Abstract
We use a representation of a graded twisted tensor product of K[x] with K[y] in \(L(K^{\mathbb {N}_{0}})\) in order to obtain a nearly complete classification of these graded twisted tensor products via infinite matrices. There is one particular example and three main cases: quadratic algebras classified in Conner and Goetz (J. Noncommut. Geom. 15(1), 41–78, 2021), a family called A(n, d, a) with the n + 1-extension property for n ≥ 2, and a third case, not fully classified, which contains a family B(a, L) parameterized by quasi-balanced sequences.
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References
Arce, J.: Representations of twisted tensor products. arXiv:1505.01232[math.RA]
Arce, J., Guccione, J.A., Guccione, J.J., Valqui, C.: Twisted tensor products of Kn with Km. Algebr. Represent. Theory 22 (6), 1599–1651 (2019). https://doi.org/10.1007/s10468-018-9833-1. MR4034796
Cap, A., Schichl, H., Vanžura, J.: On twisted tensor products of algebras. Comm. Algebra 23(12), 4701–4735 (1995). https://doi.org/10.1080/00927879508825496. MR1352565
Cibils, C.: Non-commutative duplicates of finite sets. J. Algebra Appl. 5(3), 361–377 (2006). https://doi.org/10.1142/S0219498806001776. MR2235816
Conner, A., Goetz, P.: The Koszul property for graded twisted tensor products. J. Algebra 513, 50–90 (2018). https://doi.org/10.1016/j.jalgebra.2018.07.030. MR3849878
Conner, A., Goetz, P.: Classification, Koszulity and Artin-Schelter regularity of certain graded twisted tensor products. J. Noncommut. Geom. 15(1), 41–78 (2021). https://doi.org/10.4171/jncg/395. MR4248207
Guccione, J.A., Guccione, J.J., Valqui, C.: Twisted planes. Comm. Algebra 38(5), 1930–1956 (2010). https://doi.org/10.1080/00927870903023105. MR2642035
Guccione, J.A., Guccione, J.J., Valqui, C.: Non commutative truncated polynomial extensions. J. Pure Appl. Algebra 216(11), 2315–2337 (2012). https://doi.org/10.1016/j.jpaa.2012.01.021. MR2927170
Jara, P., López Peña, J., Navarro, G., Ştefan, D.: On the classification of twisting maps between Kn and Km. Algebr. Represent. Theory 14 (5), 869–895 (2011). https://doi.org/10.1007/s10468-010-9222-x. MR2832263
López Peña, J., Navarro, G.: On the classification and properties of noncommutative duplicates. K-Theory 38(2), 223–234 (2008). https://doi.org/10.1007/s10977-007-9017-y. MR2366562
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We thank the anonymous referee for the thorough revision and numerous helpful suggestions.
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Presented by: Vyjayanthi Chari
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Christian Valqui was supported by PUCP-DGI-2019-1-0015.
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Bances, R., Valqui, C. On the Classification of Graded Twisted Planes. Algebr Represent Theor 26, 1231–1270 (2023). https://doi.org/10.1007/s10468-022-10131-8
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DOI: https://doi.org/10.1007/s10468-022-10131-8