Representations and the reduction theorem for ultragraph Leavitt path algebras (Q2025127)

This paper deals with representations of ultragraph Leavitt path algebras via branching systems. One of the key notions is that of a partial action of a group \(G\) on a set \(\Omega\). In this kind of actions there is family \(\{D_t\}_{t\in G}\) of subsets \(D_t\subset\Omega\) indexed by \(t\in G\) such that each \(t\in G\) acts bijectively on \(D_{t^{-1}}\) producing elements of \(D_t\) (plus a compatibility condition). If \(\Omega\) turns out to be a ring, then the \(D_t\) are required to be ideals and the action of \(t\) on \(D_{t^{-1}}\) to \(D_t\) gives an isomorphism. Let us denote these isomorphisms \(D_{t^{-1}}\to D_t\) by \(\alpha_t\) (\(t\in G\)). If we have a partial action of a group \(G\) on a ring \(A\) then one can construct something which resembles the group ring: the so called partial skew-ring \(A\star_{\alpha} G\). By using partial skew ring theory, it is proved the reduction theorem for these algebras. The idea of the reduction theorem is that starting with an arbitrary nonzero elements of an ultragraph Leavitt path algebras and multiplying smartly on the left and on the right by paths or ghost paths, you finally get either a nonzero multiple of a vertex or a polynomial in a cycle without exits. The reduction theorem is used to show that ultragraph Leavitt path algebras are semiprime and to completely describe faithfulness of the representations arising from branching systems, in terms of the dynamics of the branching systems. The work also studies permutative representations and provides a sufficient criteria for a permutative representation of an ultragraph Leavitt path algebra to be equivalent to a representation arising from a branching system. This criteria is applied to describe a class of ultragraphs for which every representation (under a mild condition) is permutative and has a restriction that is equivalent to a representation arising from a branching system.