Composition algebra

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In mathematics, a composition algebra $A$ over a field $K$ is a not necessarily associative algebra over $K$ together with a nondegenerate quadratic form $N$ that satisfies

N(xy)=N(x)N(y)

for all $x$ and $y$ in $A$ .

A composition algebra includes an involution called a conjugation: $x\mapsto x^{*}.$ The quadratic form $N(x)=xx^{*}$ is called the norm of the algebra.

A composition algebra (A, ∗, N) is either a division algebra or a split algebra, depending on the existence of a non-zero v in A such that N(v) = 0, called a null vector.^[1] When x is not a null vector, the multiplicative inverse of x is ${\textstyle {\frac {x^{*}}{N(x)}}}$ . When there is a non-zero null vector, N is an isotropic quadratic form, and "the algebra splits".

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Voiceover:So we have three different function definitions here. This is F of X in blue, here we map between different values of T and what G of T would be. So you could use this as a definition of G of T. And here we map from X to H of X. So for example, when X is equal to three, H of X is equal to zero. When X is equal to one, H of X is equal to two. And actually let me number this one, two, three, just like that. Now what I want to do in this video is introduce you to the idea of composing functions. Now what does it mean to compose functions? Well that means to build up a function by composing one function of other functions or I guess you could think of nesting them. What do I mean by that? Well, let's think about what it means to evaluate F of, not X, but we're going to evaluate F of, actually let's just start with a little warm-up. Let's evaluate F of G of two. Now what do you think this is going to be and I encourage you to pause this video and think about it on your own. Well it seems kind of daunting at first, if you're not very familiar with the notation, but we just have to remember what a function is. A function is just a mapping from one set of numbers to another. So for example, when we're saying G of two, that means take the number two, input it into the function G and then you're going to get an output which we are going to call G of two. Now we're going to use that output, G of two, and then input it into the function F. So we're going to input it into the function F, and what we're going to get is F of the thing that we inputted, F of G of two. So let's just take it step by step. What is G of two? Well when T is equal to two, G of two is negative three. So when I put negative three into F, what am I going to get? Well, I'm going to get negative three squared minus one, which is nine minus one which is going to be equal to eight. So this right over here is equal to eight. F of G of two is going to be equal to eight. Now, what would, using this same exact logic, what would F of H of two be? And once again, I encourage you to pause the video and think about it on your own. Well let's think about it this way, instead of doing it using this little diagram, here everywhere you see the input is X, whatever the input is you square it and minus one. Here the input is H of two, and so we're going to take the input, which is H of two, and we're going to square it and we're going to subtract one. So F of H of two is H of two squared minus one. Now what is H of two? When X is equal to two, H of two is one. So H of two is one, so since H of two is equal to one, this simplifies two one squared minus one, well that's just going to be one minus one which is equal to zero. We could have done it with the diagram way, we could have said, hey we're going to input two into H, if you input two into H you get one, so that is H of two right over here. So that is H of two, and then we're going to input that into F, which is going to give us F of one. F of one is one squared minus one, which is zero. So this right over here is F of H of two. H of two is the input into F, so the output is going to be F of our input, F of H of two. Now we can go even further, let's do a composite. Let's compose three of these functions together. So let's take, and I'm doing this on the fly a little bit, so I hope it's a good result, G of F of two, and let me just think about this for one second. So that's going to be G of F of two, and let's take H of G of F of two, just for fun. Now we're really doing a triple composition. So there's a bunch of ways we could do this. One way is to just try to evaluate what is F of two. Well F of two is going to be equal to two squared minus one. It's going to be four minus one or three. So this is going to be equal to three. Now what is G of three? G of three is when T is equal to three, G of three is four. So G of three, this whole thing, is four. F of two is three, three of G is four. What is H of four? Well we can just look back to our original graph here. When X is four, H of four is negative one. So H of G of F of two, is just equal to negative one. So hopefully this you somewhat familiar with how to evaluate the composition of functions.

Structure theorem

Every unital composition algebra over a field $K$ can be obtained by repeated application of the Cayley–Dickson construction starting from $K$ (if the characteristic of $K$ is different from $2$ ) or a 2-dimensional composition subalgebra (if $char(K) = 2$ ). The possible dimensions of a composition algebra are $1$ , $2$ , $4$ , and $8$ .^[2]^[3]^[4]

1-dimensional composition algebras only exist when $char(K) \neq 2$ .
Composition algebras of dimension 1 and 2 are commutative and associative.
Composition algebras of dimension 2 are either quadratic field extensions of $K$ or isomorphic to $K \oplus K$ .
Composition algebras of dimension 4 are called quaternion algebras. They are associative but not commutative.
Composition algebras of dimension 8 are called octonion algebras. They are neither associative nor commutative.

For consistent terminology, algebras of dimension 1 have been called unarion, and those of dimension 2 binarion.^[5]

Every composition algebra is an alternative algebra.^[3]

Using the doubled form ( _ : _ ): A × A → K by $(a:b)=n(a+b)-n(a)-n(b),$ then the trace of a is given by (a:1) and the conjugate by a* = (a:1)e – a where e is the basis element for 1. A series of exercises proves that a composition algebra is always an alternative algebra.^[6]

Instances and usage

When the field $K$ is taken to be complex numbers $C$ and the quadratic form $z 2$ , then four composition algebras over $C$ are $C itself$ , the bicomplex numbers, the biquaternions (isomorphic to the 2×2 complex matrix ring $M(2, C)$ ), and the bioctonions $C \otimes O$ , which are also called complex octonions.

