lemma 5.2 are algebraisomorphisms

µ ∈ iso(λTM, TM), λ ∈ iso(TM, λTM).

Remark 5.4. For Z-graded ideal I � TM Definition 5.1 of a deforming mapj ensure an isomorphism of the following filtered k-spaces due to the samearguments as used by Bourbaki [1959] and extended by Revoy [1977],

M⊗/µI ≈ M∧ ≡M⊗/I (5.7)

For a quadratic algebras k-linear bijection (5.7) is implicit in [Chevalley 1954]and was proved by Bourbaki [1959], Kahler [1960] and for n-ic algebras byRevoy [1977].

The algebra isomorphism µ ∈ iso(λTM, TM) and λ are polynomials in adeforming map j. We shall solve recurrent relation (5.3) for grade j = s < 0.Let

k,m, s ∈ Z, jkm,s ≡

0≤ik≤...≤i1≤m+ks

(id⊗i1 ⊗j

). . .(id⊗ik ⊗j

)if 1 ≤ k and 0 ≤ m+ ks,

id⊗m if k = 0,0 otherwise.

In particular

j1m,s =∑

0≤i≤m+s

id⊗i⊗j,

j2m,s =∑

0≤i≤l≤m+2s

(id⊗l⊗j

)◦(id⊗i⊗j

An algebra isomorphism µ ∈ iso(λjTM, TM) as the solution of recurrence(5.3) is as follows

for grade j = s < 0, µ|M⊗m = id⊗m +∑

1≤k≤−ms

(−1)kjkm,s. (5.8)

Example 5.5. For s ≡ grade j = −1, µ|k = id,

µ|M = id−j, µ|M⊗2 = id−j − idM ⊗j + j2.

For s ≡ grade j = −2, µ|(k⊕M) = id,

µ|M⊗2 = id−j,µ|M⊗3 = id−j − idM ⊗j,µ|M⊗4 = id−j − idM ⊗j − id⊗2⊗j + j2,

µ|M⊗5 = id−j − idM ⊗j − id⊗2⊗j − id⊗3⊗j+j2 + (idM ⊗j) ◦ j + (idM ⊗j)2.

6 Two Examples of Deforming Map in a Braided

Category

In this section we give two examples of a deforming map j (5.1) in terms oftensors,

σ ≡ σ1,1 ∈ End(M⊗2) a braid which is a solution of (1.1), (6.1)

η ∈ lin (M⊗2, k) a scalar product, (6.2)

m ∈ lin (M⊗2,M) an algebra structure on M. (6.3)

For m,n ∈ N, a braid σn,m ∈ EndM⊗(n+m) is defined by means of a recur-rence which is an example of the Mac Lane two ‘hexa-gons’ [Mac Lane 1963,1971],

σm+n,k ≡ (σm,k ⊗ id⊗n) ◦ (id⊗m⊗σn,k),σk,m+n ≡ (id⊗m⊗σk,n) ◦ (σk,m ⊗ id⊗n).

Figure 2: A deforming map.

j|M⊗n ≡∑

M M⊗i M M⊗(n−2−i)

The first example of a deforming map j = j(η, σ) for s = grade j = −2as a polynomial in a braid σ is defined by means of Figure 2.

j|M⊗n ≡(η ⊗ id⊗(n−2)

)◦[idM ⊗

(∑σi,1 ⊗ id⊗(n−2−i)

)]. (6.4)

j|(k⊕M) = id, j|M⊗2 = η,

j|M⊗3 = (η ⊗ idM)(id⊗3 + idM ⊗σ),

j|M⊗4 = (η ⊗ id⊗2)(id⊗4 + idM ⊗σ ⊗ idM + idM ⊗σ2,1),

j2|M⊗4 = (η ⊗ η)(id⊗4 + idM ⊗σ ⊗ idM + idM ⊗σ2,1), etc.

