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Are Grassmann numbers a concept of graded Lie algebras or is something specific to superalgebras? What are they (i.e: how are they defined, important properties, etc.)? Is there a reasonable introduction to them?

I think that what makes me wonder a little is, since there does not seem to be a sensible constructivist approach to these entities (other than accepting them as the entities that satisfy the required properties) is there nothing that stops someone from going into 'constructing' meta-superalgebras by defining 'numbers' $\kappa_{i}$, such that, e.g.,

$$\kappa_{i} \kappa_{j} = \theta_{k} \quad (\leftarrow \text{Grassmann odd}),$$ $$\kappa_{i} \kappa_{j}\kappa_{m} \kappa_{n} = \theta_{p}\quad (\leftarrow \text{Grassmann even}).$$

So I define such numbers as 'square-roots' of grassmann $a$-numbers. It seems nothing stops this process ad infinitum. Maybe there is some property I'm missing that will allow the algebra to be closed but I don't know what that could be.

Btw, I think this is a great reference Phys.SE question regarding this topic: "Velvet way" to Grassmann numbers.

Community
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lurscher
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1 Answers1

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Here I will just make a couple of general remarks.

1) Graded algebras usually refer to $\mathbb{Z}$- or $\mathbb{N}$-graded algebras, while superalgebras are $\mathbb{Z}_2$-graded algebras.

2) Grassmann numbers are oddly graded supernumbers.

Please click on the links for further information, important properties and references.

References:

1) Bryce De Witt, "Supermanifolds", Cambridge Univ. Press, 1992.

2) Deligne, Pierre and John W. Morgan, "Notes on Supersymmetry (following Joseph Bernstein)". Quantum Fields and Strings: A Course for Mathematicians (1999). American Mathematical Society. pp. 41–97. ISBN 0-8218-2012-5.

Concerning v3 of the question. The $\kappa_i$'s correspond to a $\mathbb{Z}_4$ grading, and there are indeed research works in that direction. However, many properties of numbers and supernumbers do not generalize easily to $\mathbb{Z}_n$-grading with $n>$2. For instance, I think that already Berezin showed that it is not possible to define a useful notion of (Berezin) determinant of matrices with $\mathbb{Z}_n$-graded entries if $n>$2.

peterh
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Qmechanic
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    thanks!, that property $(\theta_{i})^2 = 0$ seems to suggest these 'numbers' have zero divisors, so i'm wondering if there is a representation of these numbers using sedenions or some other algebra with zero divisors? although the infinite expansion seems to imply an infinite dimensional algebra – lurscher May 17 '11 at 18:52