NationMaster - Encyclopedia: Representations of Clifford algebras

In mathematics, the representations of Clifford algebras are also known as Clifford modules. In general a Clifford algebra C is a central simple algebra over some field extension L of the field K over which the quadratic form Q defining C is defined. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Inter. ... Clifford algebras are a type of associative algebra in mathematics. ... In mathematics, the Brauer group arose out of an attempt to classify division algebras over a given field K. It is an abelian group with elements isomorphism classes of division algebras over K, such that the center is exactly K. The group is named for the algebraist Richard Brauer. ... In abstract algebra, an extension of a field K is a field L which contains K as a subfield. ... In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ...

The abstract algebra theory of Clifford modules was founded by a paper of M. F. Atiyah, R. Bott and A. Shapiro (Clifford Modules, Topology 3 (Suppl. 1) (1964), 3–38). This article gives an explicit theory. Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... Sir Michael Francis Atiyah, OM (born 22 April 1929) is a mathematician who was born in London. ... Raoul Bott (born September 24, 1923) is a mathematician known for numerous basic contributions to geometry in its broad sense. ...

1 Matrix representations of real Clifford algebras
2 Intermezzo: the K-system for naming matrices
3 Real Clifford algebra R2,0
4 Real Cifford algebra R1,1
5 real Clifford algebra R2,1
6 Real Clifford algebra R0,2
7 real Clifford algebra R0,3
8 real Clifford algebra R3,0
9 real Clifford algebra R3,1
10 further representations with real 4x4 matrices
11 representations with real 8x8 matrices
12 representations with 16x16 real matrices

Matrix representations of real Clifford algebras

We will need to study anticommuting matrices (AB = −BA) because in Clifford algebras orthogonal vectors anticommute For the square matrix section, see square matrix. ...

$A cdot B = frac{1}{2}( AB + BA ) = 0$

For the real Clifford algebra R_p,q we need p + q mutually anticommuting matrices, of which p have +1 as square and q have −1 as square.

$begin{matrix} gamma_a^2 &=& +1 &mbox{if} &1 le a le p gamma_a^2 &=& -1 &mbox{if} &p+1 le a le p+q gamma_a gamma_b &=& -gamma_b gamma_a &mbox{if} &a ne b end{matrix}$

Such a base of gamma matrices is not unique. One can always obtain another set of gamma matrices satisfying the same Clifford algebra by means of a similarity transformation.

$begin{matrix} gamma_{a'} &=& S &gamma_{a } &S^{-1} end{matrix}$

where S is a non-singular matrix. The sets γ _a' and γ _a belong to the same equivalence class.

Intermezzo: the K-system for naming matrices

We first present a nice method for naming 2ⁿ × 2ⁿ matrices

$K_0 = begin{pmatrix} 1&00&1 end{pmatrix}, K_1 = begin{pmatrix} 0&11&0 end{pmatrix}, K_2 = begin{pmatrix} 0&-11&0 end{pmatrix}, K_3 = begin{pmatrix} 1&00&-1 end{pmatrix}.$

Notice that K₀ is the identity matrix. The names were so chosen that there is a simple rule for remembering the products:

K₁ K₂ = K₃

K₁ K₃ = K₂

K₂ K₃ = K₁

K₂ K₁ = −K₃

K₃ K₁ = −K₂

K₃ K₂ = −K₁.

Incrementing index is positive result. Decreasing index is negative result.

Attention ! These are NOT the same relations that hold for the standard basis of the quaternions. If you would name i = i₁, j = i₂ and k = i₃ you would get In linear algebra, the standard basis for an -dimensional vector space is the basis obtained by taking the basis vectors where is the vector with a in the th coordinate and elsewhere. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ...

i₁i₂=i₃

i₂i₃=i₁

i₃i₁=i₂

so the last rule is different. We will later see that the pure quaternions i,j and k can be represented by K₁₂,K₂₀and K₃₂

Remark that

$K_0^2 = K_1^2 = K_3^2 = K_0$

$K_2^2 = - K_0$

K₂ is the only one with negative square, so it can be regarded as the simplest representation of i

Then we give all possible Kronecker products a name (see matrix multiplication): In mathematics, the Kronecker product, denoted by âŠ—, is an operation on two matrices of arbitrary size resulting in a block matrix. ... This article gives an overview of the various ways to multiply matrices. ...

