Supernumbers are the generalisation of complex numbersMathworldPlanetmathPlanetmath to a commutativePlanetmathPlanetmathPlanetmath superalgebra of commuting and anticommuting “numbers”. They are primarily used in the description of in .

Let ΛN be the Grassmann algebra generated by θi, i=1N, such that θiθj=-θjθi and (θi)2=0. Denote by Λ, the Grassmann algebra of an infinite number of generatorsPlanetmathPlanetmathPlanetmath θi. A supernumber is an element of ΛN or Λ.

Any supernumber z can be expressed uniquely in the form


where the coefficients zi1in are antisymmetric in their indices.

1 Body and soul

The body of a supernumber z is defined as zB=z0, and its soul is defined as zS=z-zB. If zB0 then z has an inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath given by


2 Odd and even

A supernumber can be decomposed into the even and odd parts:

zeven = z0+12zijθiθj++1(2n)!zi1i2nθi1θi2n+,
zodd = ziθi+16zijkθiθjθk++1(2n+1)!zi1i2n+1θi1θi2n+1+.

Even supernumbers commute with each other and are called c-numbers, while odd supernumbers anticommute with each other and are called a-numbers. Note, the product of two c-numbers is even, the product of a c-number and an a-number is odd, and the product of two a-numbers is even. The superalgebra ΛN has the vector spaceMathworldPlanetmath decomposition ΛN=ca, where c is the space of c-numbers, and a is the space of a-numbers.

3 Conjugation and involution

There are two ways, one can define a complex conjugation for supernumbers. The first is to define a linear conjugationMathworldPlanetmath in completePlanetmathPlanetmathPlanetmath analogy with complex numbers:


The second way is to define an anti-linear involutionPlanetmathPlanetmathPlanetmath:


The comes down to whether the product of two real odd supernumbers is real or imaginary.

Title supernumber
Canonical name Supernumber
Date of creation 2013-03-22 13:03:27
Last modified on 2013-03-22 13:03:27
Owner mhale (572)
Last modified by mhale (572)
Numerical id 12
Author mhale (572)
Entry type Definition
Classification msc 16W55
Related topic SuperAlgebra
Defines body
Defines soul