Re: [Axiom-developer] algebras <=> groups

[Camm Maguire]

Greetings!  Bertfried already said it -- groups and algebras have an
intuitive connection through identifications of their multiplications,
but neither set is strictly contained within the other along this line
-- i.e. a famous non-associative algebra is the octonions, whereas a
group need not have a secondary combination rule which is not simply
an iteration of its multiplication (as is the case with matrix algebra
addition and multiplication).

This having been said, there are two enormous areas of practical
overlap:  

1) representation theory -- i.e. the categorization of the eigenspaces
   of an operator via its known multiplication rules with the elements
   of a 'symmetry' group

2) Lie groups, which are 'generated' by exponentiating the additive
   action of an (usually matrix vector) algebra.

It would be hard to overstate the significance of being able to
separate eigen solutions of a complex and intractable dynamic operator
from 'symmetry' arguments alone.

If one wishes to consider representation as opposed to group theory
per se, I believe the conditions for the existence of matrix algebra
representations of group over some vector space are quite broad,
perhaps even universal in the case of a finite group.

Take care,

root <address@hidden> writes:

> Bill,
> 
> The thought was triggered by this quote:
> 
>   Groups and algebras have this in common, that they each employ a process
>   of multiplication that is associative but not necessarily commutative.
>   The problem that immediately suggests itself, then, is to examine the
>   connection between the two theories. This connection is quite intimate,
>   for connected with every finite group there is an associated algebra 
>   called the group algebra. It is sometimes called after Frobenius, who
>   published a number of papers exploring this problem, the Frobenius
>   algebra of the group.
> 
> Littlewood, D.E. "The Skeleton Key of Mathematics" p107
> 
> t
> 
> 
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-- 
Camm Maguire                                            address@hidden
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