Product systems over right-angled Artin semigroups (Q2781371)

Let \(S\) be a countable semigroup and let \(\mathcal G\) be a tensor groupoid. A product system over \(S\) taking values in \(\mathcal G\) is a pair \((Y,\alpha)\) in which \(Y\) is a collection \((Y_s)_{s\in S}\) of objects in \(\mathcal G\), and \(\alpha\) is a collection \((\alpha_{s,t})_{s,t\in S}\) of isomorphisms \(\alpha_{s,t}\colon Y_s\otimes Y_t\to Y_{st}\) so that \(\alpha_{rs,t}\) is identified with \(\alpha_{r,st}\) under a given natural equivalence. Let \(\Gamma\) be a non-directed graph without loops or multiple edges with countable vertex set \(A\). Let \(\mathbb{F}_A\) be the free group on \(A\), and let \(*_\Gamma\mathbb{Z}\) be the graph product of \(A\) copies of \(\mathbb{Z}\); i.e., the quotient of \(\mathbb{F}_A\) by the normal subgroup generated by all the commutators \([a,b]\) whenever \(a,b\) are connected in \(\Gamma\). Let \(\mathbb{F}_A^+\) be the subsemigroup of \(\mathbb{F}_A\) generated by \(A\) and let \(P\) be the resulting subsemigroup of \(*_\Gamma\mathbb{Z}\). The semigroup \(P\) is called a right-angled Artin semigroup. If the graph \(\Gamma\) has no edges, then \(P\) is the free semigroup on \(A\), whereas if the graph is complete, then \(P\) is free Abelian. The authors define, given a well ordering of \(A\), a specific section \(\delta\colon P\to\mathbb{F}_A^+\) of the quotient map \(\pi\colon\mathbb{F}_A^+\to P\). Fix a collection \((X_a)_{a\in A}\) of objects in \(\mathcal G\) and define \(X_w:=X_{w_1}\otimes\cdots\otimes X_{w_{l(w)}}\) for \(w\in\mathbb{F}_A^+\). Suppose \(T=(T_{a,b})\) is a collection of isomorphisms \(T_{a,b}\colon X_a\otimes X_b\to X_b\otimes X_a\) such that \(T_{a,b}^{-1}=T_{b,a}\) whenever \(a,b\) are connected, and, whenever \(a,b,c\) form the vertices of a triangle in \(\Gamma\), \((T_{b,c}\otimes 1_a)(1_b\otimes T_{a,c})(T_{a,b}\otimes 1_c)=(1_c\otimes T_{a,b})(T_{a,c}\otimes 1_b)(1_a\otimes T_{b,c})\). The authors prove several interesting results. They characterize all product systems which take values in \(\mathcal G\). They also obtain necessary and sufficient conditions under which a collection of \(k\) \(1\)-graphs form the coordinate graphs of a \(k\)-graph. We quote here: Theorem: There is a unique product system \((Y,\alpha)= (Y^T,\alpha^T)\) over \(P\) taking values in the tensor groupoid \(\mathcal G\) such that \(Y_t=X_{\delta(t)}\) for every \(t\in P\), \(\alpha_{s,t}=1_{\delta(st)}\) if \(\delta(st)=\delta(s)\delta(t)\), and \(\alpha_{\pi(a),\pi(b)}=T_{a,b}\) if \(a,b\) are connected and \(a>b\).