Graded Algebra: Understanding Color Dirac Spinors in Space-Time

[topsquark]

I just read through a paper on a \(\displaystyle \mathbb{Z} _ 3\) graded Algebra. In this instance we are talking about color Dirac spinors in space-time. It looks like the author is talking about \(\displaystyle \left ( SU(3) \otimes L^4 \otimes \mathbb{Z}_2 \otimes \mathbb{Z} _2 \right ) \otimes \mathbb{Z} _3\). ( SU(3) is the color group, \(\displaystyle L_4 \) is the Lorentz group, \(\displaystyle \mathbb{Z} _2 \otimes \mathbb{Z} _2\) is the Dirac 4-spinor group, and \(\displaystyle \mathbb{Z} _3\) is the usual group on 3 elements.

I can (mostly) follow the paper assuming the tensor products, but what do they mean by the word "graded?"

Thanks!

-Dan

 

[HOI]

In mathematics, an "algebra" is a vector space (so we can add vectors and multiply vectors by scalars) in which we can also multiply two vectors, the result being a vector. An algebra is said to be "graded" if we can divide the vectors into "grades" that are "closed" under addition and scalar multiplication (so the sum of two vectors in one "grade" are also in that "grade" and the product of a scalar and a vector in a given "grade" is in that same "grade") but the product of two vectors in the same "grade" is not necessarily in that "grade".

The simplest example of a "graded algebra" is the algebra of all polynomials. Each "grade" is the vector space of polynomials of degree "n" or less for some positive integer "n". Adding two polynomials of degree n or less gives a polynomial of degree n or less and multiplying a polynomial of degree n or less times a scalar (real or complex number) is a polynomial of degree n or less. But while the product of two polynomials of degree n or less is a polynomial it is not necessarily of degree n or less.

 

[topsquark]

Thank you. That actually answers another question I had about the paper as well. I think I've got the idea now.

Thanks again!

-Dan