A ring or a module structure involving Grassmann variables is possible?

Question

I am interested in the algebraic properties of Grassmann variables (which are anti-commuting variables that satisfy $\theta_i \theta_j = - \theta_j \theta_i$). I know that they form an exterior algebra. However, I wonder if there is any way to define a ring or a module structure involving Grassmann variables, either by themselves or with some other ring or module.

For example, is there a ring $\theta_i \mathbb{Z}$, where $\theta_i$ is a fixed Grassmann variable and $\mathbb{Z}$ is the ring of integers? Or is there a module $\theta_i \mathbb{Z}$, where $\theta_i$ is a scalar and $\mathbb{Z}^n$ is a module over the ring $\theta_i \mathbb{Z}$ ? If not, what are the obstacles that prevent such structures from existing? And are there any other rings or modules that can be constructed with Grassmann variables?

Answer 1

They don't form any algebraic structure with an operation. They themselves certainly don't "form an exterior algebra."

As the wiki article on the topic begins:

The usage of the term "Grassmann variables" is historic; they are not variables, per se; they are better understood as the basis elements of a unital algebra.

So let's call them what they are: generators of an algebra subject to some relations.

The use of the negation symbol implies they are meant to live in some additively written abelian group where $-x$ is the inverse of $x$. So we can at least talk about $\mathbb Z$ linear combinations of these generators. If one simply starts out with symbols $\theta_i$ and freely generates an abelian group, we are still not looking at the Grassman variables: we have to introduce the relations too. This is done by taking an appropriate quotient ring:

$$ \mathbb Z\langle \theta_1,\ldots,\theta_n\rangle/(\{\theta_i\theta_j+\theta_j\theta_i\mid i,j\in 1\ldots n\}) $$

Usually it is more useful, however, to use something larger than $\mathbb Z$, such as a field $\mathbb R$. The construction of the algebra happens in the same way. When using a field $F$ we usually call the result "the $n$-dimensional exterior algebra over $F$." You could call it that for $\mathbb Z$ too but that context is probably not very common.

It's important to know that in any quotient rings of these rings, the images of the generators will still satisfy the same defining relations. In a sense, the description I gave above is for "the largest algebras" like that, because all other algebras with generators satisfying the relations have to be quotient rings (i.e. "smaller versions") of these.

It seems like, going forward, you should recognize that understanding these algebras just amounts to understanding quotients of free algebras by ideals generated by relations.

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is there a ring $\theta_i \mathbb{Z}$, where $\theta_i$ is a fixed Grassmann variable and $\mathbb{Z}$ is the ring of integers?

Not with identity, there isn't. The set $\theta_i \mathbb{Z}$ is certainly an ideal of the algebra we discuss above, but the product of any two elements is zero, so it lacks an identity element.

Or is there a module $\theta_i \mathbb{Z}$, where $\theta_i$ is a scalar and $\mathbb{Z}^n$ is a module over the ring $\theta_i \mathbb{Z}$

$\theta_i \mathbb{Z}$ is an ideal of the ring I mentioned, and that makes it a module over that ring. I'm not sure what module structure you would put on $\mathbb Z^n$. You can't just guess at them, they have to be motivated.

Answer 2

Assuming the OP is asking if Grassmann numbers form a ring, the answer is yes. Grassmann numbers can form rings of any dimension.

For instance, the ring of 4-dimensional Grassmann numbers is isomorphic to the ring of these matrices:

$1=\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} \right);$$\theta _1=\left( \begin{array}{cccc} 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \end{array} \right);$ $\theta _2=\left( \begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ \end{array} \right);$$\theta _1\theta _2=\left( \begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ \end{array} \right)$