## Main Question or Discussion Point

Hello,
I think I have an idea of what graded algebra means but when people say it has Z_2 grading etc I'm puzzled. Could someone please help me out?

By 'Z' I mean integers and '_2' means mod 2.

Last edited:

Related Linear and Abstract Algebra News on Phys.org
Hurkyl
Staff Emeritus
Gold Member
Normally, when you think of a graded algebra, you imagine each nonzero element being assigned a natural number as its degree.

But there's no reason to restrict ourselves to using the natural numbers. A Z2-graded algebra is one where the degree is an element of Z2.

For example, C is a Z2-graded algebra over R. The "even" elements (degree 0) of C are the purely real numbers, and the "odd" elements (degree 1) of C are the purely imaginary numbers.

Exercise: check that this really is a grading. For example, i is homogenous, and in the equation i * i = -1 we see that the degrees match: the degree of i * i should be 1 + 1 = 0 (remember, they're elements of Z2), and the degree of -1 is, in fact, 0.

Last edited:
I see, thank you for that information.
The example I have here is tensor algebra which it says has Z_2 grading. So I guess Z_2 grading divides tensor algebra into T+ and T- where elements of T+ has even degrees(including 0) and elements of T- has odd degrees?

Now I'm thinking if any other grading would be possible? In other words grading is not unique? Is it? or it isn't?

p.s. I referred to wikipedia first but it didn't explain Z_2 grading :D

Last edited:
Hurkyl
Staff Emeritus
Gold Member
If you scroll down, the wiki page has a section on G-graded rings and algebras; that's where it discusses the general case.

Your interpretation of the grading on the tensor algebra sounds right. And indeed, there is no reason to think that there is a unique way of turning an algebra into a graded algebra.

The Z2 grading becomes particularly when you pass to related structures. For example, when you antisymmetrize the tensor algebra, you get a "commutative" superalgebra. The qualitative behavior of the odd and even terms is quite different in that case.

matt grime
Homework Helper
The obvious way to point out there is not necessarily such a thing as a unique grading is by noting that *every* algebraic gadget is graded in infinitely many ways - just pick any grading and then put everything in degree 0.

HallsofIvy