Elementary algebraic properties of superalgebras

Here we will present the very basic ideas of Grassmann variables and polynomials over them.

Grassmann algebra
Consider a set of n odd variables \(\{ \theta^{1}, \theta^{2}, \cdots \theta^{n} \}\). By odd we will mean that they satisfy

\( \theta^{a}\theta^{b} + \theta^{b} \theta^{a}=0\).

Note that in particular this means \(\theta^{2}=0\). That is the generators are nilpotent.

The Grassmann algebra is then defined as the polynomial algebra in these variables. Thus a general function in odd variables is

\(f(\theta) = f_{0} + \theta^{a}f_{a} + \frac{1}{2!} \theta^{a} \theta^{b}f_{ba} + \cdots + \frac{1}{n!} \theta^{a_{1}} \cdots \theta^{a_{n}}f_{a_{n}\cdots a_{1}}\).

The coefficients we take as real and antisymmetric. Note that the nilpotent property of the odd variables means that the Grassmann algebra is complete as polynomials.

Example If we have the algebra generated by a single odd variable \(\theta \) then polynomials are of the form

\(a + \theta b\).

Example If we have two odd variables \(\theta\) and \(\overline{\theta}\) then polynomials are of the form

\(a + \theta b + \overline{\theta} c + \theta \overline{\theta} d\).

It is quite clear that the polynomials in odd variables forms a vector space. You can add such functions and multiply by a real number and the result remains a polynomial. It is also straightforward to see that we have an algebra. One can multiply two such functions together and get another.

The space of all such functions has a natural \(\mathbb{Z}_{2}\)-grading, which we will call parity given by the number of odd generators in each function mod 2. If the function has an even/odd number of odd variables then the function is even/odd. We will denote the parity by of a function \(\widetilde{f}= 0/1\), if it is even/odd.

Example \(a +\theta \overline{\theta} d \) is an even function and \(\theta b + \overline{\theta} c \) is an odd function.

Let us define the (super)commutator of such functions as

\([f,g] = fg -(-1)^{\widetilde{f} \widetilde{g}} gf\).

If the functions are not homogeneous, that is even or odd the commutator is extended via linearity. We see that the commutator of any two functions in odd variables vanishes. Thus we say that the algebra of functions in odd variables forms a supercommutative algebra.

Specifically note that this means the ordering of odd functions is important.

The modern approach to geometry is to define and deal with “spaces” in terms of the functions upon them. Geometrically we can think of the algebra generated by n odd variables as defining the space \(\mathbb{R}^{0|n}\). Note that no such “space” in the classical sense exists. In fact such spaces consist of only one point!

If we promote the coefficients in the polynomials to be functions of m real variables then we have the space \(\mathbb{R}^{m|n}\). We are now most of the way to defining supermanifolds, but this would be a digression from the current issues.

Noncommutative superalgebras
Of course superalgebras for which the commutator generally is non-vanishing can be defined and are naturally occurring. We will encounter such things when dealing with first order differential operators acting on functions in odd variables. Geometrically these are the vector fields. Recall that the Lie bracket between vector fields over a manifold is in general non-vanishing.

What next?
Given the basic algebraic properties of functions in odd variables we will proceed to algebraically define how to differentiate with respect to odd variables.