A Theoretic Approach of Classical Operations in Clifford Algebras

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A Theoretic Approach of Classical Operations in Clifford Algebras

Sylvain Rousseau, David Helbert

To cite this version:

Sylvain Rousseau, David Helbert. A Theoretic Approach of Classical Operations in Clifford Algebras.

[Research Report] Université de Poitiers (France). 2012. �hal-02082136�

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A Theoretic Approach of Classical Operations in Clifford Algebra

Sylvain Rousseau and David Helbert Univ. Poitiers, CNRS, Xlim

2012

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Chapter 1 Introduction

Clifford algebra concepts have a limited theoretical interest. Their properties are presented in the literature of applied mathematics by considering their existence for granted.

We find in the mathematical literature a rigorous definition of Clifford algebras but that omits to define certain concepts that have a limited theoretical interest. Similarly, works of applied mathematics that present the Clifford algebras make a list of their properties by considering their existence for granted. This report presents a definition of Clifford algebra with other interpretations and demonstrations based on [1], [2], [6] and [3] with practical notions used in [5] and [4].

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Chapter 2 Inner Product

2.1 Introduction

We will define the inner product due to Hestenes [5] from an asymmetric product called contraction which is useful in computer science [4].

We first present a definition of Clifford algebra and the universal property and secondly the Clifford algebra dimension. We then present† and α operators to finally propose a definition of the geometric algebra.

2.2 Definition and universal property

In this paper,V is a vector space on a fieldK=Ror C,Qis a quadratic form onV and B its polar form:

B(x, y) = Q(x+y)−Q(x)−Q(y)

2 , ∀x, y∈V.

The tensor algebra of V noted ⊗(V) is the unitary graded algebra:

K⊕V ⊕(V ⊗V)⊕(V ⊗V ⊗V)⊕ · · ·

The product is induced by the tensor product and the identity element is noted 1⊗. The linear subspace of elements of degreeh is noted ⊗^h(V).

Definition 1. A Clifford algebra on V with respect to Q, noted C(V, Q) is the quotient of tensor algebra⊗(V) by the two-sided ideal I_Q spanned by elements of the form:

x⊗x−Q(x), x∈V ⊂ ⊗(V).

The canonical projection of ⊗(V) onto the Clifford algebra C(V, Q) is callediQ. We set x = i_Q(x) for x ∈ ⊗(V). The quotient C(V, Q) is an associative unitary algebra which contains the identity element¯1⊗. AsV generates⊗(V),i_Q(V)generatesC(V, Q).

Using the definition, we have the fundamental relationship:

x² =Q(x) ∀x∈V, (2.1)

By developing the expression: (x+y)² =Q(x+y) withx, y∈V:

x y+y x= 2B(x, y). (2.2)

This relationship allows us to characterize orthogonal vectors.

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Property 1 (Universal property). Let A be a K-algebra and f :V →A a morphism of K-vector space such that:

∀x∈V, f(x)²=Q(x)1.

Then there exists a unique g such that the following diagram is commutive:

V

iQ

f //A

C(V, Q)

g

;; (2.3)

2.3 Dimension of C(V, Q)

Lemma 1. If u1, . . . , un span the vector space V, then

1⊗, ui1ui2· · ·uir, where 16i1< i2 <· · ·< ir6n and 16r6n, span the vector space C(V, Q).

Proof. Letu1, . . . , un be vectors generating the vector space V. All elements ofC(V, Q) are also linear combination of monomial in the u_i, 1 6 i 6 n, because i_Q(V) spans C(V, Q). On the other hand,

u_iu_j+u_ju_i= 2B(u_i, u_j), (2.4)

(u_i)²=Q(u_i). (2.5)

If we reorder these monomials using the relationship (2.4) then lowering the degree with (2.5), we can assume that all element ofC(V, Q)is a linear combination of elements of the form:

1⊗, ui1ui2· · ·uir, where16i1 < i2<· · ·< ir 6nand 16r6n.

Definition 2. Let F and G be two graded algebras. The tensor product F ×G can be endowed with an associative algebra structure by defining:

(a₁⊗a₂)(b₁⊗b₂) = (−1)^p²^q¹a₁b₁⊗a₂b₂,

where a₂ is of degree p₂ andb₁ is of degree q₁ and then extend to F ×G by linearity.

