Spahn group algebra (Rev #2, changes)

Take (i.e. the space containing just those morphisms which send only finitely many elements of a group,$G$ , as not labelling to a basis for a vector space over a field$k0$ k 0 , ) then equipped that with the multiplication defined by convolution of the functions group is will called extend to give that vector space the structure of analgebra?group algebra of $A$ over $G$ . over With pointwise addition$\mathrm{<span><del\; class="diffmod">\; k</del><ins\; class="diffmod">\; K</ins></span>}[G]$ k K[G] . This is usually an denoted associative$\mathrm{<span><del\; class="diffmod">\; k</del><ins\; class="diffmod">\; A</ins></span>}[G]$ k[G] A . -algebra.

If Equivalently instead one can think of a field we used the ring multiplication of inthe integers, following way$\mathbb{Z}$, it is usual to call the result the group ring. For this we take the free abelian group on the set of elements of the group and extend the multiplication to give a ring structure on the result. This is usually denoted $\mathbb{Z}[G]$.

As Take a further generalisation, the field elements can of be a replaced group, by any commutative ring.$G$, as labelling a module basis and define the multiplication of two elements by $x_g \cdot x_h:=x_{gh}$.

If The we notion denote of by group algebra is a special case of that of a$e_g$category algebra? , . the generator corresponding to$g\in G$, then an arbitrary element of $k[G]$ can be written as $\sum_{g\in G}n_ge_g$ where the $n_g$ are elements of $k$, and only finitely many of them are non-zero.

The multiplication is then by what is sometimes called a ‘convolution’ product, that is,

Thoughts Properties