Automorphisms of Leavitt path algebras: Zhang twist and irreducible representations (Q6564601)

The study of automorphisms of an algebra is a powerful tool for describing the symmetries of its underlying algebraic structure. But to determine the full automorphism group of a noncommutative algebra use to be an extremely difficult problem to deal with. In the particular case of Leavitt path algebras [\textit{G. Abrams} and \textit{G. Aranda Pino}, J. Algebra 293, No. 2, 319--334 (2005; Zbl 1119.16011); \textit{P. Ara} et al., Algebr. Represent. Theory 10, No. 2, 157--178 (2007; Zbl 1123.16006)], there are a few constructions, mainly inspired in previous results of graph \(C^*\)-algebras, their analytic siblings. Basically inspired in Cuntz's seminal work [\textit{J. Cuntz}, in: Quantum fields - algebras, processes, Proc. Symp., Bielefeld 1978, 187--196 (1980; Zbl 0475.46046)], \textit{J. E. Avery} et al. [Proc. Edinb. Math. Soc., II. Ser. 61, No. 1, 215--249 (2017; Zbl 1395.46048); \textit{R. Johansen} et al., Groups Geom. Dyn. 14, No. 3, 1043--1075 (2020; Zbl 1478.46056)] gave a method for constructing in some special finite graphs. \textit{S. Kuroda} and \textit{Tran Giang Nam} [J. Noncommut. Geom. 17, No. 3, 811--834 (2023; Zbl 07761257)] extended these results to arbitrary graphs, and produced Anick type automorphisms for Leavitt path algebras.\N\NIn the paper under review, the authors extend the results of [Zbl 0475.46046; Zbl 07761257], giving a construction of graded automorphisms of Leavitt path algebras of arbitrary graphs in terms of general linear groups over corners of these algebras (Theorem 2.2 and Corollary 2.3). In the specific case of Leavitt algebras (i.e., Leavitt path algebras of the \(n\)-petals rose graph \(R_n\)), they give a complete description of all (graded) automorphisms of \(L_K(R_n)\) in terms of \(\mathrm{GL}_n(L_K(R_n))\) (Propositions 2.6 and 2.7); also, they provide an alternative description in terms of \(U(L_K(R_n))\) (Corollary 2.11).\N\NA first application of these results is the construction of Zhang twists [\textit{M. Artin} et al., Invent. Math. 106, No. 2, 335--388 (1991; Zbl 0763.14001); \textit{J. J. Zhang}, Proc. Lond. Math. Soc. (3) 72, No. 2, 281--311 (1996; Zbl 0852.16005)]; these twists play a critical role in the interaction between noncommutative algebra with noncommutative projective geometry [\textit{M. Artin} and \textit{J. J. Zhang}, Adv. Math. 109, No. 2, 228--287 (1994; Zbl 0833.14002)]. In the paper under review, the authors initiate the study of Zhang twists for Leavitt path algebras as a tool for developing a geometric theory of Leavitt path algebras. In this study, the most surprising fact is that for any graph \(E\), \(L_K(E)\) embeds in any Zhang twist \(L_K(E)^{\varphi_P}\) associated to \(\varphi_P\in \text{Aut}_K^{\text{gr}}(L_K(E))\) in Corollary 2.3 (Proposition 3.2). Applied to Leavitt algebras, this result allows to characterize for which \(\varphi_P\) we have ``rigidity'' of the algebra \(L_K(R_n)\) (i.e., \(L_K(R_n)\cong L_K(R_n)^{\varphi_P}\)).\N\NAnother application of the results in Section 2 is the study of irreducible representations of Leavitt algebras. Initial results on this direction appear in [\textit{P. Ara} and \textit{K. M. Rangaswamy}, J. Algebra 417, 333--352 (2014; Zbl 1304.16002); \textit{X.-W. Chen}, Forum Math. 27, No. 1, 549--574 (2015; Zbl 1332.16006); Zbl 07761257]. In the paper under review, new classes of simple modules over Leavitt algebras are defined and classified (Theorems 4.2 and 4.5).