$K_{ab} = K_{a} otimes K_{b}$

$K_{abc} = K_{a} otimes K_{bc}= K_{a} otimes K_{b} otimes K_{c}$

Some examples

$K_{30} = begin{pmatrix} 1&0&0&0 0&1&0&0 0&0&-1&0 0&0&0&-1 end{pmatrix}, K_{11} = begin{pmatrix} 0&0&0&1 0&0&1&0 0&1&0&0 1&0&0&0 end{pmatrix}$

Each index has its level ( 2x2, 4x4, 8x8, 16x16, ...)

K₁₃ is a K₃ at the 2x2 level and a K₁ at the 4x4 level. With this notation its very easy to multiply large square matrices since

$(A otimes B)(C otimes D) = AB otimes CD$

Let's work out an example

K₁₂₃ K₂₂₂ = K₃₀₁

8x8-level 1 times 2 gives 3

4x4-level 2 times 2 gives 0 but remember the minus sign

2x2-level 3 times 2 gives 1 but with again a minus sign

( the two minus signs cancel so the result is K₃₀₁ )

We can now start to construct sets of mutually anticommuting orthogonal matrices (see orthogonal matrix), sometimes called Dirac matrices. Its obvious that two such matrices anticommute if they anticommute in an odd number of indexes (index o commutes with all the other indices). In linear algebra, an orthogonal matrix is a square matrix G whose transpose is its inverse, i. ... The Dirac equation is a relativistic quantum mechanical wave equation invented by Paul Dirac in 1928. ...

K₁₃ for example anticommutes with

K₀₁,K₀₂,K₁₁,K₁₂,K₂₀,K₂₃,K₃₀,K₃₃

and commutes with

K₀₀,K₁₀,K₁₃,K₂₁,K₂₂,K₃₁,K₃₂.

If the index 2 appears an even number of times in the name then the square of the matrix is plus the identity matrix, let's call this a Kplus

examples are K₁, K₂₂, K₃₁₁, K₂₂₂₂

If the index 2 appears an odd number of times in the name then the square of the matrix is minus the identity matrix, let's call this a Kminus

examples are K₂, K₂₂₂, K₂₁₁, K₁₂₂₂

We have now a very simple way of constructing the largest possible sets of anticommuting matrices.

Start with an existing set {K₁,K₂,K₃}

Insert a constant new index (for example a 1 in first position) and you get {K₁₁,K₁₂,K₁₃}

Then add two more matrices that anticommute in the new level and commute in the old level (by means of the zero index 0)

So you get {K₁₁, K₁₂, K₁₃, K₂₀, K₃₀}

Other examples

{K₂₁, K₂₂, K₂₃, K₁₀, K₃₀}

{K₃₁, K₃₂, K₃₃, K₁₀, K₂₀}

{K₁₁₁, K₁₁₂, K₁₁₃, K₁₂₀, K₁₃₀, K₂₀₀, K₃₀₀}

{K₂₁₁, K₂₁₂, K₂₁₃, K₂₂₀, K₂₃₀, K₁₀₀, K₃₀₀}

{K₃₁₁, K₃₁₂, K₃₁₃, K₃₂₀, K₃₃₀, K₁₀₀, K₂₀₀}

You always get a set with an odd number of matrices and there is always one Kplus more than Kminus.

Each of them can be written as the product of all the other. Example K₁₁ K₁₂ K₁₃ K₂₀ = K₃₀

Real Clifford algebra R_2,0

p = 2 and q =0 so we need 2 Kplus as basevectors

grade 0 (the scalar)

$begin{matrix} 1 = K_0 end{matrix}$

grade 1 (the vectors)

$gamma_1 = K_1 Rightarrow gamma_1^2 = K_0 = 1$

$gamma_2 = K_3 Rightarrow gamma_2^2 = K_0 = 1$

grade 2 (the pseudoscalar)

$gamma_1 land gamma_2 = frac{1}{2}(gamma_1 gamma_2 - gamma_2 gamma_1 ) = gamma_1 gamma_2 = K_2 Rightarrow (gamma_1 land gamma_2)^2 = (gamma_1 gamma_2)^2 = K_2^2 = -1$

n = p + q = 2 and we have 2² = 4 elements so it is what I. Portious calls a universal Clifford algebra.

