Adv. Studies Theor. Phys., Vol. 4, 2010, no. 8, 383 - 392

Z3-Graded

Farid Makhsoos

Department of Faculty of Sciences Islamic Azad University of Varamin(Pishva) Varamin-Tehran-Iran

Majid Bashour

Department of Mathematics Faculty of Sciences Islamic Azad University of Varamin(Pishva) Varamin-Tehran-Iran [email protected] Abstract

By considering the Z2 gradation structure, the aim of this paper is constructing Z3 gradation structure of multivectors in geometric alge- bra. Mathematics Subject Classification: 16E45, 15A66

Keywords: Graded Algebra, Geometric Algebra

1 Introduction

The foundations of geometric algebra, or what today is more commonly known as Clifford algebra, were put forward already in 1844 by Grassmann. He introduced vectors, scalar products and extensive quantities such as exterior products. His ideas were far ahead of his time and formulated in an abstract and rather philosophical form which was hard to follow for contemporary math- ematicians. Because of this, his work was largely ignored until around 1876, when Clifford took up Grassmann’s ideas and formulated a natural algebra on vectors with combined interior and exterior products. He referred to this as an application of Grassmann’s geometric algebra. Due to unfortunate historic events, such as Clifford’s early death in 1879, his ideas did not reach the wider part of the mathematics community. Hamil- ton had independently invented the quaternion algebra which was a special 384 F. Makhsoos and M. Bashour case of Grassmann’s constructions, a fact Hamilton quickly realized himself. Gibbs reformulated, largely due to a misinterpretation, the quaternion alge- bra to a system for calculating with vectors in three dimensions with scalar and cross products. This system, which today is taught at an elementary aca- demic level, found immediate applications in physics, which at that time circled around Newton’s mechanics and Maxwell’s electrodynamics. Clifford’s algebra only continued to be employed within small mathematical circles, while physi- cists struggled to transfer the three-dimensional concepts in Gibbs’ formulation to special relativity and quantum mechanics. Contributions and independent reinventions of Grassmann’s and Clifford’s constructions were made along the way by Cartan, Lipschitz, Chevalley, Riesz, Atiyah, Bott, Shapiro, and oth- ers. Only in 1966 did Hestenes identify the Dirac algebra, which had been constructed for relativistic quantum mechanics, as the geometric algebra of spacetime. This spawned new interest in geometric algebra, and led, though with a certain reluctance in the scientific community, to applications and re- formulations in a wide range of fields in mathematics and physics. More recent applications include image analysis, computer vision, robotic control and elec- tromagnetic field simulations. Geometric algebra is even finding its way into the computer game industry.[4] In this paper we consider to graded structute of geometric algebra and then formalize it to construct Z3-graded structure. Also we use this structure on two important geometric algebras: Grassmann and Clifford. This paper is organized as follows: second section contains the core struc- ture of graded spaces and consider to Z2 and Z3 gradation. In the next section we give definition of graded algebras. At last we use gradation structures to Grassmann and Clifford algebras.

2 Graded Algebras

In this section we give some preliminary definitions and results about Graded Algebras.

Definition 2.1 Let G be a finite additive and F be the real or complex field. A map σ : G × G −→ F is called the sign or commutation factor of G if it satisfies

• σ(α, β)σ(β,α)=1

• σ(α + β,γ)=σ(α, β)σ(β,γ) for any α, β, γ ∈ G. The pair (G, σ) is called signed group. Z3-Graded Geometric Algebra 385

It is easy to verify that σ(α, α)=±1 for any α ∈ G. An element α of G is called even (resp. odd)ifσ(α, α) = 1(resp. − 1). The even part {α ∈ G : σ(α, α)=1} and the odd part {α ∈ G : σ(α, α)=−1} of G are denoted by G0 and G1 respectively. G0 is a subgroup of G of index at most 2 and we have G = G0 ∪ G1(disjoint union).

Definition 2.2 A mapping φ : G×G −→ F−{0} is called a factor system on G, if for any α, β, γ ∈ G, it satisfies

(1) φ(α, β + γ)φ(β,γ)=φ(α, β)φ(α + β,γ);

(2) φ(0, 0) = 1,

It follows from (1) that

∗ φ(α, 0) = φ(0,α)=1;

∗ φ(α, −α)=φ(−α, α)=φ(α, β)φ(−α, α + β); for any α, β ∈ G.

Proposition 2.3 Let (G, σ) be an even signed group and assume that G is finitely generated. Then there is a factor system φ on G such that σ(α, β)=φ(α, β)/φ(β,α) for α, β ∈ G. Moreover, if |σ(α, β)| =1for all α, β ∈ G, we can choose φ so that |φ(α, β)| =1for all α, β ∈ G.

