# The complex algebra graded by Z-2

• PsychonautQQ
In summary, the conversation discusses the idea of grading in complex algebra, where the real part is considered 'even' and the imaginary part is considered 'odd'. The conversation also explores the concept of graded commutativity, where the multiplication is not necessarily commutative, but follows a specific formula involving the parity of the elements. It is noted that viewing complex numbers as a graded ring or algebra still results in commutative structures, and making zero the only odd element does not accurately represent the intended graded structure. The conversation concludes with the suggestion to become comfortable with the concept of graded rings before exploring graded modules and algebras.
PsychonautQQ
I'm trying to understand something in my notes here...

So if we call the real part of the complex algebra 'even' and the imaginary part 'odd' then this graded algebra is communitive but NOT graded commutative. so ab = ba for all a and b in C.

If we call the whole complex algebra 'even' and only zero (also the only element in the intersection) to be odd then it would be graded commutative.

so ab = (-1)^(|b|*|a|)*ba

but if the whole of C is even, won't the parity of |b| and |a| always be zero and therefore the multiplication would just be normal commutative?

P.S. The whole idea of grading is still uneasy with me (obviously..)

The complex numbers can be viewed as a ##\mathbb{Z}/2\mathbb{Z}## graded ring ##R_0\oplus R_1## with ##R_0=\mathbb{R}##, ##R_1=i\mathbb{R}##. This is a commutative ring.

They can also be thought of as a ##\mathbb{Z}/2\mathbb{Z}## graded algebra over the reals with the same ##A_0\oplus A_1## structure. In the algebra picture, ##A_1=i\mathbb{R}## is naturally an ##\mathbb{R}##-module, so again we see that the multiplication has to be commutative.

I don't think that your proposition to make zero odd helps too much. In that case the structure is ##A_0=\mathbb{C}## and ##A_1=\{0\}## is the trivial ##\mathbb{C}##-module. This isn't the graded structure that the example was supposed to represent.

Maybe you want to get comfortable with the idea of graded rings before adding structure to get graded modules and algebras. Graded-commutativity is a step beyond the direct sum structure that characterizes a graded ring.

## 1. What is the meaning of "complex algebra graded by Z-2"?

The complex algebra graded by Z-2 refers to a mathematical structure in which a complex algebra is divided into two subspaces based on the parity of the degree of its elements. This means that each element in the algebra is assigned either an even or an odd degree, based on which it belongs to one of the two subspaces.

## 2. How does the grading system work in a complex algebra graded by Z-2?

In a complex algebra graded by Z-2, the grading system assigns an even degree to elements that are divisible by 2, and an odd degree to elements that are not divisible by 2. This allows for a clear division of the algebra into two subspaces based on the parity of the degree of its elements.

## 3. What are the applications of a complex algebra graded by Z-2?

A complex algebra graded by Z-2 has various applications in mathematics and physics. It is commonly used in the study of Lie algebras, which are important in the theory of differential equations and symmetries. It also has applications in topological quantum field theory and superstring theory.

## 4. How is the Z-2 grading different from other grading systems?

The Z-2 grading system is different from other grading systems in the sense that it divides the algebra into two subspaces based on the parity of the degree of its elements. Other grading systems may divide the algebra into more than two subspaces, or may use different criteria to assign degrees to elements.

## 5. Can a complex algebra graded by Z-2 be extended to other number systems?

Yes, a complex algebra graded by Z-2 can be extended to other number systems, such as the real numbers or the rational numbers. However, the Z-2 grading may not always be applicable or useful in these cases, as the properties of the algebra may differ depending on the number system being used.

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