Degenerate Clifford Algebras and Their Reperesentations

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Degenerate Clifford Algebras and Their

Reperesentations

Şenay BULUT1,*

1_{Anadolu University Faculty of Science, Department of Mathematics, Yunuemre}

Campus, Eskişehir.

Abstract

In this study, we give an imbedding theorem for a degenerate Clifford algebra into non-degenerate one. By using the representations of non-non-degenerate Clifford algebra we develop a method for the representations of the degenerate Clifford algebras. We give some explicit constructions for lower dimensions.

Keywords: Degenerate Clifford algebra, representation of Clifford algebras, quadratic

form, nilpotent ideal.

Dejenere Clifford Cebirleri ve Temsilleri

Özet

Bu çalışmada dejenere Clifford cebirlerinden dejenere olmayan Clifford cebirlerine bir gömme teoremi verdik. Dejenere olmayan Clifford Cebirlerinin temsillerini kullanarak dejenere Clifford cebirlerinin temsilleri için bir metod geliştirdik. Düşük boyutlar için bazı açık temsilleri verdik.

Anahtar kelimeler: Dejenere Clifford cebri, Clifford cebirlerinin temsili, kuadratik

form, nilpotent ideal. 1 Introduction

It is well known that a non-degenerate real Clifford algebra is isomorphic to a matrix algebra (n) or direct sum (n)⊕ (n) of matrix algebras, where =ℝ,ℂ, or ℍ (see [1,2, 3]). Their structures and representations are being studied by mathematicians and physicists. On the other hand, a degenerate real Clifford algebra can not be isomorphic

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to full matrix algebras as they contain nilpotent ideal. There are some remarks about the degenerate Clifford algebra and degenerate spin groups in [4]. Exterior algebras and dual numbers are special cases of the degenerate Clifford algebras.

Definition 1The Clifford algebra Cl( QV, ) associated to a vector space V over F with quadratic form Q is defined as the quotient algebra

) ( / ) ( ) , (V Q T V I Q Cl =

where T(V) is tensor algebra T(V)=F⊕V ⊕(V ⊗V)⊕_L and I (Q) is the two-sided ideal in T(V ) generated by the elements

{

v⊗v−Q(v)⋅1

}

.

As an example when Q=0 then the Clifford algebra Cl( QV, ) is just the Grassmann algebra (also known as the exterior algebra) Λ(V).

A non-degenerate quadratic form Q on the real vector space V=ℝn_{is given by}

2 2 1 2 2 2 1 1 ) (x x x x_p x_p x_p _q Q = + +_L+ − ₊ −_L− ₊ (n= p+q).

The Clifford algebra on ℝn_{associated to Q is denoted by} q p

Cl _, . The table of non-degenerate real Clifford algebras is as follows (see [2, 3]):

) 8 )(mod (p−q p+q q p Cl _, 2 , 0 2m _ℝ₍₂m₎ 1 2m+1_ℝ₍₂m₎ _⊕_ℝ₍₂m₎ 7 , 3 2m+1 _ℂ₍₂m₎ 6 , 4 2m+2 _ℍ₍₂m₎ 5 2m+3 _ℍ₍₂m₎ _⊕_ℍ₍₂m₎

The explicit isomorphism Φ_p_,q from the Clifford algebra C lp,q to the related matrix algebra is given in [5] for all p,q .

Definition 2 Let A be a real algebra and W a real vector space, then an algebra homomorphism ρ:A→End(W) is called a representation of the algebra A.

For example, the canonical representation of the matrix group ℝ(n) on ℝn_{is ordinary}

product of a matrix with a vector. The canonical representation of the algebra of complex numbers ℂ on ℝ2_{is given}_{ρ ℂ → ℝ(2)}_: _≅_End(ℝ2₎

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − = + x y y x iy x ) ( ρ

for all x+ iy∈ℂ. One can obtain the canonical representation of the algebra ℂ(n), n by n matrices with complex entries, on the vector space ℝ2n_{as follows: Let}

∈ = z_ij _n_×n

A [ ] ℂ(n) be a matrix of complex numbers z_ij. Using the canonical representation of ℂ we obtain an algebra homomorphism

:

n

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whereby ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 1 2 22 21 1 12 11 nn n n n n n z z z z z z z z z A ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ L M L M M L L

Similarly the canonical representation of the quaternion algebra ℍ on ℝ4_{is given by} : ρ ℍ → ℝ(4) ≅End(ℝ4_), ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − − − − − = + + + x y u v y x v u u v x y v u y x kv ju iy x ) ( ρ

for all x+iy+ ju+kv∈ℍ. Likewise the complex case using the canonical representation ρ ℍ → ℝ(4) we obtain an algebra homomorphism :