Real Cifford algebra R_1,1

p=1 and q = 1 so we need one Kplus and 1 Kminus as basevectors

grade 0 (the scalar)

$begin{matrix} 1 = K_0 end{matrix}$

grade 1 (the vectors)

$gamma_1 = K_1 Rightarrow gamma_1^2 = K_0 = 1$

$gamma_2 = K_2 Rightarrow gamma_2^2 = -K_0 = -1$

grade 2 (the pseudoscalar)

$gamma_1 land gamma_2 = gamma_1 gamma_2 = K_3 Rightarrow (gamma_1 land gamma_2)^2 = (gamma_1 gamma_2)^2 = K_3^2 = K_{0} = 1$

Here again we have 2ⁿ elements in the algebra with n = p+q so it is again a universal Clifford algebra

real Clifford algebra R_2,1

p = 2 and q = 1 so 2 Kplus basevectors and 1 Kminus basevector

grade 0 (the scalar)

$begin{matrix} 1 = K_0 end{matrix}$

grade 1 ( the vectors)

$gamma_1 = K_1 Rightarrow gamma_1^2 = K_0 = 1$

$gamma_2 = K_3 Rightarrow gamma_2^2 = K_0 = 1$

$gamma_3 = K_2 Rightarrow gamma_3^2 = -K_0 = -1$

The signature is ( + + - )

grade 2 (the bivectors)

$gamma_1 land gamma_2 = gamma_3 = K_2 Rightarrow (gamma_1 land gamma_2)^2 = -1$

$gamma_1 land gamma_3 = gamma_2 = K_3 Rightarrow (gamma_1 land gamma_3)^2 = +1$

$gamma_2 land gamma_3 = -gamma_1 = -K_1 Rightarrow (gamma_2 land gamma_3)^2 = +1$

grade 3 (the pseudoscalar)

$gamma_1 land gamma_2 land gamma_3 = -1 Rightarrow (gamma_1 land gamma_2 land gamma_3)^2 = (-1)^2 = +1$

This is the first example of a non-universal Clifford algebra since p+q= 3 and we only have 2² elements and not 2³. The reason is very simple, every matrix is used twice, once as vector and once as bivector. And the pseudoscalar is real just as the scalar.

(The Hodge dual of every element is simply minus the original) In mathematics, the Hodge star operator is a linear map on the exterior algebra of an oriented inner product space which establishes a correspondence between the space of k-vectors and the space of (n-k)-vectors. ...

* A = - A

Real Clifford algebra R_0,2

Here p = 0 and q = 2 so we need 2 two anti-commuting Kminus-matrices as base vectors. This is not possible with real 2×2 matrices so we need to use 4×4 matrices, and there are many possibilities. This algebra is isomorphic with the ring H of quaternions. In mathematics, the quaternions are a non-commutative extension of the complex numbers. ...

grade 0 (the scalar)

$begin{matrix} 1 = K_{00} end{matrix}$

grade 1 (the vectors)

$gamma_1 = K_{12} Rightarrow gamma_1^2 = -K_{00} = -1$

$gamma_2 = K_{20} Rightarrow gamma_2^2 = -K_{00} = -1$

The signature is (− −)

grade 2 (the pseudoscalar)

$gamma_1 land gamma_2 = K_{12}K_{20} = K_{32} Rightarrow (gamma_1 land gamma_2)^2 = K_{32}^2 = -K_{00} = -1$

The isomorphism with the quaternions is as follows:

1 is scalar, i and j are vectors and k = ij is the pseudoscalar.

A Clifford number is a linear combination of the four elements 1, i, j and k

$begin{matrix} 1 = K_{00}, &i = K_{12}, &j = K_{20} &k = K_{32} end{matrix}$

The use of k as pseudoscalar ( i times j ) is a bit strange but perfectly sound.

real Clifford algebra R_0,3

p = 0 and q = 3 so we need 3 Kminus basevectors, this is the usual way of working with quaternions i, j and k are now basevectors and ijk = -1 is the pseudoscalar. This algebra is again isomorphic with H (the quaternions) In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ...

grade 0 (the scalar)

$begin{matrix} 1 = K_0 end{matrix}$

grade 1 (the vectors)

$gamma_1 = K_{12} = i Rightarrow gamma_1^2 = -K_{00} = -1$

$gamma_2 = K_{20} = j Rightarrow gamma_2^2 = -K_{00} = -1$

$gamma_3 = K_{32} = k Rightarrow gamma_3^2 = -K_{00} = -1$

The signature is ( - - - )

grade 2 (the bivectors)

