Abstract
We develop a comprehensive theory of algebras over a field which are locally both finite dimensional and central simple. We generalize fundamental concepts of the theory of finite dimensional central simple algebras, and introduce supernatural matrix algebras, the supernatural degree and matrix degree, and so on. We define a Brauer monoid, whose unique maximal subgroup is the classical Brauer group, and show that once infinite dimensional division algebras exist over the field, they are abundant.
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Abrams, G.: Leavitt path algebras: the first decade. Bull. Math. Sci. 5(1), 59–120 (2015)
Albert, A.A.: Structure of algebras. AMS Coll Publ XXIV (1961)
Atiyah, M. F., MacDonald, I. G.: Introduction to commutative algebra perseus. Books (1969)
Auel, A., Brussel, E., Garibaldi, S., Vishne, U.: Open problems on central simple algebras. Transform. Groups 16(1), 219–264 (2011)
Barsotti, I.: Noncountable normally locally finite division algebras. Proc. Amer. Math. Soc. 8, 1101–1103 (1957)
Brauer, R.: On algebras which are connected with the semisimple continuous groups. Ann. Math. 38, 857–872 (1937)
Bar-On, T., Gilat, S., Matzri, E., Vishne, U.: The algebra of supernatural matrices. Submitted.
Deo, T. T., Bien, M. H., Hai, B. X.: On the radicality of maximal subgroups in GLn(D). J. Algebra 365, 42–49 (2012)
Deo, T. T., Bien, M. H., Hai, B. X.: On weakly locally finite division rings. Acta Math. Vietnam. 44, 553–569 (2019)
Hai, B. X., Deo, T. T., Bien, M. H.: On subgroups in division rings of type 2. Stud. Sci. Math. Hung. 49(4), 549–557 (2012)
Gille, P. H., Szamuely, T.: Central simple algebras and galois cohomology. Cambridge Studies in Advanced Mathematics 101 (2006)
Glimm, J.G.: On a certain class of operator algebras. Trans. Amer. Math. Soc. 95, 318–340 (1960)
Haile, D. E.: The Brauer monoid of a field. J. Algebra 81(2), 521–539 (1983)
Haile, D. E., Rowen, L. H.: Weakly Azumaya algebras. J. Algebra 250(1), 134–177 (2002)
Jacobson, N.: Structure of rings, colloquium publications. XXXVII AMS (1956)
Jacobson, N.: Finite Dimensional Division Algebras. Springer, Berlin (1996)
Ježek, J., Kepka, T., Němec, P.: Commutative semigroups that are nil of index 2 and have no irreducible elements. Math. Bohem. 133(1), 1–7 (2008)
Kaplansky, I.: Rings with a polynomial identity. Bull. Amer. Math. Soc. 54, 575–580 (1948)
Knus, M. -A., Merkurjev, A., Rost, M., Tignol, J. -P.: The book of involutions, Colloquium Publications, vol. 44 American Mathematical Society (1998)
Köthe, G.: Schiefkörper unendlichen Range über dem Zentrum. Math. Ann. 105, 15–39 (1931)
Lam, T. Y.: Lectures on Modules and Rings, LNM 189, Springer-Verlag (1999)
Lam, T. Y.: Multiples, modules with isomorphic matrix rings - a survey, monographie 35 de L’Enseignement Mathématiqueu, Genève (1999)
Maltcev, V., Mazorchuk, V.: Presentation of the singular part of the Brauer monoid. Math. Bohem. 132(3), 297–323 (2007)
Matzri, E: Symbol length in the Brauer group of a field. Trans. Amer. Math. Soc. 368, 413–427 (2016)
McCrimmon, K.: Deep matrices and their Frankenstein actions, Non-associative algebra and its applications. Lect. Notes Pure Appl. Math. 246, 261–274 (2006). Chapman & Hall/CRC, Boca Raton FL
Novikov, B. V.: On the Brauer monoid (Russian). Mat. Zametki 57(4), 633–636 (1995). translation in Math. Notes 57(3–4), (1995), 440–442
Ribes, L., Zalesskii, P.: Profinite Groups. Springer, Berlin (2010)
Rowen, L.H.: Ring Theory, vol. I. Academic Press, Cambridge (1988)
Saltman, D.: Lectures on Division Algebras, CBMS Regional Conference Series in Mathematics 94, AMS (1998)
Solian, A.: Theorey of Modules. Wiley, New York (1977)
Tignol, J. P., Wadsworth, A.: Value functions on simple algebras, and associated graded rings. Springer Monographs in Mathematics (2015)
Zelmanov, E. I.: Lie algebras and torsion groups with identity. J. Combin. Alg. 1(3), 289–340 (2017)
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Presented by: Iain Gordon
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The authors are partially supported by Israeli Science Foundation grants no. 1623/16 and 630/17.
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Bar-On, T., Gilat, S., Matzri, E. et al. Locally Finite Central Simple Algebras. Algebr Represent Theor 26, 553–607 (2023). https://doi.org/10.1007/s10468-021-10103-4
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DOI: https://doi.org/10.1007/s10468-021-10103-4