Re: Definition of a algebra generated by a set ?

On Jun 10, 3:21 pm, Marc Mertens <marc.mert...@xxxxxxxxxxx> wrote:
Hello,

I'm reading a book about mathematical physics and have a lot of
troubles understanding some examples in this book.
First they define a algebra as vector space A over a field K with a
product . defined that satisfies A.(bB+cC)=bA.B+cA.C and (bB+cC).A=bB.A
+cC.A, the algebra is commutative (associative) if the product is
associative (commutative). No problem here. However at one moment they
start to give examples like:
Let Q be the associative algebra over R generated by the four elements
{1,i,j,k} satisfying i^2=j^2=k^2=-1, ij=k,jk=i,ki=j, 1^2=1,1i=i,1j=j,1k=k
which leads to the quaternions.

This multiplication table among these four elements defines
the algebra, and in the process defines what "multiplication"
will mean for these four objects. Nothing is implied about
how you might represent them. It's just a set of symbols with
a "multiplication" table.

Let V be a real vector space with the inner product u.v and e1,...en a
orthonormal basis, the Clifford algebra associated with the inner product
space is the associative algebra generated by 1,e1,..en with product
rules eiej+ejei=2gij 1 (gij=ei.ej)
Now I understand that you can create the quaternions as a set of tuples
of real numbers with the correct multiplication, but I fail to see how
you can take just 4 elements {1,i,j,k} and then generate a vector space
over a field together with a product (unless you define 1=(1,0,0,0) , i=
(0,1,0,0), j=(0,0,1,0) and k=(0,0,0,1) and define addition in the usual
way (product is of course a little bit more complex), but you can hardly
call this generating.

That happens to be a representation which, with "multiplication"
suitably defined, gives the same algebra. But the multiplication
table itself is the algebra, regardless of the representation.

Also when I look up Clifford algebras on the internet, It is never
defined as the associative algebra generated by ....

There seems to be a pattern here where given a set of undefined elements
{1,a1,...an) you go on and generate a algebra out of this. So how do you
proceed to define a vector space (definition of the sum, multiplication
by a scalar (what is the field used) and the algebra product. I looked
really hard but could not find such a definition in the book neither in
other books I have on algebra neither on the internet. Either I'm missing
something obvious or my book is based on some loosy mathematical
principles.


I'm not familiar with Clifford algebras, but this looks like
group theory, and you seem to be hung up on the distinction between
an abstract group and a representation of that group (there may
be more than one natural representation). I'd suggest looking
under keywords like "group theory" and "representation", such
as here:
http://en.wikipedia.org/wiki/Group_representation

- Randy

.

Relevant Pages

  • Exterior algebras as Hopf algebras
    ... It's well-known that the exterior algebra on a vector space is a Hopf ... with the antipode given by the Hodge star. ...
    (sci.math)
  • Re: Representation theory basic questions..
    ... >> finite-dimensional division algebra over C? ... then the whole algebra CG would be infinite-dimensional over ... But why is D_i a vector space over C? ... direct product of matrix rings over division rings D_i, ...
    (sci.math)
  • Re: algebraic number
    ... any of them is a root of a polynomial with it's cofficient are ... The roots of polynomials with algebraic number coefficients are ... when considered as a vector space over Q. Conversely, ... contained in a finite dimensional extension K of Q, then a is algebra ...
    (sci.math)
  • Definition of a algebra generated by a set ?
    ... I'm reading a book about mathematical physics and have a lot of ... the algebra is commutative if the product is ... Let Q be the associative algebra over R generated by the four elements ... Let V be a real vector space with the inner product u.v and e1,...en a ...
    (sci.math)
  • Re: Definition of a algebra generated by a set ?
    ... algebra as vector space A over a field K with a product. ... Let Q be the associative algebra over R generated by the four elements ... That happens to be a representation which, ...
    (sci.math)