$gamma_1 land gamma_2 = K_{12} K_{20} = K_{32} = gamma_3$

$gamma_3 land gamma_1 = K_{32} K_{12} = K_{20} = gamma_2$

$gamma_2 land gamma_3 = K_{20} K_{32} = K_{12} = gamma_1$

grade 3 (the pseudoscalar)

$gamma_1 land gamma_2 and gamma_3 = K_{12} K_{20} K_{32} = -K_{00} = -1$

A Clifford number is here again a linear combination of the 4 elements 1 i j and k. The use of -1 as pseudoscalar (ijk)is as we are used to, but it makes the algebra a new example of a non-universal Clifford algebra, since p + q = 3 and we only have 2² elements.

real Clifford algebra R_3,0

This is the famous Pauli algebra, if you think of K₀₂ as i and K₀₀ as 1. We have tree Kplus as basevectors. The Pauli matrices are a set of 2 × 2 complex Hermitian matrices developed by Pauli. ...

grade 0 (the scalar)

$begin{matrix} 1 = K_0 end{matrix}$

grade 1 (the vectors)

$gamma_1 = K_{10} = sigma_1 Rightarrow gamma_1^2 = K_{00} = +1$

$gamma_2 = K_{22} = sigma_2 Rightarrow gamma_2^2 = K_{00} = +1$

$gamma_3 = K_{30} = sigma_3 Rightarrow gamma_3^2 = K_{00} = +1$

The signature is ( + + + )

grade 2 (the bivectors)

$sigma_1 land sigma_2 = K_{10} K_{22} = K_{32} = K_{02} K_{30}= i sigma_3$

$sigma_3 land sigma_1 = K_{30} K_{10} = -K_{20} = K_{02}K_{22} = i sigma_2$

$sigma_2 land sigma_3 = K_{22} K_{30} = K_{12} = K_{02} K_{10} = i sigma_1$

grade 3 (the pseudoscalar)

$sigma_1 land sigma_2 and sigma_3 = K_{10} K_{22} K_{30} = K_{02} = i$

So i is the pseudoscalar and the equations for the bivectors mean in fact that each bivector is the Hodge star of the one vector not part of the bivector.

real Clifford algebra R_3,1

This to me is the most interesting real Clifford algebra because it enables the construction of a Dirac-like equation without complex numbers. Majorana discovered it. Hence real spinors are called Majorana spinors. The algebra is also known as the Majorana-algebra. It makes use of all the 16 4x4 real matrices. The four basevectors are in fact the tree Pauli matrices (Kplus) completed with a fourth antihermitian (Kmin) matrix. The signature is ( + + + - ) See sign convention For the signature ( + - - - ) or ( - - - + ) often used in physics you need 4x4 complex matrices or 8x8 real matrices because you can not form 3 anticommuting Kmin 4x4 matrices. See R_1,3 for several representations. In some physics textbooks and articles, certain quantities are defined with the opposite sign from that which is used in other publications. ...

grade 0 (the scalar)

$begin{matrix} 1 = K_0 end{matrix}$

grade 1 (the vectors)

$gamma_1 = K_{10} Rightarrow gamma_1^2 = K_{00} = +1$

$gamma_2 = K_{22} Rightarrow gamma_2^2 = K_{00} = +1$

$gamma_3 = K_{30} Rightarrow gamma_3^2 = K_{00} = +1$

$gamma_4 = K_{23} Rightarrow gamma_4^2 = -K_{00} = -1$

The signature is ( + + + - )

grade 2 (the bivectors, tree rotations and tree boosts)

$gamma_1gamma_2 = K_{10}K_{22} = K_{32} Rightarrow (gamma_1gamma_2)^2 = -K_{00}= -1$

$gamma_1gamma_3 = K_{10}K_{30} = K_{20} Rightarrow (gamma_1gamma_3)^2 = -K_{00}= -1$

$gamma_2gamma_3 = K_{22}K_{30} = K_{12} Rightarrow (gamma_2gamma_3)^2 = -K_{00}= -1$

$gamma_1gamma_4 = K_{10}K_{23} = K_{33} Rightarrow (gamma_1gamma_4)^2 = K_{00}= +1$

$gamma_2gamma_4 = K_{22}K_{23} = -K_{01} Rightarrow (gamma_1gamma_2)^2 = K_{00}= +1$

$gamma_3gamma_4 = K_{30}K_{23} = -K_{13} Rightarrow (gamma_1gamma_2)^2 = K_{00}= +1$

grade 3 (the pseudovectors, the Hodge duals of the vectors)