With the help of Example 5.4 I can calculate an algebra isomorphism µin Corollary 5.3 as a polynomial in two variables µ = µ(η, σ),

µ|M⊗2 = id⊗2− η. (6.5)

µ|M⊗3 = id⊗3−η ⊗ idM −(η ⊗ idM) ◦ (idM ⊗σ)− idM ⊗ η. (6.6)

µ|M⊗4 = id⊗4−η ⊗ id⊗2− id⊗2⊗ η − idM ⊗ η ⊗ idM

−(idM ⊗ η ⊗ idM) ◦ (id⊗2⊗σ) + η ⊗ η+(η ⊗ η − η ⊗ id⊗2) ◦ (idM ⊗σ ⊗ idM + idM ⊗σ2,1). (6.7)

µ|M⊗5 = id⊗5−η ⊗ id⊗3− id⊗3⊗η − idM ⊗η ⊗ id⊗2− id⊗2⊗η ⊗ idM

−(η ⊗ id⊗3)(idM ⊗σ ⊗ id⊗2)− (idM ⊗η ⊗ id⊗2)(id⊗2⊗σ ⊗ idM)

−(id⊗2⊗η ⊗ idM)(id⊗3⊗σ)

−(η ⊗ id⊗3)(idM ⊗σ2,1 ⊗ idM + idM ⊗σ3,1)

+(idM ⊗η ⊗ η − idM ⊗η ⊗ id⊗2)(id⊗2⊗σ2,1)

+η ⊗ η ⊗ idM +η ⊗ idM ⊗η + idM ⊗η ⊗ η+(idM ⊗η ⊗ η)(id⊗2⊗σ ⊗ idM) + (η ⊗ idM ⊗η)(idM ⊗σ ⊗ id⊗2)

+(η ⊗ η ⊗ idM)(idM ⊗σ ⊗ id⊗2 + id⊗3⊗σ)

+(η ⊗ η ⊗ idM +η ⊗ idM ⊗η)(idM ⊗σ2,1 ⊗ idM)

+(η ⊗ η ⊗ idM)(idM ⊗σ ⊗ σ)

+(η ⊗ η ⊗ idM +η ⊗ idM ⊗η)(idM ⊗σ3,1)

+(η ⊗ η ⊗ idM)(id⊗3⊗σ)(idM ⊗σ2,1 ⊗ idM + idM ⊗σ3,1). (6.8)

The second example of a deforming map j = j(m,σ) is for a k-algebra{M,m} (6.3). This example has s = grade j = −1 and is a polynomial inan invertible braid (6.1) defined as follows

j|M⊗n ≡∑[

(σ−1)1,i ⊗ id⊗(n−2−i)]◦(m⊗ id⊗(n−2)

idM ⊗σi,1 ⊗ id⊗(n−2−i)),

j|(k⊕M) ≡ 0, j|M⊗2 ≡ m and µ|M⊗2 = id−m.

Theorem 6.1 (Oziewicz, Paal, Rozanski 1995, Theorem 5.3). Let

∀ v ∈M, x ≡{η ◦ ev ∈ lin (M,k) orm ◦ ev ∈ lin (M,M).

Let σ be a braid operator (6.1). Then due to the braid equation (1.1)

Dx(σ) ≡ j(·, σ) ◦ ev ∈ der TM.

Example 6.2. Consider two ideals

I∓ ≡ gen{v ⊗ w ± w ⊗ v, ∀v, w ∈M} � TM, (6.9)

then ∀v ∈M, {j(η,−s) ◦ ev}I− < I− and {j(m,+s) ◦ ev}I+ < I+.(6.10)

The condition (6.10) is necessary and sufficient that exists the Dirac matrixγ = ∧+ iη or γ = ∧+ im (4.3). The Dirac matrix is a µI∓ map if the Diraccondition (4.9) holds,

γv ◦ γw + γw ◦ γv = γη(v⊗w+w⊗v), (6.11)

γv ◦ γw − γw ◦ γv = γm(v⊗w−w⊗v). (6.12)

The deformation (5.2) with (6.4) goes back to the work of Hermann Graß-mann [1877]. The factored deformed creator γ (4.3-4.8) for (5.8) fulfillingthe Dirac condition (4.9-6.11) generalize the Chevalley map for an ideal (6.9)[Chevalley 1954, pp. 38-42]. Chevalley introducing his map rediscoveredDirac matrices from a deeper level.

