A talented monoid view on Lie bracket algebras over Leavitt path algebras
Abstract
In this paper, we study properties such as simplicity, solvability and nilpotency for Lie bracket algebras arising from Leavitt path algebras, based on the talented monoid of the underlying graph. We show that graded simplicity and simplicity of the Leavitt path algebra can be connected via the Lie bracket algebra. Moreover, we use the Gelfand–Kirillov dimension for the Leavitt path algebra for a classification of nilpotency and solvability.
Communicated by P. Ara
References
- 1. , Leavitt path algebras: The first decade, Bull. Math. Sci. 5(1) (2015) 59–120. Crossref, Web of Science, Google Scholar
- 2. , Leavitt Path Algebras,
Lecture Notes in Mathematics , Vol. 2191 (Springer Verlag, 2017). Crossref, Google Scholar - 3. , Simple Lie algebras arising from Leavitt path algebras, J. Pure Appl. Algebra 219 (2012) 2302–2313. Crossref, Web of Science, Google Scholar
- 4. , The Leavitt path algebra of a graph, J. Algebra 293(2) (2005) 319–334. Crossref, Web of Science, Google Scholar
- 5. , Finite-dimensional Leavitt path algebras, J. Pure Appl. Algebra 209(3) (2007) 753–762. Crossref, Web of Science, Google Scholar
- 6. , On the simplicity of the Lie algebra of a Leavitt path algebra, Commun. Algebra 44 (2016) 4114–4120. Crossref, Web of Science, Google Scholar
- 7. , Leavitt path algebras of finite Gelfand-Kirillov dimension, J. Algebra Appl. 11(6) (2012) 1250225. Link, Web of Science, Google Scholar
- 8. , Towards a K-theoretic characterization of graded isomorphisms between Leavitt path algebras, J. K-Theory 14(2) (2014) 203–245. Crossref, Google Scholar
- 9. L. G. Cordeiro, D. Gonçalvez and R. Hazrat, The talented monoid of a directed graph with applications to graph algebras, preprint (2020), arXiv:2003.09911. Google Scholar
- 10. , Normal Monoids and Factor Monoids of Commutative Monoids (The University of Michigan, 1963). Google Scholar
- 11. ,
Lie groups, Lie algebras, and representations , in Quantum Theory for Mathematicians (Springer, New York, 2013), pp. 333–366. Crossref, Google Scholar - 12. , The graded Grothendieck group and the classification of Leavitt path algebras, Math. Ann. 355 (2013) 273–325. Crossref, Web of Science, Google Scholar
- 13. , The talented monoid of a Leavitt path algebra, J. Algebra 547 (2020) 430–455. Crossref, Web of Science, Google Scholar
- 14. , Graphs with disjoint cycles classification via the talented monoid, J. Algebra 593 (2022) 319–340. Crossref, Web of Science, Google Scholar
- 15. , Introduction to Lie Algebras and Representation Theory, Vol. 9 (Springer Science & Business Media, 2012). Google Scholar
- 16. , Growth of Algebras and Gelfand-Kinillow Dimension,
Graduate studies in Mathematics , Vol. 22, Revised edn. (American Mathematical Society, Providence, RI, 2000). Google Scholar - 17. , Simple Lie algebras arising from Steinberg algebras of Hausdorff ample groupoids, J. Algebra 595 (2022) 194–215. Crossref, Web of Science, Google Scholar
- 18. , Lie solvable Leavitt path algebras, J. Algebra Appl. 21 (2021), https://doi.org/10.1142/S0219498822502036. Web of Science, Google Scholar
- 19. , The Jordan-Hölder Theorem for monoids with group action, J. Algebra Appl., https://doi.org/10.1142/S0219498823500883. Web of Science, Google Scholar
- 20. , Solvability and nilpotency of Novikov algebras, Commun. Algebra 48(12) (2020) 5412–5420. Crossref, Web of Science, Google Scholar
- 21. , Every graded ideal of a Leavitt path algebra is graded isomorphic to a Leavitt path algebra, Bull. Austral. Math. Soc. 105 (2021) 248–256. Crossref, Web of Science, Google Scholar
- 22. L. Vas, Graded irreducible representations of Leavitt path algebras: A new type and complete classification, preprint (2022), arXiv:2201.02446. Google Scholar