Lemma 1. Let Qand Q⁰ be two quadratic forms on vector spaces of finite dimension V andV⁰ respectively. Then there exists a surjective morphism of algebra ofC(V⊕V⁰, Q⊕ Q⁰) onto C(V, Q)⊗C(V⁰, Q⁰).

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Proof. Letu:V ⊕V⁰ −→C(V, Q)⊗C(V⁰, Q⁰) be defined by u(x, x⁰) =iQ(x)⊗1 + 1⊗i_Q⁰(x⁰), forx∈V and x⁰ ∈V⁰. We have:

(u(x, x⁰))²= (i_Q(x)⊗1 + 1⊗i_Q⁰(x⁰))²

= (i_Q(x))²⊗1 + 1⊗(i_Q⁰(x))²+i_Q(x)⊗i_Q⁰(x⁰)−i_Q⁰(x)⊗i_Q(x)

=Q(x) +Q⁰(x⁰) = (Q⊕Q⁰)(x, x⁰).

Using the universal property, there exists a uniqueψsuch that:

V ⊕V⁰

i_Q⊕Q0

u //C(V, Q)⊗C(V⁰, Q⁰)

C(V ⊕V⁰, Q⊕Q⁰)

ψ

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is commutative. On the other hand, asi_Q(V)spansC(V, Q)andi_Q⁰(V⁰)spansC(V⁰, Q⁰), C(V, Q)⊗C(V⁰, Q⁰)is spanned by elements of form1⊗i_Q⁰(x⁰) =u(0, x⁰)andiQ(x)⊗1 = u(x,0)for x∈V andx⁰∈V⁰. The map ψis then surjective.

Theorem 1. We have the following structure theorem:

i) dimC(V, Q) = 2ⁿ and for all basis (u₁, . . . , u_n) of V, the following elements

1⊗, ui1ui2· · ·uir, where 16i1 < i2 <· · ·< ir 6n and16r 6n, (2.6) form a basis for C(V, Q).

ii) The morphism iQ:V −→C(V, Q) is injective.

2.4 † and α operators

2.4.1 α operator

Property 2. There exists a unique endomorphism of algebra ofC(V, Q) denotedα such that:

α(v) =−v, ∀v∈V

This endomorphism is an involution called the main involution of the Clifford algebra C(V, Q).

We note the set of even elements C⁺(V, Q) ={x∈C(V, Q), α(x) =x} and the set of odd elementsC⁻(V, Q) ={x ∈ C(V, Q), α(x) =−x}. We have C(V, Q) =C⁺(V, Q)⊕ C⁻(V, Q) and C⁺(V, Q) is a subalgebra of C(V, Q).

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2.4.2 † operator

Property 3. There exists a unique anti-automorphism of C(V, Q), noted† such that:

v^†=v ∀v∈V This anti-automorphism is an involution.

2.5 Geometric Algebra

2.5.1 Canonical Isomorphism between ∧(V) and C(V, Q)

Vector subspace representation using the exterior product in the Grassmann algebra allows for example to express the belonging relation to a point in a space or the character of a linearly dependent vector subset by simple algebraic relations. We would like to equip a Clifford algebra with such a product. It is sufficient to exhibit an isomorphism between the two spaces and transport the product of an algebra on the other. The question is which isomorphism to choose so that both products perform well with respect to the other. A solution would be to choose aQ-orthogonal base(e1, . . . , en) ofV and to send the element e₁∧ · · · ∧e_n on e₁· · ·e_n. We can then prove that an isomorphism between vector spaces is defined. We consider here an independent definition of any choice of a base inspired by [1] and [3].

Definition 3. Letx∈V, we define a creation operator for alla∈ ⊗(V)byex(a) =x⊗a.

Definition 4. Let h and j be integers >0 such that j6h, (e1, . . . , en) a base of V and ϕ an element ofV^∗. We define the contraction operatorCϕ^j of ⊗^h(V) in ⊗^h−1(V) by:

C_ϕ^j(ei1⊗ · · · ⊗ei_h) =ϕ(eij)(ei1 ⊗ · · · ⊗ecij ⊗ · · · ⊗ei_h).

Property 4. Let ϕ ∈ E^∗, there exists a unique vector space morphism iϕ : ⊗(V) −→

⊗(V) such that:

i_ϕ(1) = 0, (2.7)

∀x∈V, i_ϕ◦e_x+e_x◦i_ϕ =ϕ(x) Id. (2.8) Proof. i_ϕ is already defined on K. It is defined on ⊗ⁱ(V) by induction.