The Dirac condition (6.12) for a skew-symmetrical multiplication m implythe Jacobi condition (if 1

2∈ k) and in particular holds if {M,m} is a Lie k-

algebra.

Example 6.3 (The Dirac Condition for n-ic Algebra). Condition (4.9) gener-alize the Dirac definition of the gamma matrices introduced in 1928. Let likein (6.5),

η ∈ lin (M⊗n,k), λ = λ(η) ∈ iso(TM, λTM), λ|M⊗n ≡ id +η,

P ∈ End(M⊗n), I = gen im P � TM and I � λTM.

The composition γEnd ◦ P : M⊗n −→ End(M∧) is said to be anti-multi-commutator of the Dirac matrix γ, and γEnd − γEnd ◦ P is said to be thecommutator. The Dirac condition (4.9) has a form

γEnd ◦ µ ◦ P = 0, µ ◦ P = P − η ◦ P.

Then for anti-commutator we have

γEnd ◦ P = γEnd ◦ η ◦ P = idM∧ ·(η ◦ P ).

If P is a projector then the above Dirac condition holds for arbitrary ‘skewsymmetric’ part η − η ◦ P.

7 Involutive Braid

The technical Sections 6-7 illustrate an example of a deformation of a tensoralgebra and results will be used in the next Section for a calculation of theminimum polynomial of the tensor product of the Dirac matrices γµ ⊗ γµ.

The formulas of the previous Section holds for arbitrary (pre)-braid σ.For further needs it would be convenient to have these expressions for aninvolutive braid σ2 = id . In this section we assume that a scalar productη ∈ lin (M⊗2,k) is a σ-morphism

σ2 = id, idM ⊗η = (η ⊗ idM) ◦ σ1,2. (7.1)

The generalization of condition (7.1) for non-involutive pre-braid and for agiven pre-braid-dependent ideal I = I(σ) was derived from condition (4.1)in [Durdevic and Oziewicz 1996].

With help of (7.1) an algebra isomorphism (5.8) for a deforming map(6.4) is a polynomial in a scalar product η,

µ|M⊗m = id⊗m +∑

1≤k≤m2

(−1)k(η⊗k ⊗ id⊗(m−2k)

)◦ Ak(σ). (7.2)

In particular for no more than five letters we have

A1(σ)|M⊗3 = id⊗3 + idM ⊗σ + σ1,2, (7.3)

A1(A1 − 1)(A1 − 3)|M⊗3 = 0,

A1(σ)|M⊗4 = id⊗4 + idM ⊗σ ⊗ idM +σ1,2 ⊗ idM + idM ⊗σ2,1

+(σ1,2 ⊗ idM) ◦ (id⊗2⊗σ) + σ2,2, (7.4)

A2(σ)|M⊗4 = id⊗4 + idM ⊗σ ⊗ idM + idM ⊗σ2,1, (7.5)

A1(σ)|M⊗5 = id⊗5 + idM ⊗σ ⊗ id⊗2 +σ1,2 ⊗ id⊗2

+ idM ⊗σ2,1 ⊗ idM + idM ⊗σ3,1 + (σ1,2 ⊗ id⊗2)(id⊗2⊗σ ⊗ idM)

+σ2,2 ⊗ idM +(σ1,2 ⊗ id⊗2)(id⊗2⊗σ2,1)

+(σ2,2 ⊗ idM)(id⊗3⊗σ) + σ3,2, (7.6)

A2(σ)|M⊗5 = id⊗5 + id⊗3⊗σ + idM ⊗σ ⊗ id⊗2

+ id⊗2⊗σ1,2 + idM ⊗σ2,1 ⊗ idM + idM ⊗σ ⊗ σ+ idM ⊗σ1,3 + idM ⊗σ3,1 + (id⊗3⊗σ)(idM ⊗σ2,1 ⊗ idM)

+(id⊗3⊗σ)(idM ⊗σ3,1) + (id⊗2⊗σ1,2)(idM ⊗σ2,1 ⊗ idM)

+σ1,4 + σ1,4(id⊗2⊗σ ⊗ idM) + (id⊗2⊗σ1,2)(idM ⊗σ3,1)

+σ1,4(id⊗2⊗σ2,1). (7.7)

Let T denote the transposition, then

(a⊗ b)T = aT ⊗ bT and [(σT )m,n]T = σn,m.

