symmetric monoidal (∞,1)-category of spectra
A division algebra is a possibly non-associative algebra $A$, typically over a field $k$, which has the property that for any nonzero element $a$ and any element $b$, the equations $a x=b$ and $x a=b$ each have a unique solution for $x$ in the algebra.
Usually the algebra is assumed to have a multiplicative identity, in which case this condition implies that each element has a left inverse and a right inverse. These inverses necessarily coincide if the algebra is associative, but this may fail in the absence of associativity.
A sub-topic of interest is the existence of a multiplicative norm for the algebra, see at normed division algebra.
An alternative definition is used in Baez 02: The axioms given there require that the algebra is finite-dimensional over $k$ and that for any $a$ and $b$ in the algebra, $a b = 0$ implies at least one of $a$ or $b$ is already $0$. This coincides with the definition given above for finite dimensional algebras.
Perhaps the most famous division algebras are the real numbers $\mathbb{R}$, the complex numbers $\mathbb{C}$, the quaternions $\mathbb{H}$, and the octonions $\mathbb{O}$. According to the Hurwitz theorem, these are the only normed, finite dimensional, real division algebras.
Any division ring is an associative division algebra over its center and has identity, but it may not be finite dimensional over its center.
The ring of rational polynomials $\mathbb{R}(X)$ is an infinite dimensional real associative division algebra.
There exists a division algebra with identity that does not have two-sided inverses for every nonzero element.
There exists a (nonassociative) algebra $A$ such that for each nonzero $a \in A$ there exists $a^{-1} \in A$ with $a a^{-1} = a^{-1} a = 1$, but there are nonzero zero divisors.
The sedenions $\mathbb{S}$ are a finite-dimensional normed real algebra that is not a division algebra (it has zero divisors.)
The ideal generated by $Y$ in the polynomial ring $\mathbb{R}[Y]$ is an associative, commutative, infinite dimensional real algebra without identity and without zero divisors. The equation $Yx=Y$ does not have a solution for $Y$ in the algebra.
For many applications (also to physics) the most interesting division algebras are the normed division algebras over the real numbers: By the Hurwitz theorem these are the real numbers, complex numbers, quaternions and octonions. These have important relations to supersymmetry.
exceptional spinors and real normed division algebras
Lorentzian spacetime dimension | $\phantom{AA}$spin group | normed division algebra | $\,\,$ brane scan entry |
---|---|---|---|
$3 = 2+1$ | $Spin(2,1) \simeq SL(2,\mathbb{R})$ | $\phantom{A}$ $\mathbb{R}$ the real numbers | super 1-brane in 3d |
$4 = 3+1$ | $Spin(3,1) \simeq SL(2, \mathbb{C})$ | $\phantom{A}$ $\mathbb{C}$ the complex numbers | super 2-brane in 4d |
$6 = 5+1$ | $Spin(5,1) \simeq$ SL(2,H) | $\phantom{A}$ $\mathbb{H}$ the quaternions | little string |
$10 = 9+1$ | $Spin(9,1) {\simeq}$ “SL(2,O)” | $\phantom{A}$ $\mathbb{O}$ the octonions | heterotic/type II string |
See at division algebra and supersymmetry.
See also
Wikipedia, Division algebra
John Baez, Section 1.1 Preliminaries In: The octonions, Bull. Amer. Math. Soc. 39 (2002), 145-205. (doi:10.1090/S0273-0979-01-00934-X, website)
Last revised on November 9, 2020 at 12:00:26. See the history of this page for a list of all contributions to it.