: n ρ ℍ(n) → ℝ(4n) ≅End(ℝ4n₎ whereby ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 1 2 22 21 1 12 11 nn n n n n n q q q q q q q q q A ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ L M L M M L L

which is the canonical representation of ℍ(n), the algebra of n× n−matrices with quaternion entries. By using the canonical representations of matrix algebras one can obtain representation of non-degenerate Clifford algebras C lp,q for all p,q. The representations of the C lp,q are being studied by various authors (see [6, 7, 8]). There are various applications of the matrix representations of Clifford algebras ([5, 9, 10]). For the explicit expression of Dirac operator the representation of the Clifford algebra is also critical (see [11]).

2 Degenerate Clifford algebras on ℝn

In this section we investigate Clifford algebras associated to a degenerate quadratic form Q on ℝn_{. From Sylvester theorem it is enough to consider the following}

degenerate quadratic form

2 2 1 2 2 2 1 1 ) (x x x x_p x_p x_p _q Q = + +_L+ − ₊ −_L− ₊ (n= p+q+r,r>0).

The Clifford algebra associated to the degenerate quadratic form Q(x) is denoted by

r q p

Cl _, _, . We can determine the degenerate Clifford algebra Cl_p_,_q_,_r as follows: Let

{

e1,e2,K,ep,ε1,ε2,K,εq,θ1,K,θr

}

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) (x

Q .Then, we have xy=−yx for the basis elements x and y and x≠ y. Also, the following identities hold:

1 1 ) ( 2 ₌ _⋅ ₌ i i Q e e for 1≤i≤ p 1 1 ) ( 2 ₌ _⋅ ₌₋ j j Q ε ε for 1≤ j≤q 0 1 ) ( 2 ₌ _⋅ ₌ k k Q θ θ for 1≤k ≤r

Definition 3 An ideal ℑ of the algebra A is called nilpotent if its power ℑk_is

{ }

₀ _for

some positive integer k. Such least k is called order of ℑ. All elements of the form

∑

+

∑

+

∑

+

∑

I JK LM PRS S R P PRS M L LM K J JK I I b e c d e a , , , , θ ε θ ε θ θ

constitute an ideal of Cl_p_,_q_,_r, where a_I,b_JK,c_LM,d_PRS are real numbers and we use multiple index notation, i.e.,

k i i I θ θ θ _K 1 = for I =

{

i₁,_K,i_k

} {

⊂ 1,_K,r

}

and k i i

i₁ < ₂ <_L< . We denote this ideal by ℑ, some authors call it the Jacobson radical [4]. The set

{

θ_I,e_Jθ_K,ε_Lθ_M,e_Pε_Rθ_S

}

is a basis for the ideal ℑ and also note that the all elements are nilpotent. Then, ℑ is nilpotent ideal (see 13.28 Theorem in [12] on page 479 ).

Nilpotent ideals are important to the characterization of an algebra.

Theorem 4 Let A be an algebra with unit. Then, A is semi-simple if and only if there is no nilpotent ideal different from zero.

r q p

Cl _, _, can not be a matrix algebra for r >0, because ℑ is nilpotent ideal. By the Theorem 4 the Clifford algebra Cl_p_,_q_,_r is not semi-simple algebra. On the other hand,

r q p

Cl _, _, can be embedded into the matrix algebras.

2.1 Embeddings of degenerate Clifford algebras into matrix algebras

Now let us explain a theorem concerning the degenerate Clifford algebras.

Theorem 5 The degenerate Clifford algebra Cl_p_,_q_,_r can be embedded into the non-degenerate Clifford algebra Cl_p_{+ ,}_r_q₊_r.

Proof It is enough to give a one to one algebra homomorphism from Cl_p_,_q_,_r to

r q r p

Cl _{+ ,} ₊ . It is possible to extend it to whole algebra when the homomorphism is defined on the basis elements. This homomorphism is defined on the basis elements in the following way:

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r q p r q p, , :Cl , , Ψ → Clp+ ,rq+r 1 e _a e ₁ M M p e _a e_p 1 ε _a ε₁ M M q ε _a ε _q 1 θ _a ep+1+ε q+1 M M r θ _a ep+r +εq+r

Moreover, there are following equalities:

2 2 , , , , 2 , ,qr( i ) pqr(1) 1 [ pqr( i)] i p e =Ψ = = Ψ e =e Ψ 2 2 , , , , 2 , ,qr( i ) pqr( 1) 1 [ pqr( i)] i p ε =Ψ − =− = Ψ ε =ε Ψ ) ( 2 , ,qr i p θ Ψ = Ψ_p_,_q_,_r(0)=0 2 , , ( )] [Ψ_p_q_r θ_i = _[ _]2 i q i p e ₊ +ε ₊ 2 2 i q i p i q i q i p i p e e e ₊ + ₊ ₊ + ₊ ₊ + ₊ = ε ε ε = 1+e_p₊_iε_q₊_i −e_p₊_iε_q₊_i−1=0

From this theorem, degenerate Clifford algebra Cl_p_,_q_,_r can be embedded into the matrix algebras such as ℝ(m), ℂ(m), ℍ(m), ℝ(m) ⊕ℝ(m), ℍ(m) ⊕ℍ(m).

Remark 6 There is a general embedding theorem of an algebra into an endomorphism algebra (see [13] Theorem 2.11 on page 49). In this case, the dimension of the

representation space became larger and it is not useful for explicit constructions. Let us give some examples in lower dimensions:

Example 7 The full degenerate Clifford algebra Cl₀_,₀_,₂ which is equivalent to the Grass- mann algebra Λ( ℝ2_{) can be embedded into the non-degenerate Clifford algebra}

2 , 2

Cl

in the following way:

Λ = 2 , 0 , 0 : Cl f ( ℝ2_{) →} 2 , 2 Cl 1 θ a e1+ε1 2 θ a e2 +ε2

Since Cl₂_,₂ is isomorphic to the matrix algebra ℝ (4), one can also give an imbedding from Cl₀_,₀_,₂ into the matrix algebra ℝ (4) as follows:

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2 , 0 , 0 Cl → Cl₂_,₂ ≅ ℝ (4) 1 θ _a ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = + 0 2 0 0 0 0 0 0 0 0 0 2 0 0 0 0 1 1 ε e 2 θ _a ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − = + 0 0 2 0 0 0 0 2 0 0 0 0 0 0 0 0 2 2 ε e

Example 8 Degenerate Clifford algebra Cl₁_,₀_,₁ can be embedded into the non-degenerate Clifford algebra Cl₂_,₁. We know that Cl₂_,₁ is isomorphic to the matrix algebra ℝ(2) ⊕ℝ(2),. The embedding f can be given on the basis elements as follows:

1 , 0 , 1 : Cl f → Cl₂_,₁ 1 e a e 1 1 θ a e2 +ε1

The map f defined above can be seen that it is an one to one algebra homomorphism. If we use the composition of the homomorphism f and the isomorphism Ψ₂_,₁, then we get an one to one algebra homomorphism as follows:

1 , 0 , 1 Cl → Cl₂_,₁ ≅ ℝ(2) ⊕ℝ(2) 1 e _a _⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = 0 1 1 0 , 0 1 1 0 1 e 1 θ _a _⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡− − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − = + 1 1 1 1 , 1 1 1 1 1 2 ε e where _⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − = 1 1 1 1 , 1 1 1 1 1 1θ e .

Any x∈Cl₁_,₀_,₁ can be written as x=x₀⋅1+x₁e₁+x₂θ₁+x₃e₁θ₁. By writing the values of

1 1 1 1, ,

,

1e θ eθ we get the following identity:

x = x0 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ 1 0 0 1 , 1 0 0 1 +x₁ _⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ 0 1 1 0 , 0 1 1 0 +x₂ _⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡− − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − 1 1 1 1 , 1 1 1 1 +x₃ _⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − 1 1 1 1 , 1 1 1 1

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= _⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − + − + + − + − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − + + − − + + 3 2 0 3 2 1 3 2 1 3 2 0 3 2 0 3 2 1 3 2 1 3 2 0 _, x x x x x x x x x x x x x x x x x x x x x x x x

Hence, the degenerate Clifford algebra Cl₁_,₀_,₁ is isomorphic to the following subalgebra of the matrix algebra ℝ(2) ⊕ℝ(2).

⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ = ∈ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − + − + + − + − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − + + − − + + 3 , 2 , 1 , 0 , , 3 2 0 3 2 1 3 2 1 3 2 0 3 2 0 3 2 1 3 2 1 3 2 0 _x _R _i x x x x x x x x x x x x x x x x x x x x x x x x i Example 9 Let 2 1 ) (x x

Q =− be a degenerate quadratic form on V = ℝ2_and 1 , 1 , 0 Cl its Clifford algebra. The degenerate Clifford algebra Cl₀_,₁_,₁ is embedded into

non-degenerate Clifford algebra Cl₁_,₂. Moreover, we know that the Clifford algebra Cl₁_,₂

isomorphic to the matrix algebra ℂ(2). The embedding map from Cl₀_,₁_,₁ to Cl₁_,₂ is defined on basis elements as follows:

1 , 1 , 0 Cl → Cl₁_,₂ ≅ ℂ(2) 1 ε _a ε₁ 1 θ _a e₁+ε₂

It can be seen that this map is a one to one algebra homomorphism. By composing this homomorphism and the isomorphism Ψ₁_,₂ we get the following homomorphism:

1 , 1 , 0 Cl → Cl₁_,₂ ≅ ℂ(2) 1 ε _a _⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − = 0 1 1 0 1 ε 1 θ _a _⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − = + i i e 1 1 2 1 ε where _⎥ ⎦ ⎤ ⎢ ⎣ ⎡− = 1 1 1 1 _i i θ

ε . Any x∈Cl₀_,₁_,₁ can be written as

1 1 3 1 2 1 1 0 1 xε xθ xεθ x x= ⋅ + + + .

Then, by writing the values of 1,ε₁,θ₁,ε₁θ₁we have the following identity:

x = x0 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ 1 0 0 1 +x₁ _⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − 0 1 1 0 +x₂ _⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − i i 1 1 +x₃ _⎥ ⎦ ⎤ ⎢ ⎣ ⎡− 1 1 i i = _⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − + + + + + − + − i x x x i x x x i x x x i x x x 2 3 0 3 2 1 3 2 1 2 3 0 Hence, ⊂ ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ = ∈ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − + + + + + − + − ≅ , 0,1,2,3 2 3 0 3 2 1 3 2 1 2 3 0 1 , 1 , 0 x R i i x x x i x x x i x x x i x x x Cl _i ℂ(2).

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Example 10 Let 2 2 2 1 ) (x x x

Q = − be a degenerate quadratic form on V = ℝ3_and 1 , 1 , 1 Cl

its Clifford algebra. In similar way it is possible to be embedded into a suitable matrix algebra. As above examples the degenerate Clifford algebra Cl₁_,₁_,₁ is embedded into non-degenerate Clifford algebra Cl₂_,₂. The embedding is defined on the basis elements as follows: 1 , 1 , 1 Cl → Cl₂_,₂ ≅ ℝ (4) 1 e _a e ₁ 1 ε _a ε₁ 1 θ _a e₂+ε₂

By composing this homomorphism and the isomorphism Ψ₂_,₂ we get a homomorphism from Cl₁_,₁_,₁ to the matrix algebra ℝ (4) as follows:

1 , 1 , 1 Cl → Cl₂_,₂ ≅ ℝ (4) 1 e _a ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 1 e 1 ε _a ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − = 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 1 ε 1 θ _a ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − = + 0 0 2 0 0 0 0 2 0 0 0 0 0 0 0 0 2 2 ε e We have ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 1 1ε e , ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − = 0 0 0 2 0 0 2 0 0 0 0 0 0 0 0 0 1 1θ e , ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = 0 0 0 2 0 0 2 0 0 0 0 0 0 0 0 0 1 1θ ε , ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = 0 0 2 0 0 0 0 2 0 0 0 0 0 0 0 0 1 1 1εθ e .

Then, any element x in Cl₁_,₁_,₁ can be written in the following way:

1 1 1 7 1 1 6 1 1 5 1 1 4 1 3 1 2 1 1 0 1 xe xε xθ x eε xεθ xeθ xeεθ x x= ⋅ + + + + + + +

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We know the values of e₁,ε₁,θ₁,e₁ε₁,ε₁θ₁,e₁θ₁,e₁ε₁θ₁. If these values are used, then we get x = x0 ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 +x₁ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 +x₂ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 = +x3 ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ −2 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 + ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 4 x + ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ 0 0 0 2 0 0 2 0 0 0 0 0 0 0 0 0 5 x + ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − + 0 0 0 2 0 0 2 0 0 0 0 0 0 0 0 0 6 x ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ 0 0 2 0 0 0 0 2 0 0 0 0 0 0 0 0 7 x . Then, ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − + + − + − + − + − + − + = 4 0 2 1 7 3 6 5 2 1 4 0 6 5 7 3 4 0 2 1 2 1 4 0 2 2 2 2 2 2 2 2 0 0 0 0 x x x x x x x x x x x x x x x x x x x x x x x x x .

Hence, the degenerate Clifford algebra Cl₁_,₁_,₁ is isomorphic to the following subalgebra of the matrix algebra ℝ(4).

