Semi-group algebra

2010 Mathematics Subject Classification: Primary: 20E35 [MSN][ZBL]

An algebra $\Phi(S)$ over a field $\Phi$ with a basis $S$ that is at the same time a multiplicative semi-group. In particular, if $S$ is a group, one obtains a group algebra. If the semi-group $S$ contains a zero, this zero is usually identified with the zero of the algebra $\Phi(S)$. The problem of describing all linear representations of a semi-group $S$ over a field $\Phi$ (cf. Linear representation; Representation of a semi-group) is equivalent to that of describing all representations of the algebra $\Phi(S)$. The importance of semi-group algebras in the theory of semi-groups is the possibility they offer of utilizing the richer tools of the theory of algebras to study linear representations of semi-groups. An example of this kind of result is: The algebra $\Phi(S)$ of a finite semi-group $S$ is semi-simple if and only if all linear representations of the semi-group $S$ over the algebra $\Phi$ are reducible.

More precisely, let $S$ be a semi-group and $\Phi$ a field. Consider the vector space $V$ of all formal finite sums $V=\{\sum_{s\in S} a_s s\}$, i.e. the vector space over $\Phi$ with basis $S$. The semi-group multiplication $(s,t)\mapsto st$ extends linearly to define an algebra structure on $V$. This is the semi-group algebra $\Phi[S]$.
If $S$ is a semi-group with zero $z$, the subspace $\Phi_z$ is an ideal in $\Phi[S]$ and the contracted semi-group algebra $\Phi_0[S]$ is the quotient algebra $\Phi_0[S] = \Phi[S]/\Phi_z$.
For an inverse semi-group (cf. Inversion semi-group) one has the following analogue of Maschke's theorem (cf. Group algebra). The semi-group algebra $\Phi[S]$ of a finite inverse semi-group $S$ is semi-simple if and only if the characteristic of $\Phi$ is zero or is a prime that does not divided the order of any sub-semi-group of $S$.