For determination of the minimum polynomials of the tensor product of theDirac matrices in the next Section we need the products Ak(σ)[Al(σ

T )]T foran involutive braid σ2 = id . Examples

A1AT1 |M⊗3 = 3 · id⊗3 +2 · idM ⊗σ+σ2,1 +σ1,2 +2(σ⊗ idM)(idM ⊗σ)(σ⊗ idM),

A1AT1 |M⊗4 = 6 · id⊗4 +2 · idM ⊗σ ⊗ idM

+(σ1,2 ⊗ idM + idM ⊗σ2,1

) (id⊗4 +σ2,2

)+(id⊗4 +4 · idM ⊗σ ⊗ idM +σ2,2

) (σ2,1 ⊗ idM + idM ⊗σ1,2

)+2 ·

(id⊗2⊗σ

) (σ2,1 ⊗ idM

)+ 2 ·

(σ1,2 ⊗ idM

) (id⊗2⊗σ

)+6 · σ2,2 + 2 · σ1,3

(idM ⊗σ2,1

8 Minimum Polynomial of the Tensor Pro-

duct of the Dirac Matrices

Let η = ηT ∈ lin (M,M∗) 'M∗⊗2 be a symmetric scalar product on vectorsand let ξ = ξT ∈ lin (M∗,M) ' M⊗2 be a symmetric scalar product onco-vectors. We suppose that these products are invertible i.e. they are notdegenerate,

ξ ◦ η = idM , η ◦ ξ = idM∗ , ηξ = dimM. (8.1)

Let {eµ ∈ M} be any basis in k-space M, not necessarily η-orthogonal,and let {eµ ∈M∗} be a basis in a dual k-space, dual to a basis {eµ}, eµeν ≡

δµν . Then {ξµ ≡ ξeµ ∈M} is ξ-dependent reciprocal basis in M and

ξ =∑

ξµ ⊗ eµ = ξT =∑

eµ ⊗ ξµ ∈M⊗2. (8.2)

The Dirac matrix (4.3) can be tensored

γ ∈ lin [M,F End(M∧)], Tγ ∈ alg [TM, TF End(M∧)].

We wish to know the spectrum of an operator,

Γ ≡ (Tγ)ξ =∑

γξµ ⊗ γeµ ∈ [End(M∧)]⊗2 ' End[(M∧)⊗2]. (8.3)

An operator (8.3) enter the game if we consider a tensor product of C`-modules or a tensor product of spinor or twistor spaces. In particular theCasimir operators of the Lorentz Lie algebra are represented in a tensorproduct of C`-module in terms of (8.3). A spectrum of (Tγ)ξ allows todecompose the tensor product of modules into a sum of simple modules.

An operator (8.3) (Tγ)ξ = γµ ⊗ γµ a priori depends on a scalar productξ and on a braid σ and his spectrum depends on dimM ≡ dimkM.

In the sequel we assume that

(i) a braid is minus the switch σ = −s i.e. ∀ v, w ∈M, σ(v ⊗ w) = −w ⊗ v,(ii) a quadratic form Q generate two-sided ideal I � TM,

I ≡ gen{v ⊗ v −Q(v),∀ v ∈M}(iii) a symmetric scalar product η (8.1) is associated to a quadratic form Q.

Lemma 8.1. The above assumptions (i-iii) imply condition (4.1), i.e. implythat exists the Dirac matrix γ.

Proof is omitted.Let for a fixed dimM, pdimM(x) denotes the minimum polynomial of (8.3).

Theorem 4.3 imply that pdimM(x) is of the order ≤ 1 + dimM.

