BAÜ Fen Bil. Enst. Dergisi Cilt 15(1) 48-58 (2013)

Degenerate Clifford Algebras and Their Reperesentations

Şenay BULUT 1,* 1 Anadolu University Faculty of Science, Department of Mathematics, Yunuemre Campus, Eski şehir.

Abstract

In this study, we give an imbedding theorem for a degenerate Clifford algebra into non- degenerate one. By using the representations of 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 çalı şmada dejenere Clifford cebirlerinden dejenere olmayan Clifford cebirlerine bir gömme teoremi verdik. Dejenere olmayan Clifford Cebirlerinin temsillerini kullanarak dejenere Clifford cebirlerinin temsilleri için bir metod geli ştirdik. Dü şük 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

* Şenay BULUT, [email protected], Tel: (222) 335 05 80.

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

Cl (V ,Q) = T (V /) I(Q) 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(V ,Q ) 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= ℝ n is given by

1 2 L 2 2 L 2 Q(x) = x1 + x2 + + x p − x p+1 − − xp+q (n = p + q ). n The Clifford algebra on ℝ associated to Q is denoted by Cl p,q . The table of non- degenerate real Clifford algebras is as follows (see [2, 3]):

p + q ( p − q)(mod )8 Cl p,q 2,0 2m ℝ 2( m ) 1 2m +1 ℝ 2( m ) ⊕ ℝ 2( m ) 7,3 2m +1 ℂ 2( m ) 6,4 2m + 2 ℍ 2( m ) 5 2m + 3 ℍ 2( m ) ⊕ ℍ 2( m )

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 ℝ n is ordinary product of a matrix with a vector. The canonical representation of the algebra of complex numbers ℂ on ℝ 2 is given ρ :ℂ → ℝ(2) ≅ End( ℝ 2 ) x − y ρ(x + iy ) =   y x  for all x + iy ∈ℂℂℂ. One can obtain the canonical representation of the algebra ℂ(n), n by n matrices with complex entries, on the vector space ℝ 2n as follows: Let

A = [zij ]n×n ∈ℂ(n) be a matrix of complex numbers zij . Using the canonical representation of ℂ we obtain an algebra homomorphism 2n ρn : ℂ(n) → ℝ(2n) ≅ End( ℝ )

49 BAÜ Fen Bil. Enst. Dergisi Cilt 15(1) 48-58 (2013) whereby L ρ(z11 ) ρ(z12 ) ρ(z1n )   ρ(z ) ρ(z ) L ρ(z ) ρ (A) =  21 22 2n  n  M M L M    L ρ(zn1) ρ(zn2 ) ρ(znn ) Similarly the canonical representation of the quaternion algebra ℍ on ℝ 4 is given by ρ :ℍ → ℝ(4) ≅ End( ℝ 4 ), x − y − u − v   y x − v u ρ(x + iy + ju + kv ) =   u v x − y   v − u y x  for all x + iy + ju + kv ∈ℍ. Likewise the complex case using the canonical representation ρ :ℍ → ℝ(4) we obtain an algebra homomorphism 4n ρn : ℍ(n) → ℝ(4n) ≅ End( ℝ ) whereby L ρ(q11 ) ρ(q12 ) ρ(q1n )   ρ(q ) ρ(q ) L ρ(q ) ρ (A) =  21 22 2n  n  M M L M    L ρ(qn1) ρ(qn2 ) ρ(qnn ) 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 ℝ n

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

1 2 L 2 2 L 2 Q(x) = x1 + x2 + + x p − x p+1 − − xp+q (n = p + q + r,r > ).0

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

Cl p,q,r . We can determine the degenerate Clifford algebra Cl p,q,r as follows:

Let K K K {e1 , e2 , , e p ,ε1 ,ε 2 , ,ε q ,θ1 , ,θ r } be an orthonormal basis for ℝ n with respect to the degenerate quadratic form