Property 5. We have iϕ=Ph

j=1(−1)^j+1Cϕ^j on ⊗^h(V).

Proof. The proof can be found by induction on h. By linearity, it is sufficient to prove the equality for a term x⊗awithx∈V and a∈ ⊗^h(V).

Property 6. Let ϕ, ψ∈V^∗, we have the following equalities:

(i_ϕ)² = 0 and i_ϕ◦i_ψ+i_ψ◦i_ϕ = 0.

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Proof. We first remark that(i_ϕ)² ande_x commute using (2.8). We then prove by induction onhthat (iϕ)² = 0 on⊗^h(V). On the other hand,

i_ϕ◦i_ψ +i_ψ◦i_ϕ = (i_ϕ+ψ)²−(i_ϕ)²−(i_ψ)²

= 0

Property 7. Let a∈ ⊗^h(V) andb∈ ⊗(V), we have the following equality:

iϕ(a⊗b) =iϕ(a)⊗b+ (−1)^ha⊗iϕ(b). (2.9) Proof. By linearity, we can suppose thatb∈ ⊗^k(V). We have successively:

iϕ(a⊗b) =

h+k

X

j=1

(−1)^j−1C_ϕ^j(a⊗b)

=





h

X

j=1

(−1)^j−1C_ϕ^j(a)



⊗b+a⊗





h+k

X

j=h+1

(−1)^j−1C_ϕ^j−h(b)





=i_ϕ(a)⊗b+ (−1)^ha⊗i_ϕ(b).

Property 8. The mapiϕ descends to ∧(V) by:

iϕ(x1∧x2∧ · · · ∧x_h) =

h

X

j=1

(−1)^j+1ϕ(xj)(x1∧ · · · ∧xbj∧ · · · ∧x_h).

Proof. It is sufficient to prove that idealI is stable byi_ϕ. By linearity, it is sufficient to prove that for alla∈ ⊗^h(V), x∈V and b∈ ⊗(V), we have: iϕ(a⊗x⊗x⊗b) ∈I. By applying (2.9) twice and given thatiϕ(x⊗x) = 0, we find:

iϕ(a⊗x⊗x⊗b) =iϕ(a)⊗x⊗x⊗b+ (−1)^ha⊗x⊗x⊗iϕ(b)∈I.

Lemma 2. Let F be a bilinear form on V, there exists a unique linear endomorphism λF on ⊗(V) such that:

λF(1) = 1, (2.10)

λF ◦ex= (ex+i^F_x)◦λF, (2.11) wherei^F_x =i_x^∗ and x^∗(y) =F(x, y).

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Proof. λ_F is defined on K by (2.10). Moreover, for all x ∈ V and a ∈ ⊗^h(V), we necessarily have:

λ_F(x⊗a) = (e_x+i^F_x(x))◦λ_F(a) =x⊗λ_F(a) +i^F_x(λ_F(a)). (2.12) This defines λF on ⊗^h+1(V).

Lemma 3. For allϕ∈E^∗, λF andiϕ commute.

Proof. By linearity, it is sufficient to prove that λ_F and i_ϕ commute on ⊗^h(V) ∀h >0 and by induction onh. It is obvious forh= 0 becauseλ_F(1) = 1. Leth>0,a∈ ⊗^h(V) and x∈V, we have:

λ_F(i_ϕ(x⊗a)) =λ_F (i_ϕ(x)⊗a−x⊗i_ϕ(a))

=ϕ(x)λ_F(a)−(e_x+i^F_x)(i_ϕ(λ_F(a)))

=i_ϕ(λ_F(x⊗a)).

Lemma 4. Let F andF⁰ be two bilinear form onE, we have the relationshipλF◦λF⁰ = λ_F_+F⁰. In particular, λ_F is an isomorphism of⊗(V).

Proof. By linearity, it is sufficient to prove thatλ_F◦λ_F⁰ =λ_F_+F⁰ on⊗^h(V)for allh>0.

We prove this by induction onh.

The case h = 0 is obvious because λF(1) = λ_F⁰(1) = 1. Let h > 0, a ∈ ⊗^h(V) and x∈V, we have:

λF(λ_F⁰(x⊗a)) =λF(λ_F⁰(ex(a)))

= (ex+i^F_x)(λF(λ_F⁰(a))) +i^F_x⁰(λF(λ_F⁰(a)))

=λF+F⁰(x⊗a), This proves the relationship on ⊗^h+1(V).