Conjecture 8.2. The minimum polynomial of Γ is independent of invertiblescalar product ξ and

pdimM(x) =

{(x2 − 1)(x2 − 32) . . . (x2 − n2) if dimM = n is odd,

x(x2 − 22) . . . (x2 − n2) if dimM = n is even.

Conjecture 8.2 holds true for dimM ≤ 4. For dimM ≤ 3 this was shownby hand made calculation with the help of Theorem 4.3. To calculate aminimum polynomial of (8.3) for dimM = 4 in the same manner needs tosum (10+15)2 = 625 terms. Adan Ruben Rodrıguez Domınguez proved withthe help of the computer program Maple that Conjecture 8.2 holds true fordimM = 4.

Table II contain some technical details of the hand made calculationsof the minimum polynomials. The tensor powers of a scalar product (8.2)supplemented by an appropriate permutations with the help of the switchs(v⊗w) ≡ (w⊗v) are considered as k-linear maps with values in End(M∧⊗M∧) (Table II) as in the examples

γEndξ−→ γEndξ = dimM · idM∧ ,

η ⊗ η (id⊗s⊗id)(ξ⊗ξ)−−−−−−−−−→ dimM · idM∧⊗M∧ .

Table 2: The examples of the action of ‘ξ⊗n’ with values in End(M∧ ⊗M∧)

η ⊗ γEnd → dimM · id[η ⊗ γ]⊗ [γ ⊗ η] → Γ

[(η ⊗ γ)(id⊗σ)]⊗ [η ⊗ γ] → −Γ[γEnd(id⊗σ)]⊗ [η ⊗ γ] → (dimM − 2) · Γ

[(η ⊗ γ)(σ ⊗ id)]⊗ [η ⊗ γ] → −(dimM) · Γ[(η ⊗ γ)σ1,2]⊗ [η ⊗ γ] → Γ[(η ⊗ γ)σ2,1]⊗ [η ⊗ γ] → Γ

[(η ⊗ γ)(1.1)]⊗ [η ⊗ γ] → −Γ[(η ⊗ γEnd)(id⊗2⊗σ ⊗ id)]⊗ [γ ⊗ η ⊗ η] → (dimM − 2) · Γ

[(η ⊗ γEnd)(id⊗2⊗σ)]⊗ [γ ⊗ η ⊗ γ] → Γ2 − 2 · dimM · id[(η ⊗ γEnd)(id⊗σ1,2)]⊗ [η ⊗ γEnd] → −Γ2 + 2 · dimM · id[(η ⊗ γEnd)(id⊗σ1,3)]⊗ [η ⊗ γEnd] → −Γ3 + 4(dimM − 1)Γ

9 Opposite Exterior Algebra

Let a k-subspace I < M⊗ be Z-graded then a factor k-space M∧ ≡ M⊗/Iis Z-graded. In a particle physics Z-gradation corresponds to a number of

particles.A principal unipotent automorphism (a Z2-graded involution) ψ ∈ aut⊗

of tensor k-algebra TM factors to involutive automorphism ψ ∈ aut∧ of anexterior k-algebra

∧M and extends to unipotent Ψ ∈ End(M∧ ⊗M∧),

∧ ∈ lin (M∧ ⊗M∧,M∧), (9.1)

ψ|M ≡ − idM , ∧ ◦ (ψ ⊗ ψ) = ψ ◦ ∧, ψ2 ≡ idM∧ ,

Ψ ≡ 12(idM∧ ⊗ idM∧ −ψ⊗ψ+ψ⊗ idM∧ + idM∧ ⊗ψ), Ψ2 = idM∧⊗M∧ . (9.2)

A Z2-gradation of the spacesM⊗ and ofM∧ is given by projectors 12(idM∧ ±ψ)

on even (of grade 0 ∈ Z2) and on odd (of grade 1 ∈ Z2) multivectors.A tensor and exterior k-algebras are Z2-graded and an exterior algebra

for an ideal (6.9) is Z2-commutative. In order to express this condition onexterior product (9.1) in argument-free way we need the switch, cf. with(1.2), and an involutive braid for which the Artin braid equation (1.1) holds,

s, σ ∈ End(M∧ ⊗M∧), ∀α, β ∈M∧, s(α⊗ β) ≡ β ⊗ α, s2 = id = σ2.(9.3)