50 BULUT Ş.

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

2 for 1≤ i ≤ p ei = Q(ei ) ⋅1 = 1 2 for 1≤ j ≤ q ε j = Q(ε j ) ⋅1 = −1 2 for 1≤ k ≤ r θk = Q(θk ) ⋅1 = 0

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

∑aIθI + ∑bJK eJθK + ∑cLM ε LθM + ∑dPRS ePε RθS IKJM,,,LSP,R

constitute an ideal of Cl p,q,r , where aI ,bJK ,cLM ,dPRS are real numbers and we use multiple index notation, i.e., θ = θ Kθ for I = {i ,K,i }⊂ { ,1 K,r} and I i1 ik 1 k L i1 < i2 < < ik . We denote this ideal by ℑ, some authors call it the Jacobson radical

[4]. The set {θI ,eJθK ,ε LθM ,ePε RθS } 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.

Cl p,q,r 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,

Cl p,q,r 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

Cl p+r,q+r . 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:

51 BAÜ Fen Bil. Enst. Dergisi Cilt 15(1) 48-58 (2013)

Ψp,q,r :Cl p,q,r → Cl p+r,q+r a e1 e1 M M

ep a ep a ε1 ε1 M M

ε q a ε q a θ1 ep+1 + ε q+1 M M a θr ep+r + ε q+r

Moreover, there are following equalities:

2 2 2 Ψp,q,r (ei ) = Ψp,q,r )1( =1 = [Ψp,q,r (ei )] = ei 2 2 2 Ψp,q,r (ε i ) = Ψp,q,r (− )1 = −1 = [Ψp,q,r (εi )] = ε i

2 = Ψp,q,r (θi ) Ψp,q,r )0( = 0 2 = 2 2 2 [Ψp,q,r (θi )] [ep+i + ε q+i ] = ep+i + ep+iε q+i + ε q+iep+i + ε q+i

= 1+ ep+iε q+i − ep+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 Cl 2,0,0 which is equivalent to the Grass- 2 mann algebra Λ ( ℝ ) can be embedded into the non-degenerate Clifford algebra Cl 2,2 in the following way:

2 → f : Cl 2,0,0 = Λ ( ℝ ) Cl 2,2 a θ1 e1 + ε1 a θ2 e2 + ε 2

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

52 BULUT Ş.

Cl 2,0,0 → Cl 2,2 ≅ ℝ (4) 0 0 0 0   2 0 0 0 θ a e + ε =   1 1 1 0 0 0 0   0 0 2 0 0 0 0 0   0 0 0 0 θ a e + ε =   2 2 2 2 0 0 0   0 − 2 0 0

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

f : Cl 1,0,1 → Cl 1,2 a e1 e1 a θ1 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 Ψ 1,2 , then we get an one to one algebra homomorphism as follows:

Cl 1,0,1 → Cl 1,2 ≅ ℝ(2) ⊕ ℝ(2)  0 1 0 1 a   e1 e1 =   ,    1 0 1 0 1 −1 −1 −1 a   θ1 e2 + ε1 =  ,  1 −1  1 1 

 1 −1  1 1    where e1θ1 =   ,  .  1 −1 −1 −1

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

,1 e1,θ1,e1θ1 we get the following identity:

 1 0 1 0  0 1 0 1     x = x0   ,  + x1   ,    0 1 0 1  1 0 1 0  1 −1 −1 −1  1 −1  1 1      + x2   ,   + x3   ,    1 −1  1 1   1 −1 −1 −1

53 BAÜ Fen Bil. Enst. Dergisi Cilt 15(1) 48-58 (2013)

 x + x + x x − x − x  x − x + x x − x + x   0 2 3 1 2 3 0 2 3 1 2 3  =   ,   x1 + x2 + x3 x0 − x2 − x3  x1 + x2 − x3 x0 + x2 − x3 