⎪ ⎪ ⎭ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ = ∈ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − + + − + − + − + − + − + 7 , , 1 , 0 , 2 2 2 2 2 2 2 2 0 0 0 0 4 0 2 1 7 3 6 5 2 1 4 0 6 5 7 3 4 0 2 1 2 1 4 0 K i R x x x x x x x x x x x x x x x x x x x x x x x x x i

If we continue in similar way, we have the following imbeddings: ≅ → ₁_,₄ 1 , 3 , 0 Cl Cl ℍ(2)⊕ ℍ(2) Cl₁_,₁_,₁ →Cl₂_,₂ ≅ ℝ(4) ≅ → ₁_,₅ 1 , 4 , 0 Cl Cl ℍ(4) Cl₁_,₂_,₁ →Cl₂_,₃ ≅ ℂ(4) ≅ → ₂_,₃ 2 , 1 , 0 Cl Cl ℂ(4) Cl₁_,₃_,₁→Cl₂_,₄ ≅ ℍ(4) ≅ → ₂_,₄ 2 , 2 , 0 Cl Cl ℍ(4) Cl₁_,₄_,₁ →Cl₂_,₅ ≅ ℍ(4) ⊕ ℍ(4) ≅ → ₂_,₅ 2 , 3 , 0 Cl Cl ℍ(4)⊕ ℍ(4) Cl₁_,₀_,₂ →Cl₃_,₂ ≅ℝ(4) ⊕ ℝ(4)

We can easily determine the degenerate Clifford algebras Cl_p_,_q_,_r for the higher values of p ,,q r in the similar way.

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3 Spinor Representations of Degenerate Clifford Algebras

A representation of degenerate Clifford algebra can be given as follows: Let

) ( :Cl_p₊_r_,_q₊_r → End V

ρ

be the spinor representation of Cl_p_{+ ,}_r_q₊_r which is widely known, and

r q r p r q p r q p Cl →Cl + +

Ψ _, _, : _, _, _, be the imbedding defined above, then the composition

) ( : _, _, , ,qr Clpqr End V p → Ψ o ρ

is an algebra homomorphism, so it is a representation of the degenerate Clifford algebra

r q p

Cl _, _, and we call it spinor representation of Cl_p_,_q_,_r. Now we give some example in lower dimension. Example 11 Example 12 Example 13 2 , 0 , 0 Cl → ℝ(4) 1 θ _a ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ 0 2 0 0 0 0 0 0 0 0 0 2 0 0 0 0 2 θ _a ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ −2 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 1 , 0 , 1 Cl → ℝ(2) 1 e a ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ 0 1 1 0 1 θ a ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − 1 1 1 1

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Example 14 1 , 1 , 1 Cl → ℝ(4) 1 e a ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 1 ε a ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 1 θ a ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ −2 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 References

[1] T. Friedrich, Dirac Operators in Riemannian Geometry, AMS, (2000). [2] R. Harvey, Spinors and Calibrations, Academic Press, (1990).

[3] H.B. Lawson and M.L. Michelsohn, Spin Geometry, Princeton Univ., (1989). [4] R. Ablamowicz, Structure of spin groups associated with degenerate Clifford

algebras, Journal of Mathematical Physics 27, (1986).

[5] N. Değirmenci and Ş. Karapazar, Explicit Isomorphisms of Real Clifford Algebras, International Journal of Mathematics and Mathematical Sciences, (2006). [6] M. F. Atiyah, R. Bott and V.K. Shapiro, Clifford Modules, Topology, 3, 1, 3-38,

(1964).

[7] A.Dimaskis, A new representation of Clifford Algebras, Journal of Physics A: Mathematical and Theoretical., 22, 3171-3193 (1989).

[8] A.Trautman, Clifford Algebras and Their Representations, Encyclopedia of Mathematical Physics, (2005). 1 , 1 , 0 Cl → ℝ(4) 1 ε a ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 1 θ a ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0

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[9] N. Değirmenci and Ş. Koçak, Generalized Self-Duality of 2-Forms, Advance in Applied Clifford Algebras, 13,107-113, (2003).

[10] N. Değirmenci and N. Özdemir, The Construction of Maximum Independent Set of Matrices via Clifford Algebras, Turkish Journal of Mathematics, 31, 193-205, (2007).

[11] H. Baum and I. Kath, Parallel spinors and holonomy groups on

pseudo-Riemannian spin manifolds, Annals of Global Analysis and Geometry, 17, 1-17, (1999).

[12] C. Faith, Algebra I: Rings, Modules, and Categories, Springer-Verlag, (1973). [13] D. R. Farenick, Algebras of Linear Transformations, Springer-Verlag, (2001).