The algebraic small dimension lemma with an anti-homomorphism on semigroups
The main result of the paper is an extension of the ``algebraic small dimension'' lemma recently proved by \textit{H. Stetkær} [Aequationes Math. 95, No. 2, 281--299 (2021; Zbl 1467.39018)] as a tool for solving d'Alembert's functional equation. Let \(S\) be a semigroup, \(\mathbb{K}\) be an algebraically closed field of characteristic different from \(2\), \(\mathbb{K}^*\) denotes the multiplicative group \(\mathbb{K}\setminus \{0\}\) and \(\mathbb{K}[S]\) is the vector space over \(\mathbb{K}\) of \(\mathbb{K}\)-valued functions on \(S\). With \(\pi\) we denote a representation of \(S\) on the vector space \(V\) and the dimension of \(\pi\) is the dimension of the space \(V\). Moreover, \(\tau:S \to S\) denotes an involutive anti-homomorphism of \(S\) (i.e., \(\tau(xy)=\tau(y)\tau(x)\)). The above mentioned theorem by \textit{H. Stetkær} is the following: Theorem. Let \(\pi\) be a finite-dimensional irreducible representation of \(S\) on \(V\) such that \(\mathrm{dim} (V)\ge 2\) and let \(\mu :S \to \mathbb{K}^*\) be a homomorphism satisfying \(\mu(x\tau(x))=1\) for all \(x \in S\). Suppose that there exist a non-zero vector \(v_0 \in V\) and a function \(g \in \mathbb{K}[S]\) such that \[ [\pi(x)+\mu(x)\pi(\tau(x))]v_0=2g(x)v_0, \quad x\in S. \] Then \(\mathrm{dim} (V)=2\) and \(\mu(x)\pi(\tau(x))=\mathrm{adj}\, (\pi(x))\) for all \(x \in S\), where \(\mathrm{adj}\, A=(\mathrm{tr} \, A)I-A\). The extension proved here replaces the involutive anti-homomorphism \(\tau\) with an anti-homomorphism \(\psi\) and assumes that the function \(g\) is \(\psi\)-even, i.e., \(\mu(x)g(\psi(x))=g(x)\). Moreover, it is proved that \(g=\frac{1}{2}\chi_{\pi},\) where \(\chi_{\pi}:=tr(\pi(x))\), \(g(xy)g(yx)\) and \(g\) is a non-zero solution of d'Alembert \(\mu\)-functional equation \[ g(xy)+\mu(y)g(x\psi(xy))=2g(x)g(y), \quad x,y \in S.\] This result is then used for the investigation of the relation between the d'Alembert \(\mu\)-functional equation and the Wilson \(\mu\)-functional equation \[ w(xy)+\mu(y)w(x\psi(xy))=2w(x)g(y), \quad x,y \in S.\] In the last section the d'Alembert \(\mu\)-functional equation and the Wilson \(\mu\)-functional equation are solved when the domain is a compact group.
scientific article; zbMATH DE number 7369078
scientific article