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

x + x + x x − x − x  x − x + x x − x + x    0 2 3 1 2 3 0 2 3 1 2 3   ,   xi ∈ R,i = 3,2,1,0  x1 + x2 + x3 x0 − x2 − x3  x1 + x2 − x3 x0 + x2 − x3  

2 2 Example 9 Let Q(x) = −x1 be a degenerate quadratic form on V = ℝ and Cl 1,1,0 its

Clifford algebra. The degenerate Clifford algebra Cl 1,1,0 is embedded into non- degenerate Clifford algebra Cl 2,1 . Moreover, we know that the Clifford algebra Cl 2,1 isomorphic to the matrix algebra ℂ(2). The embedding map from Cl 1,1,0 to Cl 2,1 is defined on basis elements as follows:

Cl 1,1,0 → Cl 2,1 ≅ ℂ(2) a ε1 ε1 a θ1 e1 + ε 2

It can be seen that this map is a one to one algebra homomorphism. By composing this homomorphism and the isomorphism Ψ 2,1 we get the following homomorphism:

Cl 1,1,0 → Cl 2,1 ≅ ℂ(2) 0 −1 a ε ε1 =   1 1 0  i 1  a θ e1 + ε 2 =   1 1 − i

−1 i where ε1θ1 =   . Any x ∈ Cl 1,1,0 can be written as  i 1

x = x0 ⋅1+ x1ε1 + x2θ1 + x3ε1θ1 .

Then, by writing the values of ,1 ε1,θ1,ε1θ1 we have the following identity:

1 0 0 −1 i 1  −1 i x = x0   + x1   + x2   + x3   0 1 1 0  1 − i  i 1

x0 − x3 + x2i − x1 + x2 + 3ix  =   x1 + x2 + 3ix x0 + x3 − x2i 

Hence,

x0 − x3 + x2i − x1 + x2 + x3i  Cl 1,1,0 ≅   xi ∈ R,i = 3,2,1,0  ⊂ ℂ(2). x1 + x2 + x3i x0 + x3 − x2i  

54 BULUT Ş.

2 2 3 Example 10 Let Q(x) = x1 − x2 be a degenerate quadratic form on V = ℝ and Cl 1,1,1 its Clifford algebra. In similar way it is possible to be embedded into a suitable matrix algebra. As above examples the degenerate Clifford algebra Cl 1,1,1 is embedded into non-degenerate Clifford algebra Cl 2,2 . The embedding is defined on the basis elements as follows:

Cl 1,1,1 → Cl 2,2 ≅ ℝ (4) a e1 e1 a ε1 ε1 a θ1 e2 + ε 2

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

Cl 1,1,1 → Cl 2,2 ≅ ℝ (4) 0 1 0 0   1 0 0 0 e a e =   1 1 0 0 0 1   0 0 1 0 0 −1 0 0    1 0 0 0 ε a ε =   1 1 0 0 0 −1   0 0 1 0  0 0 0 0   0 0 0 0 θ a e + ε =   1 2 2 2 0 0 0   0 − 2 0 0

We have 1 0 0 0  0 0 0 0 0 0 0 0       0 −1 0 0 0 0 0 0 0 0 0 0 e ε =   , eθ =   , ε θ =   , 1 1 0 0 1 0  1 1 0 − 2 0 0 1 1 0 2 0 0       0 0 0 −1 2 0 0 0 2 0 0 0 0 0 0 0   0 0 0 0 e ε θ =   . 1 1 1 2 0 0 0   0 2 0 0

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

x = x0 ⋅1+ x1e1 + x2ε1 + x3θ1 + x4e1ε1 + x5ε1θ1 + x6e1θ1 + x7e1ε1θ1

55 BAÜ Fen Bil. Enst. Dergisi Cilt 15(1) 48-58 (2013)

We know the values of e1,ε1,θ1,e1ε1,ε1θ1,e1θ1,e1ε1θ1 . If these values are used, then we get

