Conjugation in semigroups and finite dimensional algebras (Q794775)

Let \(S\) be a finite semigroup. Then, for \(s\in S\), the idempotent power of \(s\) is denoted by \(e_ s\) and \(e_ ss\) is denoted by \(s^*\). Two elements \(s\) and \(t\) in \(S\) are said to be conjugate if there exist \(p,q\in S\) such that \(s^*=pq\), \(t^*=qp\). It is shown that \(s\) and \(t\) are conjugate if and only if there are mutually inverse elements \(x,y\in S\) such that \(s^*=xt^*y\), \(t^*=ys^*x\) and that the number of conjugacy classes of \(S\) is the same as the number of irreducible representations of \(S\) over the complex numbers, and that \(s\) and \(t\) are conjugate if and only if \(h(s)=h(t)\) for all irreducible characters \(h\) of \(S\). These ideas are generalized to finite dimensional complex algebras \(A\). Let \(T(A)\) be the subspace generated by all commutators \(ab-ba\), \(a,b\in A,\) and let \(J(A)\) denote the Jacobson radical of \(A\). Then it is shown that \(h(a)=h(b)\) for all irreducible characters of \(A\) if and only if \(a-b\in T(A)+J(A)\). Thus the number of nonisomorphic simple \(A\)-modules is \(\dim A/(T(A)+J(A))\).