From: kramsay@aol.commangled (Keith Ramsay)
Subject: Re: graded algebra
Date: 22 Nov 1999 04:23:52 GMT
Newsgroups: sci.math

In article <38373E99.4DFAF5CE@new.ox.ac.uk>, Julius Ross
<julius.ross@new.ox.ac.uk> writes:
>I am reading about tensors and exterior algebras.  Could anyone tell me
>what a graded algebra is?

It's an algebra A which, as a vector space, is a sum of subspaces
A_0, A_1, A_2,..., where the product of an element of A_m and an
element of A_n lies in A_{m+n}. The elements of the A_i are called
the homogeneous elements of A of degree i. Every element is uniquely
a sum of homogeneous elements.

It is of course possible that the algebra can be decomposed in more
than one such way; a graded algebra is an algebra taken together with
such a grading.

The algebra of polynomials in variables x1,...,xn can be graded by the
degree of the terms, and the homogeneous elements are simply the
homogeneous polynomials in the usual sense of the term.

Tensors are extended to an algebra by allowing the operation of
abstractly adding together tensors of different types. The notion of
degree, for the grading, is the same notion of degree as you should
have seen defined already for tensors. The multiplication is taking
the tensor product-- extended by the distributive law to sums of
tensors of different types. The homogeneous elements are the sums of
tensors of the same degree.

Sometimes people grade algebras by some semigroup other than the
natural numbers. For instance, a Z-graded algebra is one which is a
sum of ...A_{-2}, A_{-1}, A_0, A_1, A_2,..., with the analogous
condition that products of elements from A_m and A_n are in A_{m+n}.
The generalization to G-graded algebras where G is a semigroup is
obvious enough. But in this context, I'm pretty sure that it's a
matter of an N-graded algebra (which is common enough anyway-- the
first book I checked defined "graded algebra" to mean N-graded
algebra).

Keith Ramsay
==============================================================================

From: Robin Chapman <rjc@maths.ex.ac.uk>
Subject: Re: graded algebra
Date: Mon, 22 Nov 1999 08:32:47 GMT
Newsgroups: sci.math

In article <38373E99.4DFAF5CE@new.ox.ac.uk>,
  Julius Ross <julius.ross@new.ox.ac.uk> wrote:
> Hi
>
> I am reading about tensors and exterior algebras.  Could anyone tell me
> what a graded algebra is?

The most basic type of graded algebra is that which is the direct sum
of abelian groups A_0, A_1, A_2, and such that if a in A_n and b in A_m
then ab in A_{n+m}. For example let M be a module over the commutative
ring K. Then T(M) and Lambda(M), the tensor algebra and exterior
algebra respectively, are graded algebras. The gradings are defined
by T(M)_k is the k-foled tensor product of M with itself, and
Lambda(M)_k = Lambda^k(M) is the k-th exterior power of M.

One can generalize the idea by indexing the summands by elements of
any group or monoid. For example, Clifford algebras, a generalization
of exterior algebras, are graded by the additive group of two
elements. So a Clifford algebra C = C_0 (+) C_1 where C_j C_k is
contained in C_{j+k} with j+k considered modulo 2. For the basics
on Clifford algebras see volume 3 of Paul Cohn's _Algebra_.

--
Robin Chapman
 http://www.maths.ex.ac.uk/~rjc/rjc.html
  "`Well, I'd already done a PhD in X-Files Theory at UCLA, ...'"
    Greg Egan, _Teranesia_


Sent via Deja.com http://www.deja.com/
Before you buy.