Talk:Graded algebra
From Wikipedia, the free encyclopedia
Here a link should be included to Clifford-algebras and to matrix representations of these algebras.
The article Graded algebra twice uses the five-character notation "∧" (whose characters are ampersand, a, n, d, and semicolon). For some users, this produces a square box, like that used for "character not in font". Presumably "^" cannot be used instead; if not, IMO an explanatory note, or simple, reversible instructions for overcoming this (probably in an article like Wikipedia:problems displaying special characters) are needed. --Jerzy 20:54, 2003 Dec 3 (UTC)
This article is about the exterior algebra, not about graded rings/algebras in general. I propose that most of the material should go to exterior algebra; and that this be redirected to a new graded ring page.
Charles Matthews 12:54, 16 Dec 2003 (UTC)
- I agree completely, although I think this page should remain here with the proper definition instead of just a redirect. I'll see if I can put something together. -- Fropuff 07:00, 2004 Feb 15 (UTC)
I've replaced the content of this page with what I believe is the correct definition. The old content is below. Much of this should be moved to the page on exterior algebra. -- Fropuff 07:57, 2004 Feb 15 (UTC)
A graded algebra is an algebra generated when an outer product (wedge product) is defined in a vector space Vn over the scalars F.
The outer product generates a set of new entities: the UNIQ4706244f761f2742-math-00000E61-QINU-vectors. As they are obtained by the outer product of k linearly independent vectors, they are said to be of step or grade k. k-vectors are vectors in nature, so any k-vector is a member of a vector subspace known as subspace of grade k, denoted by ∧kVn. Each of this has a dimension of C(n,k) where C(n,k) is the binomial coefficient.
Vectors are said to have step 1, so
with dimension n, and scalars are considered as the 0-step vector space ∧0Vn, and have dimension 1. The n-vectors also generate a 1-dimensional vector space, so all n-vectors are scalar multiples of a arbitrarily-chosen unitary n-vector. Given that essentially behave as scalars, they are often referred to as pseudoscalars. Similarly, (n − 1)-vectors are also called pseudovectors.
In order to achieve closure, all these spaces are combined by considering the direct sum of all of them. The resulting space is a new vector space called the graded algebra:
and we call multivectors to its elements.
The dimension of the graded algebra is 2n, and the structure of the grades subspaces is that of the Pascal triangle (see binomial coefficient).
Can it be explained more why Clifford algebras are Z2 graded and not N graded. After all, all grades up to the grade of the pseudoscalar exists.
- Clifford multiplication only respects the Z/2Z grading. For example, the product of two vectors is, in general, the sum of a scalar and a bivector, which is not homogeneous in what you would call the N grading. In the exterior algebra, however, the product of two vectors (degree 1) is always a bivector (degree 2 = 1 + 1). -- Fropuff 18:55, 2004 Jun 2 (UTC)
I don't understand what has been added about discrete valuation rings (certainly filtered, I see), and the monoid ring bit. This all requires more leisurely exposition; and I think starting with G-graded in general is too steep, anyway. Charles Matthews 15:03, 4 Jul 2004 (UTC)
Why does Hilbert Function redirect here, if this says almost nothing about Hilbert Functions? There should be a page about Hilbert Functions and Hilbert Polynomials. 24.239.168.205 21:22, 24 March 2006 (UTC)
[edit] Graded rings reorganization
I have reorganized the page to reflect the fact that one can have a graded algebra over a graded ring, so both A-multiplication and E-multiplication respect the gradations. Even though most graded A-algebras in the wild are graded with respect to the trivial gradation on A, these more general sorts of graded algebras do occur from time to time. Silly rabbit 07:43, 8 June 2006 (UTC)
- it should be stated in the given examples what exactly are the graded rings. i think like: the algebra of polinomials over ring K is a graded algebra over the trivial gradation of K. --itaj 23:06, 8 October 2006 (UTC)