$gamma_2gamma_3gamma_4 = K_{22}K_{30}K_{23} = K_{31} Rightarrow (gamma_2gamma_3gamma_4)^2 = K_{00} = +1$

$gamma_1gamma_3gamma_4 = K_{10}K_{30}K_{23} = -K_{03} Rightarrow (gamma_1gamma_3gamma_4)^2 = K_{00} = +1$

$gamma_1gamma_2gamma_4 = K_{10}K_{22}K_{23} = -K_{11} Rightarrow (gamma_1gamma_2gamma_4)^2 = K_{00} = +1$

$gamma_1gamma_2gamma_3 = K_{10}K_{22}K_{30} = K_{02} = i Rightarrow (gamma_1gamma_2gamma_3)^2 = -K_{00} = -1$

the last one was the pseudoscalar in R_3,0

grade 4 (the pseudoscalar)

$gamma_1gamma_2gamma_3gamma_4 = K_{10}K_{22}K_{30}K_{23} = K_{21} Rightarrow (gamma_1gamma_2gamma_3gamma_4)^2 = -K_{00} = -1$

further representations with real 4x4 matrices

$begin{matrix} R_{2,2} &K_{10} &K_{22} &K_{21} &K_{23} end{matrix}$

$begin{matrix} R_{3,2} &K_{10} &K_{22} &K_{30} &K_{21} &K_{23} end{matrix}$

representations with real 8x8 matrices

$begin{matrix} R_{4,0} &K_{110} &K_{122} &K_{130} &K_{300} end{matrix}$

$begin{matrix} R_{4,1} &K_{110} &K_{122} &K_{130} &K_{300} &K_{200} end{matrix}$

$begin{matrix} R_{4,2} &K_{110} &K_{122} &K_{130} &K_{300} &K_{200} &K_{121} end{matrix}$

$begin{matrix} R_{4,3} &K_{110} &K_{122} &K_{130} &K_{300} &K_{200} &K_{121} &K_{123} end{matrix}$

$begin{matrix} R_{3,3} &K_{110} &K_{122} &K_{130} &K_{200} &K_{121} &K_{123} end{matrix}$

$begin{matrix} R_{2,3} &K_{110} &K_{122} &K_{200} &K_{121} &K_{123} end{matrix}$

$begin{matrix} R_{1,3} &K_{110} &K_{200} &K_{121} &K_{123} end{matrix}$

This last one is very important in physics since it is the most used Clifford algebra for working in Minkowski space-time. Signature ( + - - - ) see sign convention More used representations are In some physics textbooks and articles, certain quantities are defined with the opposite sign from that which is used in other publications. ...

$begin{matrix} R_{1,3} &K_{100} &K_{210} &K_{222} &K_{230} end{matrix}$

Interestingly with 8x8 real matrices one can form 7 anticommuting Kmin matrices. They form a baseset for the non-universal real Clifford algebra R_0,7

$begin{matrix} R_{0,7} &K_{302} &K_{102} &K_{230} &K_{210} &K_{023} &K_{021} &K_{222} end{matrix}$

$begin{matrix} R_{0,6} &K_{302} &K_{102} &K_{230} &K_{210} &K_{023} &K_{021} end{matrix}$

$begin{matrix} R_{0,5} &K_{302} &K_{102} &K_{230} &K_{210} &K_{023} end{matrix}$

$begin{matrix} R_{0,4} &K_{302} &K_{102} &K_{230} &K_{210} end{matrix}$

( For R_0,3 we showed one only needs 4x4 real matrices)

representations with 16x16 real matrices

$begin{matrix} R_{5,0} &K_{1110} &K_{1122} &K_{1130} &K_{1300} &K_{3000} end{matrix}$

$begin{matrix} R_{5,1} &K_{1110} &K_{1122} &K_{1130} &K_{1300} &K_{3000}&K_{1121} end{matrix}$

$begin{matrix} R_{5,2} &K_{1110} &K_{1122} &K_{1130} &K_{1300} &K_{3000}&K_{1121} &K_{1123} end{matrix}$

$begin{matrix} R_{5,3} &K_{1110} &K_{1122} &K_{1130} &K_{1300} &K_{3000}&K_{1121} &K_{1123} &K_{1200} end{matrix}$