1 0 0 0 0 1 0 0 0 −1 0 0        0 1 0 0 1 0 0 0 1 0 0 0 x = x   + x   + x   0 0 0 1 0 1 0 0 0 1 2 0 0 0 −1       0 0 0 1 0 0 1 0 0 0 1 0  0 0 0 0 1 0 0 0  0 0 0 0       0 0 0 0 0 −1 0 0 0 0 0 0 = + x   + x   + x   3 2 0 0 0 4 0 0 1 0  5 0 2 0 0       0 − 2 0 0 0 0 0 −1 2 0 0 0 0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 + x   + x   . 6 0 − 2 0 0 7 2 0 0 0     2 0 0 0 0 2 0 0

Then,

 x0 + x4 x1 − x2 0 0    x + x x − x 0 0 x =  1 2 0 4  . 2x + 2x 2x − 2x x + x x − x   3 7 5 6 0 4 1 2  2x5 + 2x6 − 2x3 + 2x7 x1 + x2 x0 − x4 

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

 x0 + x4 x1 − x2 0 0       x + x x − x 0 0   1 2 0 4  x ∈ R,i = ,1,0 K 7,    i  2x3 + 2x7 2x5 − 2x6 x0 + x4 x1 − x2     2x5 + 2x6 − 2x3 + 2x7 x1 + x2 x0 − x4  

If we continue in similar way, we have the following imbeddings:

Cl 1,3,0 → Cl 4,1 ≅ ℍ(2) ⊕ ℍ(2) Cl 1,1,1 → Cl 2,2 ≅ ℝ(4)

Cl 1,4,0 → Cl 5,1 ≅ ℍ(4) Cl 1,2,1 → Cl 3,2 ≅ ℂ(4)

Cl 2,1,0 → Cl 3,2 ≅ ℂ(4) Cl 1,3,1 → Cl 4,2 ≅ ℍ(4)

Cl 2,2,0 → Cl 4,2 ≅ ℍ(4) Cl 1,4,1 → Cl 5,2 ≅ ℍ(4) ⊕ ℍ(4)

Cl 2,3,0 → Cl 5,2 ≅ ℍ(4) ⊕ ℍ(4) Cl 2,0,1 → Cl 2,3 ≅ 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.

56 BULUT Ş.

3 Spinor Representations of Degenerate Clifford Algebras

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

ρ : Cl p+r,q+r → End (V ) be the spinor representation of Cl p+r,q+r which is widely known, and

Ψp,q,r :Cl p,q,r → Cl p+r,q+r be the imbedding defined above, then the composition o ρ Ψp,q,r : Cl p,q,r → End (V ) is an algebra homomorphism, so it is a representation of the degenerate Clifford algebra

Cl p,q,r and we call it spinor representation of Cl p,q,r .

Now we give some example in lower dimension.

Example 11 ℝℝℝ(4) Cl 2,0,0 → 0 0 0 0   2 0 0 0 θ a   1 0 0 0 0   0 0 2 0

0 0 0 0   0 0 0 0 θ a   2 2 0 0 0   0 − 2 0 0   Example 12

→ ℝℝℝ Cl 1,0,1 (2) a e1 0 1   1 0

a θ1 1 −1   1 −1

Example 13

→ ℝℝℝ Cl 1,1,0 (4) a ε 1 0 0 −1 0    0 0 0 −1   1 0 0 0    0 1 0 0  a θ1 0 −1 1 0   1 0 0 1   1 0 0 1   0 1 −1 0

57 BAÜ Fen Bil. Enst. Dergisi Cilt 15(1) 48-58 (2013)

Example 14

ℝ Cl 1,1,1 → ℝℝ(4) a e1 0 1 0 0   1 0 0 0   0 0 0 1   0 0 1 0 a ε1 0 −1 0 0    1 0 0 0   0 0 0 −1   0 0 1 0  a θ1 0 0 0 0   0 0 0 0   2 0 0 0   0 − 2 0 0

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