If m is a multiplication then m ◦ σ is said to be σ-opposite multiplicationand a k-space with m◦σ is said to be a σ-opposite algebra. A multiplicationm is σ-commutative if m = m ◦ σ. The exterior product (9.1) is (Ψ ◦ s)-commutative (Z2-graded commutative),

∧ = ∧ ◦Ψ ◦ s, Ψ ◦ s = s ◦Ψ. (9.4)

The condition (9.4) is reduced to not graded (skew)-commutativity if ψ =∓ idM∧ . An s-opposite exterior k-algebra

∧oppM has an associative multi-plication ∧ ◦Ψ, ∧opp

M ≡ {M∧,∧ ◦Ψ}.

9.1 Deformed Exterior Algebra

We will use the following k-linear forms

φ ∈ lin (M⊗2, k), η ≡ 12(φ+ φT ), ω ≡ 1

2(φ− φT ) ∈ lin (M∧2,k), (9.5)

φ = ω + η.

A form φ extends to an algebra map φ ∈ alg (∧M,∧M∗).

An ω-deformed exterior algebra∧ωM ≡ {M∧,∧ω} is a family of associa-

tive Z2-commutative multiplications {∧ω} (Theorem 4.3) parametrized by a‘two-particle correlation’ ω (9.5),

∧ω = ∧ω ◦Ψ ◦ s. (9.6)

An ω-deformation of an exterior algebra do not respect Z-gradation. A Z-gradation of a space M∧ is not unique. For each deformed exterior product∧ω (9.6) exists a unique ω-deformed Z-gradation preserved by ∧ω. In a sequelan initial, not deformed, Z-gradation of an exterior space M∧ is said to bea bare Z-gradation and an ω-deformed Z-gradation is said to be a cloudedZ-gradation,

M∧ ≡M∧ω = ⊕M∧ωi, dimM∧ωi ≡ dimM∧i,

M∧ω0 ≡M∧0 ≡ k, M∧ω1 ≡M∧1 ≡M,

M∧ωi < M∧i ⊕M∧(i−2) ⊕ . . .⊕ (k orM).

If a deformation of an exterior product is accompanied with a deformation ofa Z-gradation then a deformation is trivial and exists ω-dependent algebraisomorphism µ ∈ iso(∧ω,∧) of Z-graded and Z2-commutative algebras.

10 Opposite Clifford Algebra

For a scalar product φ ∈ lin (M⊗2,k) (9.5), which need not to be necessarilysymmetric, a Clifford k-algebra of multivectors C`(M,φ,−s) ≡ {M∧,∧φ} isan associative η-deformation of an exterior k-algebra C`(M,ω,−s) = {M∧,∧ω}or an associative φ-deformation of an exterior k-algebra C`(M, 0,−s) ≡

A k-algebra C`(M,φ,−s) is a φ-family of associative Z2-graded multiplica-tions {∧φ} (grade ∧φ = 0 ∈ Z2) parametrized by a tensor φ (9.5).

A Clifford associative product ∧φ is a φ-dependent deformation of exteriorproduct

∧φ(M∧k ⊗M∧l) < M∧(k+l) ⊕M∧(k+l−2) ⊕ . . .⊕M∧|k−l|,

∀φ, ψ ∈ aut ∧φ, ∧φ ◦ (ψ ⊗ ψ) = ψ ◦ ∧φ.

A Clifford algebra of multivectors with a pair of products ∧ and ∧η is saidto be the Kahler-Atiyah algebra [Kahler 1960, Atiyah 1970].

Exists a unique clouded φ-dependent Z-gradation preserved by a Cliffordproduct ∧φ.

An η-deformation is not trivial because (Ψ◦s)-commutativity is violated.A deviation from (Ψ ◦ s)-commutativity (9.6) is measured by a symmetricscalar product η (9.5).