The matrix ring $M(2, C)$ has long been an object of interest, first as biquaternions by Hamilton (1853), later in the isomorphic matrix form, and especially as Pauli algebra.

The squaring function $N (x) = x 2$ on the real number field forms the primordial composition algebra. When the field $K$ is taken to be real numbers $R$ , then there are just six other real composition algebras.^[3]^: 166 In two, four, and eight dimensions there are both a division algebra and a split algebra:

binarions: complex numbers with quadratic form

x 2 + y 2

and split-complex numbers with quadratic form

x 2 - y 2

quaternions and split-quaternions,

octonions and split-octonions.

Every composition algebra has an associated bilinear form B(x,y) constructed with the norm N and a polarization identity:

B(x,y)\ =\ [N(x+y)-N(x)-N(y)]/2.

^[7]

History

The composition of sums of squares was noted by several early authors. Diophantus was aware of the identity involving the sum of two squares, now called the Brahmagupta–Fibonacci identity, which is also articulated as a property of Euclidean norms of complex numbers when multiplied. Leonhard Euler discussed the four-square identity in 1748, and it led W. R. Hamilton to construct his four-dimensional algebra of quaternions.^[5]^: 62 In 1848 tessarines were described giving first light to bicomplex numbers.

About 1818 Danish scholar Ferdinand Degen displayed the Degen's eight-square identity, which was later connected with norms of elements of the octonion algebra:

Historically, the first non-associative algebra, the Cayley numbers ... arose in the context of the number-theoretic problem of quadratic forms permitting composition…this number-theoretic question can be transformed into one concerning certain algebraic systems, the composition algebras...^[5]^: 61

In 1919 Leonard Dickson advanced the study of the Hurwitz problem with a survey of efforts to that date, and by exhibiting the method of doubling the quaternions to obtain Cayley numbers. He introduced a new imaginary unit $e$ , and for quaternions $q$ and $Q$ writes a Cayley number $q + Q e$ . Denoting the quaternion conjugate by $q'$ , the product of two Cayley numbers is^[8]

(q+Qe)(r+Re)=(qr-R'Q)+(Rq+Qr')e.

The conjugate of a Cayley number is $q' - Q e$ , and the quadratic form is $qq' + QQ'$ , obtained by multiplying the number by its conjugate. The doubling method has come to be called the Cayley–Dickson construction.

In 1923 the case of real algebras with positive definite forms was delimited by the Hurwitz's theorem (composition algebras).

In 1931 Max Zorn introduced a gamma (γ) into the multiplication rule in the Dickson construction to generate split-octonions.^[9] Adrian Albert also used the gamma in 1942 when he showed that Dickson doubling could be applied to any field with the squaring function to construct binarion, quaternion, and octonion algebras with their quadratic forms.^[10] Nathan Jacobson described the automorphisms of composition algebras in 1958.^[2]

The classical composition algebras over $R$ and $C$ are unital algebras. Composition algebras without a multiplicative identity were found by H.P. Petersson (Petersson algebras) and Susumu Okubo (Okubo algebras) and others.^[11]^: 463–81

References

Wikibooks has a book on the topic of: Associative Composition Algebra

^ Springer, T. A.; F. D. Veldkamp (2000). Octonions, Jordan Algebras and Exceptional Groups. Springer-Verlag. p. 18. ISBN 3-540-66337-1.
^ ^a ^b Jacobson, Nathan (1958). "Composition algebras and their automorphisms". Rendiconti del Circolo Matematico di Palermo. 7: 55–80. doi:10.1007/bf02854388. Zbl 0083.02702.
^ ^a ^b ^c Guy Roos (2008) "Exceptional symmetric domains", §1: Cayley algebras, in Symmetries in Complex Analysis by Bruce Gilligan & Guy Roos, volume 468 of Contemporary Mathematics, American Mathematical Society, ISBN 978-0-8218-4459-5
^ Schafer, Richard D. (1995) [1966]. An introduction to nonassociative algebras. Dover Publications. pp. 72–75. ISBN 0-486-68813-5. Zbl 0145.25601.
^ ^a ^b ^c Kevin McCrimmon (2004) A Taste of Jordan Algebras, Universitext, Springer ISBN 0-387-95447-3 MR2014924
^

Associative Composition Algebra/Transcendental paradigm#Categorical treatment at Wikibooks
^ Arthur A. Sagle & Ralph E. Walde (1973) Introduction to Lie Groups and Lie Algebras, pages 194−200, Academic Press
^ Dickson, L. E. (1919), "On Quaternions and Their Generalization and the History of the Eight Square Theorem", Annals of Mathematics, Second Series, 20 (3), Annals of Mathematics: 155–171, doi:10.2307/1967865, ISSN 0003-486X, JSTOR 1967865
^ Max Zorn (1931) "Alternativekörper und quadratische Systeme", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 9(3/4): 395–402
^ Albert, Adrian (1942). "Quadratic forms permitting composition". Annals of Mathematics. 43 (1): 161–177. doi:10.2307/1968887. JSTOR 1968887. Zbl 0060.04003.
^ Max-Albert Knus, Alexander Merkurjev, Markus Rost, Jean-Pierre Tignol (1998) "Composition and Triality", chapter 8 in The Book of Involutions, pp. 451–511, Colloquium Publications v 44, American Mathematical Society ISBN 0-8218-0904-0

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