Definition 2.4 A V is said to be G-graded if we are given a family (Vα)α∈G of subspaces of V such that V is their direct sum, V = Vα. α∈G

An element of V is said to be homogeneous of grade α ∈ G if it is an element of Vα. Let V and W be two G-graded vector spaces. A linear mapping T : V → W is said to be homogeneous of grade α ∈ G if T (Vβ) ⊂ Wα+β for all β ∈ G. Let L(V,W) denote the vector space of all linear mappings of V into W and let Lα(V,W) denote the subspace of those linear mappings of V into W which are homogeneous of grade α. We define Lgr(V,W) to be the sum of these subspaces, obviously this sum is directed:  Lgr(V,W)= Lα(V,W). α∈G

Thus Lgr(V,W)isaG-graded vector space. Note that Lgr(V,W) is equal to L(V,W) if (for example) Vα = {0} and Wα = {0} for all but a finite number of degrees. In the case where V = W and Vα = Wα for all α ∈ G, we shall simplify 386 F. Makhsoos and M. Bashour

the notations and write L(V ) and Lgr(V ) instead of L(V,V ) and Lgr(V,V ), respectively. Let U, V and W be three G-graded vector spaces and let h : U → V and k : V → W be two linear mappings. If h is homogeneous of grade α and k is homogeneous of grade β, then koh is homogeneous of degree α + β.

Definition 2.5 An algebra A is called G-graded algebra if A has direct sum decomposition A = Aα where Aα is a subalgebra of A of grade α for α∈G ∈ A A ⊂A ∈ any α G with additional condition that α β α+β for all α, β G. A G-graded (associative) algebra A = Aα over F is called σ-commutative α∈G algebra if ab = σ(α, β)ba holds for any a ∈Aα,b∈Aβ and α, β ∈ G. It is important to note that in the case char F=2 we must add the condition 2 2 a = σ(α, α)a for any a ∈Aα. Definition 2.6 Graded For two G-graded algebras A and B over F, the G-graded vector space A⊗B = ( (Aβ ⊗Bγ )) is a G-graded algebra if, we define the multiplica- α∈G β+γ=α tion by (a⊗b).(c⊗d)=σ(β,γ)(ac⊗bd) for β,γ ∈ G and a ∈A,b∈Bβ,c∈Aγ and d ∈B. The algebra A⊗B is called the graded tensor product of A and B over F.IfA and B are σ-commutative, so is A⊗B.  Let V = Vα be a G-graded vector space over F. The G-graded vector α∈G space     V ⊗ V = (Vβ ⊗ Vγ) α∈G β+γ=α is a G-graded algebra if, we define the multiplication by (u ⊗ v).(w ⊗ s)=σ(β,γ)(uw ⊗ vs) for β,γ ∈ G and u, s ∈ V, v ∈ Vβ,w ∈ Vγ . For any n ∈ Z, let

⊗ ⊗ ···⊗ Tn(V )=V V V V = V n n times

T0(V )=F, T1(V )=V and Tn(V )={0} for n ≤−1. Definition 2.7 Tensor Algebra For a G-graded vector space, tensor algebra of V over F is defined as ∞ T (V )= Tn(V ) n=0 Z3-Graded Geometric Algebra 387

As is well-known, T (V ) has a natural Z×G-gradation which is fixed by the ⊗···⊗ ∈ ∈ condition that the grade of a tensor v1 vn, with vi Vαi ,αi G for 1 ≤ i ≤ n, is equal to (n, α1,...,αn). Tn(V ) is a subspace of T (V ) consisting of the homogeneous tensors of order n ∈ Z.

2.1 Z2 and Z3 gradation

Now we consider to the most important groups, Z2 and Z3 which are abelian. To construct gradation using these groups, we have to define sign or commu- tation factor on them. Z2 gradation: Let F be complex number field. A mapping σ : Z2 × Z2 → F defined by

αβ σ(α, β)=(−1) , for all α, β ∈ Z2 (1) is a sign on Z2 which satisfies conditions of definition 2.1. In this case a vector space V is called Z2-graded if it can be decomposed to direct sum of two subspace as V = V0 ⊕ V1. Also for a Z2-graded algebra A = A0 ⊕A1, the multiplication is said to be anticommutative if it satisfies

αβ ab =(−1) ba, for all a ∈Aα and b ∈Aβ,α,β∈ Z2

Z3 gradation: Consider the cyclic group Z3. It can be represented in the complex plane as 2πi √ multiplication by primary cubic roots of unity q = e 3 (i = −1)

q3 =1, and q2 + q +1=0 .