In particularλ_F◦λ_−F =λ0 and it is not difficult to show thatλ0 = Id. Henceλ_F◦λ_−F = λ−F ◦λ_F = Id and λ_F is an isomorphism of ⊗(V).

Property 9. LetI_Qbe the two-sided ideal of⊗(V)spanned byx⊗x−Q(x)forx∈V. We setI =I0. TheλB operator sends the idealIQ in the idealI and defines an isomorphism of vector space between C(V, Q) and ∧(V).

Proof. It is sufficient to prove that λ_B(I_Q) ⊂I. Indeed the induced morphism will be surjective andC(V, Q)and∧(V)have the same dimension. This is done by induction.

We obtain by passing to the quotient in (2.11) the relationshipλB(xu) =x∧λB(u) + ix^∗(λ_B(u)). Then by identifying an element ofC(V, Q) and its image byλ_B, we find:

xu=x∧u+i_x^∗(u). (2.13)

An exterior product on C(V, Q) compatible with the geometric product in the sense of (2.13) is then defined.

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Remark 1. By applying the † operator to the relationship (2.13) and by assuming that u is ak-vector, we can establish the similar equality:

ux=u∧x−(−1)^kix^∗(u). (2.14) 2.5.2 Definition of an inner product.

We consider the extension ofB on the whole Clifford algebraC(V, Q). We first observe that for all x, y ∈ V, ix^∗(y) = B(x, y). The map ix^∗ extends the scalar product to V ×C(V, Q). It exists many possible extensions: the left contraction is an asymmetric product which is useful in computer science[4]. We will define the inner product due to Hestenes [5] from the contraction rather than from property (19).

Definition 5. Let k >0,x=v₁∧ · · · ∧v_k be a decomposable k-vector and y∈C(V, Q).

We fix lx = iv^∗₁ ◦ · · · ◦iv^∗_k. We call left contraction of x and y and we note it xcy the element l_x(y). We then extend this contraction by linearity by fixing λcy=λy.

Definition 6. Let x be a r-vector and y a s-vector. We define the inner product of x andy and we note itx·y the element:

x·y=







0 if r= 0 or s= 0

xcy if r6s

(−1)^s(r−1)ycx if r > s.

We then extend this product by linearity.

Remark 2. In fact, if r = s we have xcy = ycx. The last case in previous definition need not be strict.

Property 10. Let x be a r-vector, y a s-vector, z a t-vector. We have:

i) x·y= (−1)^r(s−1)y·x forr 6s.

ii) x·(y·z) = (x∧y)·z forr+s6t and r, s >0.

iii) (x·y)·z=x·(y∧z) fors+t6r and s, t >0.

iv) x·(y·z) = (x·y)·z for r+t6s.

Proof. i). It comes directly from the definition.

ii). By inner product bilinearity, we can suppose that x and y are decomposable, i-e x = x₁∧ · · · ∧x_r and y =y₁ ∧ · · · ∧y_s. By using the notations of the definition 5, as r6t ands6t, we have:

x·(y·z) =xc(ycz) becauser, s >0

=l_x(l_y(z) =lx∧y(z) = (x∧y)·z because r+s6t.

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iii). It comes from (i) by exchanging the roles of x andz in (ii).

iv). For r= 0 or t= 0, the relationship is obvious. We thus suppose r, t >0. By using (i) and (ii), we successively have:

x·(y·z) = (−1)^s(t−1)x·(z·y) = (−1)^t(s−1)(x∧z)·y

= (−1)^t(s−1)(−1)^rt(z∧x)·y= (−1)^t(s−1)(−1)^rtz·(x·y)

= (−1)^t(s−1)(−1)^rt(−1)^t(s−r−1)(x·y)·z= (x·y)·z.

2.5.3 Geometric Algebra

The Clifford algebra is now equipped with an exterior product. We can use concepts introduced in Grassmann algebras. We say that an element of C(V, Q) isdecomposable if it can be written as the exterior product of any number of element of V. We call the set ofk-vecteurs the set C^k(V, Q) = Vect{x₁∧ · · · ∧xk, x1, . . . , xk∈V}. By convention C⁰(V, Q) = K. The C^k(V, Q) spaces are in direct sum C(V, Q) =

n

M

i=0

Cⁱ(V, Q) and we have the equality dimC^r(V, Q) = ⁿ_k

.

