What is a Graded Algebra with Z_2 Grading?

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Discussion Overview

The discussion revolves around the concept of graded algebras, specifically focusing on Z2 grading. Participants explore the definition, examples, and implications of this type of grading within algebraic structures, including tensor algebras and polynomial algebras.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant seeks clarification on the meaning of Z2 grading in graded algebras.
  • Another participant explains that Z2 grading involves assigning degrees from the integers modulo 2, with an example of complex numbers illustrating even and odd elements.
  • A participant connects Z2 grading to tensor algebra, suggesting it divides the algebra into even (T+) and odd (T-) elements.
  • There is a question raised about the uniqueness of grading in algebras, with participants suggesting that grading is not necessarily unique.
  • One participant notes that every algebra can be graded in infinitely many ways, indicating flexibility in grading approaches.
  • A general definition of graded algebras is provided, highlighting that grades form vector spaces under addition but are not closed under multiplication.
  • An example of a graded algebra with Z2 grading is proposed, involving polynomials with exponents in Z2.

Areas of Agreement / Disagreement

Participants generally agree on the basic concept of Z2 grading and its implications but express differing views on the uniqueness of grading methods and the nature of graded algebras.

Contextual Notes

Some participants reference external sources, such as Wikipedia, indicating that definitions and explanations may vary or be incomplete. There is an acknowledgment of the complexity and variety in grading approaches within algebraic structures.

precondition
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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.
 
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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.


See Wikipedia for more info.
 
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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
 
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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.
 
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.
 
A "graded algebra", in general, is an algebra made up of a number of subsets (the "grades") such that each "grade" is a vector space under addition but not closed under multiplication. The most important example is the algebra of all polynomials. The set of all polynomials of degree less than or equal to a given n forms a vector space but is not closed under multiplication. The entire set of polynomials is closed under multiplication and so is an algebra.

A example of a graded algebra "with Z2 grading" might be the set of all polynomial with exponents in Z2.
 

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