Proposition 10.1 (Opposite Clifford algebra). Let φ ∈ lin (M⊗2,k). Then

∧φ = ∧−φT ◦Ψ ◦ s.

proof. By induction on Z grade of multivectors from M∧. 2

An s-opposite Clifford algebra C`opp(M,φ,−s) is a k-space of multivectorsM∧ with an s-opposite associative multiplication ∧−φT ◦Ψ,

C`opp(M,φ,−s) = {M∧,∧−φT ◦Ψ}.

The above s-opposite Clifford algebra is different from (Ψ◦s)-opposite definedby Chevalley [1954], [C`(M, η)]opp = C`(M,−η).

An (M, η)-universal Clifford Z2-graded algebra can be realized as a µφ-quantization of an exterior algebra and is determined by a symmetic scalarproduct η whereas ω parametrize isomorphic Clifford algebras. An ω-quantizationviolate a Z-gradation and this violation is forcing to forget or to ‘quantize’bare Z-gradation of a space M∧. In a present note a realization C`(M,φ,−s)is a φ-deformation of an exterior algebra, i.e. C`(M,φ,−s) is a Clifford alge-bra of multivectors with a distinguished bare Z-gradation of M∧ and with amultiplication ∧φ (Theorem 4.3).

In µφ-quantized or φ-deformed symmetric algebra a role of η and ω isreversed. Fauser [1996] and Fauser & Stumpf in this volume demonstratedthat in a quantum field theory of fermions a symmetric scalar product η canbe identified with an (anti)-commutator and ω with a propagator.

References

Bourbaki Nicolas, Algebre, chap. 9: formes sesquilinearies et formes quadra-tiques”, Paris, Hermann 1959

Chevalley Claude [1954], “The Algebraic Theory of Spinors”, Columbia Uni-versity Press, New York

Crumeyrolle Albert [1990], “Orthogonal and Symplectic Clifford Algebras,Spinor Structures”, Kluwer Academic Publishers, Dordrecht

Durdevic Mico and Zbigniew Oziewicz [1996], Spinors in braided geome-try, in “Generalizations of Complex Analysis and Their Applications inPhysics”, edited by Julian Lawrynowicz, Banach Center Publications,volume 37, Warszawa, pp. 315–325, ISSN 0137-6934

Fauser Bertfried, Clifford - algebraische Formulierung und Regularitat in derQuantenfeldtheorie, Doctoral Thesis, Universitat Tubingen 1996

Fauser Bertfried, Dirac theory from a field theoretic point of view, in “Clif-ford Algebras and Their Applications in Mathematical Physics”, KluwerAcademic Publishers, Dordrecht 1997, hep-th/9606048

Fauser Bertfried and Harald Stumpf[1997] Positronium as an example of algebraic composite calculations,Advances in Applied Clifford Algebras this volume

Graßmann Hermann [1877], Der Ort der Hamilton’schen Quaternionen inder Ausdehnungslehre, Math. Ann. 12 375

Kan D. M. [1958], Adjoint functors, Trans. Amer. Math. Soc. 87 294–329

Kahler Erich, Innerer und ausserer Differentialkalkul, Abh. Dt. Akad. Wiss.Berlin, Kl. fur Mathematik, Physik und Technologie, Jahrg. 1960, Nr. 4

Kahler Erich, Der innere Differentialkalkul, Rendiconti di Matematica e dellesue Applicazioni, Roma, 21 (3-4) (1962) 425–523

Lawson H. Blaine Jr. and Marie-Louise Michelsohn[1989] “Spin Geometry”, Princeton University Press

Mac Lane Saunders, Natural associativity and commutativity, Rice Univer-sity Studies 49 (1963) 28–46

Mac Lane Saunders, Categories for Working Mathematician. GraduateTexts in Mathematics # 5, Springer-Verlag 1971 MR50#7275

Oziewicz Zbigniew, Eugen Paal and Jerzy Rozanski, Derivations in braidedgeometry, Acta Physica Polonica B 26 (7) (1995) 1253–1273

Revoy Philippe[1977] Algebras de Clifford et algebres exterieures, Journal of Algebra46 268–277

Revoy Philippe, Nouvelles algebres de Clifford, C R Acad. Sci. Paris284(1977) 985–988

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