We noteh·i_i the projection ofC(V, Q) onto Cⁱ(V, Q), so that:

x=

n

X

i=1

hxi_i. (2.15)

We call degree of element xof C(V, Q)the integer max_06i6n{0} ∪ {i, hxi_i 6= 0}. A non- nullk-vector is this of degree k.

We now establish two useful lemmas relying vector product and geometric product.

Lemma 2. If x1, . . . , xr is an orthogonal family of element of V, then x₁· · ·x_r =x₁∧ · · · ∧x_r.

Proof. By induction onr. The caser= 1is obvious. We supposex₁· · ·x_r =x₁∧ · · · ∧x_r for all 1 6r < n. Let xr+1 ∈ V such that the family (x1, . . . , xr+1) is orthogonal. We fixu=x₂· · ·x_r+1. According to (2.13) we havex₁· · ·x_r+1 =x₁u=x₁∧u+i_x^∗(u). Yet i_x^∗(u) =Pr+1

j=2(−1)^jB(x₁, x_j)(x₂∧ · · · ∧xb_j∧ · · · ∧x_r+1) = 0, becauseB(x₁, x_j) = 0 for all j= 2, . . . , r+ 1. We thus have the equalityx1· · ·xr =x1∧ · · · ∧xr.

Lemma 3. Let F be a vector subspace of V, (v₁, . . . , v_r) a base of F and (u₁, . . . , u_r) an orthogonal base of F. If A denotes the transfer matrix of the base (u1, . . . , ur) to the base (v1, . . . , vr), we have the equality:

v₁∧ · · · ∧v_r= det(A)u₁· · ·u_r. (2.16)

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Proof. We havev_i =Pr

k=1a_kiu_k,∀16i6r . We can then write:

v1∧ · · · ∧vk=

r

X

i=0

ai1ui

!

∧ · · · ∧

r

X

i=0

airui

!

= X

σ∈Sr

a_σ(1)1· · ·a_σ(r)ru_σ(1)∧ · · · ∧u_σ(r)

= det(A)u₁∧ · · · ∧u_r = det(A)u₁· · ·u_r,

Property 11. Let v₁, . . . v_r be vectors,x a k-vector and y∈C(V, Q). Also:

i) (v₁∧ · · · ∧v_r)^†=v_r∧ · · · ∧v₁ = (−1)^r(r−1)² (v₁∧ · · · ∧v_r).

ii) x^†= (−1)^k(k−1)² x.

iii) y^†=

n

X

i=0

(−1)ⁱ⁽ⁱ⁻¹⁾² hyi_i.

Property 12. Let x be k-vector and y, z ∈C(V, Q). We have the following equalities:

i) α(x) = (−1)^kx.

ii) α(y∧z) =α(y)∧α(z).

iii) α(y) =

n

X

i=0

(−1)ⁱhyi_i. iv) C⁺(V, Q) = M

06i6n i pair

Cⁱ(V, Q).

v) C⁻(V, Q) = M

06i6n i impair

Cⁱ(V, Q).

Lemma 5. Let x ∈ V and ∧_x :C(V, Q) −→ C(V, Q) such that ∧_x(u) = x∧u for all u∈C(V, Q). Then:

h·i_i+1◦ ∧_x=∧_x◦ h·i_i ∀isuch that 06i6n−1. (2.17) Proof. h·i_i and ∧_x are indeed linear operators, it is sufficient to check (2.17) on each spaceC^k(V, Q)where06k6n. Let u∈C^k(V, Q) andisuch as 06i6n−1.

Ifk6=i,h·i_i+1◦ ∧_x(u) =hx∧ui_i+1 = 0and ∧_x◦ h·i_i(u) =∧_x(hui_i) = 0.

Ifk=r,h·i_i+1◦ ∧_x(u) =hx∧ui_i+1 =x∧uand ∧_x◦ h·i_i(u) =∧_x(hui_i) =x∧u.

Property 13. Let x be a r-vector and y a s-vector. If r 6= 0 or s 6= 0, we have x∧y=hxyi_r+s.

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Proof. If r= 0 or s= 0, the equality is obvious. We now suppose r, s >0. We proceed by induction on n = r +s. The case r = s = 1 for n = 2 is obvious. We suppose n >2and r+s=n. By linearity, we can suppose thatxis decomposable, we thus have x=x₁∧x⁰ and then:

x∧y=x1∧x⁰∧y

=x1∧ hx⁰yi_r+s−1, by recurrence hypothesis,

=∧_x₁◦ h·ir+s−1(x⁰y) =h·i_r+s◦ ∧_x₁(x⁰y), according lemma 5,

=hx₁∧(x⁰y)i_r+s=hx₁x⁰y−i_x^∗

1(x⁰y)i_r+s =hx₁x⁰yi_r+s

=h(x₁∧x⁰)y+i_x^∗

1(x⁰)yi_r+s=hxyi_r+s, Hence the relationship at rank n.