A mapping σ : Z3 × Z3 → F defined by

αβ σ(α, β)=q , for all α, β ∈ Z3 (2) is called sign (or comutation factor)ofZ3 which also satisfies the conditions of definition 2.1. Let V be an n dimensional vector space which is Z3-graded over F, this means that we can write V as direct sum of three sets, V = V0 ⊕ V1 ⊕ V2. Also for a Z3-graded algebra A = A0 ⊕A1 ⊕A2, the multiplication is said to be ternary anticommutative if it satisfies

αβ ab = q ba, for all a ∈Aα and b ∈Aβ,α,β∈ Z3

3 Geometric Algebra

This section is devoted to construct Z3-graded geometric algebra according to the previous concepts. 388 F. Makhsoos and M. Bashour

For an n dimensional Z3-graded vector space V over F, the Z3-graded vector space     V ⊗ V = (Vβ ⊗ Vγ )

α∈3 β+γ=α is a Z3-graded algebra if, we define the multiplication by

(u ⊗ v).(w ⊗ s)=σ(β,γ)(uw ⊗ vs) for β,γ ∈ Z3 and u, s ∈ V, v ∈ Vβ,w ∈ Vγ. The algebra V ⊗ V is σ- commutative, which is called Z3-graded tensor product. For every two elements u, v ∈ V , which u = u1 ⊕ u2 ⊕ u3 and v = v1 ⊕ v2 ⊕ v3, the element of V V has the explicit form:

u ⊗ v = {u0 ⊗ v0}⊕{u0 ⊗ v1 ⊕ u1 ⊗ v0}⊕{u0 ⊗ v2 ⊕ u1 ⊗ v1 ⊕ u2 ⊗ v0}

In this manner we can see that for k ∈ Z+, the set k V of k-tensors is a 0 1 Z3-graded σ-commutative algebra. We define V = FF and V = V . The elements of this space will be called k-vectors. Let

∞  k T (V )= V (3) k=0 be the tensor algebra of V over F, the elements of which are finite sums of tensors of arbitrary finite grades on V . As is well-known, T (V ) has a natural Z × Z3-gradation which is fixed by the condition that the grade of a tensor ⊗···⊗ ∈ ∈ Z ≤ ≤ ··· v1 vk , with vi Vαi ,αi 3 for 1 i k, is equal to (n, α1, ,αk).

Consider the two-sided ideal generated by all elements of the form vαvβ − σ(α, β)vβvα where α, β ∈ Z3 and vα ∈ Vα,vβ ∈ Vβ. This means that    I(V ):= Ak⊗ vαvβ−σ(α, β)vβvα ⊗Bk,vα ∈ Vα,vβ ∈ Vβ,Ak,Bk ∈ T (V ) k

We define the geometric algebra over V by quoting out this ideal from T (V ).

Definition 3.1 The geometric algebra GA(V ) over the vector space V is defined by

GA(V ):=T (V )/I(V ). Z3-Graded Geometric Algebra 389

This quotient algebra, is a σ-commutative Z3-graded algebra. When it is clear from the context what vector space we are working with, we will often denote by G. The product in GA, called the geometric or Clifford product, is inherited from the tensor product in T (V ) and we denote it by juxtaposition (or · if absolutely necessary),

GA × GA → GA, (A, B) → AB := [A ⊗ B]=A ⊗ B + I. Note that this product is bilinear and associative. Geometric Algebra is a general case for two important algebras, Grass- mann and clifford algebra. Here we consider there structure as special cases of geometric algebra. Consider to 3. The subspace of T (V ) consisting of the homogeneous tensors of order k ∈ Z will be deoted by Tn(V ); indeed

n  k Tn(V )= V k=0 of course, Tn(V )={0} if n ≤−1. T0(V )=F and T1(V )=V . This subspace is the 2n dimensional vector space of multivectors over V . When equeped with the exterior product ∧, the vector space Tn(V )=Λn(V ) is called Grassmann algebra over V . Note that Λn(V ) is another example of a Z×Z3-gradation, with a Z-graded structure inherited from the usual Z-grading of T (V ).

4 graded Graßmann Algebra

In this section we consider to an important algebra which has wide usage in theoretical physics and Mathematics. Of course physicists and mathemati- cians use this algebra instead of number fields. So it’s elements is also called supernumbers. Here, we consider to structure of this algebra in details. It is important to note that a Grassmann algebra is a geomeric algebra with special multiplication which is called outer product.

