Groups acting on tensor products. (Q392434)

Let \(U\), \(V\) and \(W\) be Abelian groups. A function \(\circ\colon U\times V\to W\) is called a bimap if it is surjective and \((u+u')\circ v=u\circ v+u'\circ v\) and \(u\circ(v+v')=u\circ v+u\circ v'\) for all \(u,u'\in U\) and \(v,v'\in V\). The autotopisms of a bimap \(\circ\) are the elements \((f,g,h)\in\Aut(U)\times\Aut(V)\times\Aut(W)\) such that \(uf\circ gv=(u\circ v)^h\) for all \(u\in U\) and \(v\in V\) (the actions are, respectively, right, left, and exponential); the set \(\Aut(\circ)\) of all autotopisms is a group.NEWLINENEWLINE The authors are interested in giving a description of \(\Aut(\circ)\) and computing this in the case where \(U,V\) and \(W\) are vector spaces. The ring of adjoints of a bimap \(\circ\) is defined to be \(\text{Adj}(\circ):=\{(f,g)\in\text{End}(U)\times\text{End}(V)\mid uf\circ v=u\circ gv\text{ for all }u\in U\text{ and }v\in V\}\) which can be shown to be the largest ring \(R\) acting faithfully on \(U\) and \(V\) such that \(\circ\colon U\times V\to W\) factors through the tensor product \(U\otimes_RV\) [see \textit{J. B. Wilson}, J. Algebra 322, No. 8, 2642-2679 (2009; Zbl 1205.20019)].NEWLINENEWLINE The main results of the present paper are the following. For any bimap \(\circ\colon U\times V\to W\) the autotopism group \(\Aut(\circ)\) embeds in \(N(\text{Adj}(\circ)):=\{(f,g)\in\Aut(U)\times\Aut(V)\mid\text{Adj}(\circ )^{(f,g)}=\text{Adj}(\circ)\}\). Moreover the following are equivalent: this embedding is an isomorphism; the linear map \(U\otimes_{\text{Adj}(\circ)}V\to W\) is an isomorphism; \(\circ\) is a tensor product. Similarly, if \(\circ\colon V\times V\to W\) is a nondegenerate alternating or symmetric bimap, then \(\Psi\text{Isom}(\circ):=\{f\mid(f,f,g)\in\Aut(\circ)\}\) can be embedded naturally in \(N^*(\text{Adj}(\circ)):=\{f\mid(f,f)\in N(\text{Adj}(\circ))\}\), and there is equality precisely when \(\circ\) is a symmetric or exterior tensor product. Some of these results have been applied in a project of the authors and E. A. O'Brien to construct generators for the automorphism groups of \(p\)-groups of class \(2\) and exponent \(p\).