Property 14. Let x be a r-vector and y a s-vector. If r 6= 0 and s 6= 0, we have x·y =hxyi_|r−s|.

Proof. By linearity, we can suppose that x and y are decomposable. We use induction on r.

1. Ifr = 1, by using (2.13) we have:

x·y=i_x^∗(y) =i_x^∗(y) +hx∧yi_|s−1|=hi_x^∗(y)i_s−1+hx∧yi_|s−1|

=hi_x^∗(y) +x∧yi_|s−1|=hxyi_|s−1|.

2. We supposer >1, there are two cases:

a) Ifr 6s, we fixx=x1∧ · · · ∧xr with orthogonal vectorsx1, . . . xr . By using (2.13), we have:

x·y= (x1∧ · · · ∧xr)·y= (x1∧ · · · ∧xr−1)·(xr·y)

= (x1· · ·xr−1)·(xr·y) =h(x₁· · ·xr−1)(xr·y)i|r−1−(s−1)|

=h(x₁· · ·xr−1)(x_ry−x_r∧y)i_|r−s|

=hx₁· · ·xryi_|r−s|− h(x₁∧ · · · ∧xr∧y)i_|r−s|=hxyi_|r−s|.

b) If r>s, we fix y=y₁∧ · · · ∧y_s with orthogonal vectors y₁, . . . , y_s. By using (2.14), we have:

x·y =x·(y1∧ · · · ∧ys) = (x·y1)·(y2∧ · · · ∧ys)

=h(x·y1)y2· · ·ysi|r−1−(s−1)|=h(xy₁−x∧y1)y2· · ·ysi_|r−s|

=hxy₁· · ·y_si_|r−s|− hx∧y₁∧ · · · ∧y_si_|r−s|=hxyi_|r−s|,

Hence the relationship at rank r.

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2.6 Conclusion

We have presented a definition of the Clifford Algebra with a theoretical viewpoint.

Defined products will then allow us to write linear transformations of the spaceV. We had to be interested in the case whereK=R.

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Chapter 3 Geometric Transformations

3.1 Introduction

We are interested in the case whereK=R. We assume that the quadratic formQis non- degenerated and has the signature (p, q). For simplicity we also note Rp,q the Clifford algebra C(V, Q) and R^kp,q the k-vectors of this algebra. 0-vectors, 1-vectors, 2-vectors and 3-vectors are also called respectively scalars, vectors, bivectors and trivectors. We will introduce the concept of pseudo-vectors and the operation of duality that allows to algebrize the transition to the orthogonal of a subspace of V. Products defined in the part I will then allow us to write linear transformations of the space V.

In a first time we define the invertibility then the duality. We secondly present the orthogonal projection onto a subspace, the orthogonal symmetry and the rotation.

3.2 Inversibility

An interest of Clifford algebras is that you can take the inverse of a non-isotropic vector.

Indeed, letv be a non-isotropic vector. We have v² =Q(v) thenv_Q(v)^v = _Q(v)^v v= 1and v⁻¹= _Q(v)^v .

You can also notice that everyk-decomposable nonzero vectorubeing a regular vector subspace is invertible. Such a vector can be writtenλv₁· · ·v_kwhere thev_i anticommute, Q(vi) =i,i =±1and λ∈R^∗. We have:

uu^†=u^†u=λ²Q(v1)· · ·Q(vk) =λ², with=1· · ·_k and thus

u⁻¹= λ²u^†.

3.3 Duality

We choose an orthonormal base (e1, . . . , en) ofV. We call pseudo-vector relative to the base (e1, . . . , en) the element:

I =e₁· · ·e_n. We have the following relationships:

I⁻¹ =I^†= (−1)ⁿ⁽ⁿ⁻¹⁾² I,

A Theoretic Approach of Classical Operations in Clifford Algebras

A Theoretic Approach of Classical Operations in Clifford Algebras

A Theoretic Approach of Classical Operations in Clifford Algebra

Contents

Chapter 1 Introduction

Chapter 2

Inner Product

Chapter 3

Geometric Transformations