Z2 Graßmann algebra

The Graßmann algebra(or exterior algebra)Λn with n generators is the (over C) generated by a set of n anticommuting generators { }n ∈ C ξi i=1 and by 1 with the property

ξiξj = −ξjξi for all i, j, (4) 390 F. Makhsoos and M. Bashour

• The second power (or higher) of any generator vanishes :

2 (ξi) = 0 (5)

• Any product of three or more generators also vanishes :

ξiξjξk =0 i, j, k =0, 1, ..., N (6)

It follows from 4 that any element of Λn is linear combination of ≤ ··· ≤ monomials ξm1 ξm2 ...ξmk with 1 m1

λ = λ(ξ)= λm1,...,mkξm1 ξm2 ...ξmk . (7) k≥0 m1,...,mk

The term corresponding to k = 0 is proportional to the unit. The relation λ = λ(ξ) shows the fact that λ is expressed in the form of a in ξm. The expression in elements of Λn in the above form is not unique in general. This becomes unique if supplementary conditions are imposed on coefficients λm1,...,mk. For instance, we may require that λm1,...,mk = 0 whenever the relation m1

Mn = {(m1,...,mk)| ;1≤ k ≤ n ;1≤ m1 < ···

Therefore with this condition any element λ ∈ Λn can be written uniquely as

λ = λm1,...,mkξm1 ξm2 ...ξmk k≥0 m1,...,mk m1

= λ0 + λm1,...,mkξm1 ξm2 ...ξmk

(m1,...,mk)∈Mn such that λ0 ∈ C is the number for k =0. Z3-Graded Geometric Algebra 391

Z3-graded Grassmann algebra

Z3 is the cyclic group of three elements. It can be represented in the complex 2πi 2 plane as multiplication by the primary cubic root of unity q = e 3 , q and 3 q = 1. The analog of the Z2-graded Grassmann algebra can be introduced as follows:

Consider an associative algebra spanned by N generators ξi; i =0, = 1, ...., N, between which only ternary relations exist. This means that the binary products of any two of such elements are considered as independent entities, i.e. ξiξj are independent of ξjξi where i, j =0, 1..., N). In this case the ternary commutation is given by the following ternary relations:

2 ξiξjξk = qξjξkξi = q ξkξiξj i, j, k =0, 1, ..., N (10)

Two important properties follow automatically:

• The third power (or higher) of any generator vanishes :

3 (ξi) = 0 (11)

• Any product of four or more generators also vanishes :

ξiξjξkξl =0 i, j, k, l =0, 1, ..., N (12)

we can associate grade 0 to the identity element and grade 1 to the gener- ators ξ’s. By defining ξ¯ = ξ2, for every generator ξ, we can see that

• Grade-0 : I, ξξ,¯ ξξξ, ξ¯ξ¯ξ¯

• Grade-1 : ξ, ξ¯ξ¯

• Grade-2 : ξ,¯ ξξ

3+4N +9N 2 +2N 3 The dimension of resulting algebra is: D = 3 392 F. Makhsoos and M. Bashour

5 Graded Clifford Algebra  ∞ k Let V be an n-dimensional real vector space and T (V )= k=0 V be the k tensor algebra over V . Denote the space of antisymmetric k-tensors by V , n k which its elements are called k-vectors. Let Tn(V )= k=0 V denote the 2n-dimensional real vector space of multivectors over V . The clifford product between a vector v ∈ V and a multivector a in Tn(V ) is given by va = v ∧ a + v.a. The resulting algebra is the so called Clifford algebra C (V ). + According to [5], the usual Z2- grading of C (V ) is given by C (V ) ⊕ C − C ⊕C C k C k (V )= 0 1 where 0 = k even V and 1 = k odd V .In this way C (V ) is given by a direct sum of subspaces C i,i=0, 1 which satisfy C C ⊆C i j i+j(mod 2). AlsoZ3 -grading of C (V ) is given by C (V,g)=C 0 ⊕C 1 ⊕C 2 where C k C k C k 0 = 3k V , 1 = 3k+ V and 2 = 3k+2 V .

References

[1] J. S. R. Chrisholm, A. K. Common (eds.). Clifford Algebras and their Ap- plications in Mathematical Physics. D. Reidel Publishing Co., Dordrecht, 1986.

[2] M. EL Baz, Y. Hassouni, F. Madouri, Z3-graded Grassmann Variables, Parafermions and their Cohrent States, arXiv:math-ph/0206017v1

[3] Y. Kobayashi and S. Nagamachi, Analysis on generalized superspace, J. Math. Phys. 21,2247(1986)

[4] Douglas Lundholm, Geometric (Clifford) algebra and its applications, arXiv:math/0605280v1

[5] R. A. Mosna, D. Miralles and J. Vaz Jr, Z2gradings of Clifford algebras and multivector structures, arXiv:math-ph/0212020

Publishers Group, Dordrecht, 1999.

Received: January, 2010