$begin{matrix} R_{5,4} &K_{1110} &K_{1122} &K_{1130} &K_{1300} &K_{3000}&K_{1121} &K_{1123} &K_{1200} &K_{2000} end{matrix}$

$begin{matrix} R_{4,4} &K_{1110} &K_{1122} &K_{1130} &K_{1300} &K_{1121} &K_{1123} &K_{1200} &K_{2000} end{matrix}$

$begin{matrix} R_{3,4} &K_{1110} &K_{1122} &K_{1130} &K_{1121} &K_{1123} &K_{1200} &K_{2000} end{matrix}$

$begin{matrix} R_{2,4} &K_{1110} &K_{1122} &K_{1121} &K_{1123} &K_{1200} &K_{2000} end{matrix}$

$begin{matrix} R_{1,4} &K_{1110} &K_{1121} &K_{1123} &K_{1200} &K_{2000} end{matrix}$

$begin{matrix} R_{0,4} &K_{1121} &K_{1123} &K_{1200} &K_{2000} end{matrix}$

$begin{matrix} R_{1,8} &K_{1302} &K_{1102} &K_{1230} &K_{1210} &K_{1023} &K_{1021} &K_{1222} &K_{2000} &K_{3000} end{matrix}$

$begin{matrix} R_{1,7} &K_{1302} &K_{1102} &K_{1230} &K_{1210} &K_{1023} &K_{1021} &K_{1222} &K_{3000} end{matrix}$

$begin{matrix} R_{1,6} &K_{1302} &K_{1102} &K_{1230} &K_{1210} &K_{1023} &K_{1021} &K_{3000} end{matrix}$

$begin{matrix} R_{1,5} &K_{1302} &K_{1102} &K_{1230} &K_{1210} &K_{1023} &K_{3000} end{matrix}$

$begin{matrix} R_{1,4} &K_{1302} &K_{1102} &K_{1230} &K_{1210} &K_{3000} end{matrix}$

R_1,3 only needs 4x4 real matrices

$begin{matrix} R_{0,8} &K_{1302} &K_{1102} &K_{1230} &K_{1210} &K_{1023} &K_{1021} &K_{1222} &K_{2000} end{matrix}$

R_0,7 only needs 8x8 real matrices

$begin{matrix} R_{9,0} &K_{2302} &K_{2102} &K_{2230} &K_{2210} &K_{2023} &K_{2021} &K_{2222} &K_{1000} &K_{3000} end{matrix}$

$begin{matrix} R_{8,0} &K_{2302} &K_{2102} &K_{2230} &K_{2210} &K_{2023} &K_{2021} &K_{2222} &K_{1000} end{matrix}$

$begin{matrix} R_{7,0} &K_{2302} &K_{2102} &K_{2230} &K_{2210} &K_{2023} &K_{2021} &K_{2222} end{matrix}$

$begin{matrix} R_{6,0} &K_{2302} &K_{2102} &K_{2230} &K_{2210} &K_{2023} &K_{2021} end{matrix}$

R_5,0 only needs 8x8 real matrices

Categories: Representation theory | Clifford algebras

Results from FactBites:

Clifford algebra (457 words)
	Clifford algebras are associative algebras of importance in mathematics, in particular in the theories of quadratic forms and of orthogonal groups, and in physics.
	The associated graded algebra is canonically isomorphic to the exterior algebra Λ V of the vectorspace.
	In case the field k is the field of real numbers the Clifford algebra of a quadratic form of signature p,q is usually denoted C(p,q).

Representations of Clifford algebras - definition of Representations of Clifford algebras in Encyclopedia (1064 words)
	In general a Clifford algebra C is a central simple algebra over some field extension L of the field K over which the quadratic form Q defining C is defined.
	The abstract algebra theory of Clifford modules was founded by a paper of M.
	This algebra is isomorphic with the ring H of quaternions.

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Contents

Intermezzo: the K-system for naming matrices

Real Clifford algebra R2,0

Real Cifford algebra R1,1

real Clifford algebra R2,1

Real Clifford algebra R0,2

real Clifford algebra R0,3

real Clifford algebra R3,0

real Clifford algebra R3,1

further representations with real 4x4 matrices

representations with real 8x8 matrices

representations with 16x16 real matrices

Real Clifford algebra R_2,0

Real Cifford algebra R_1,1

real Clifford algebra R_2,1

Real Clifford algebra R_0,2

real Clifford algebra R_0,3

real Clifford algebra R_3,0

real Clifford algebra R_3,1