|
|
|
|
|
Category Of Graded Vector SpacesIn mathematics, K-Z2Vect is the category with objects are Z2-graded vector spaces over the givenfield K and with morphisms the even and odd linear transformations between two Z2-graded vector spaces. An even linear transformation is a linear transformation which maps the even part to the even part and the odd part to the odd part; while an odd linear transformation is a linear transformation which maps the even part to the odd part and the odd part to the even part. Note that any linear transformation can the expressed uniquely as the sum of an even and an odd linear transformation. K-Z2Vect is an example of a Z2-graded category. The even morphisms map to 0 while the odd morphisms map to 1. K-Z2Vect is a monoidal category since the tensor product of two Z2 graded vector spaces is another Z2 graded vector space. It's also a braided monoidal category with the involutive braiding operator given by for pure elements. In this sense, it is a symmetric monoidal category. There is also a parity reversing functor from this category to itself which interchanges the even and odd subspaces. This category is the foundation for the study of "superobjects" like supervector spaces, superalgebras, Lie superalgebras, supergroups, supermanifolds, superspace etc. We can also have more general groups to grade the vector spaces by. for example, in the study of anyonic objects, we use the category of ZN-graded vector spaces instead. This time, instead.
|
 |
|
| Copyright 2005-2009 OnPedia.com. All Rights Reserved |
|
|