Title | Derivations and automorphisms of locally matrix algebras and groups |
Publication Type | Journal Article |
Year of Publication | 2020 |
Authors | Bezushchak, OO |
Abbreviated Key Title | Dopov. Nac. akad. nauk Ukr. |
DOI | 10.15407/dopovidi2020.09.019 |
Issue | 9 |
Section | Mathematics |
Pagination | 19-23 |
Date Published | 9/2020 |
Language | English |
Abstract | We describe derivations and automorphisms of infinite tensor products of matrix algebras. Using this description, we show that, for a countable–dimensional locally matrix algebra A over a field F, the dimension of the Lie algebra of outer derivations of A and the order of the group of outer automorphisms of A are both equal to | F |ℵ0 , where |F| is the cardinality of the field F. Let A* be the group of invertible elements of a unital locally matrix algebra A. We describe isomorphisms of groups [A*, A*]. In particular, we show that inductive limits of groups SLn(F) are determined by their Steinitz numbers. |
Keywords | automorphism, derivation, locally matrix algebra |
1. Kurosh, A. (1942). Direct decompositions of simple rings. Rec. Math. [Mat. Sbornik] N.S., 11, No. 3, pp. 245-264.
2. Steinitz, E. (1910). Algebraische Theorie der Körper. J. Reine Angew. Math., 137, pp. 167-309.
3. Glimm, J.G. (1960). On a certain class of operator algebras. Trans. Amer. Math. Soc., 95, No. 2, pp. 318-340.
4. Bezushchak, O. & Oliynyk, B. (2020). Unital locally matrix algebras and Steinitz numbers. J. Algebra Appl. https://doi.org/10.1142/S0219498820501807
5. Bezushchak, O. & Oliynyk, B. (2020). Primary decompositions of unital locally matrix algebras. Bull. Math. Sci., 10, No. 1. https://doi.org/10.1142/S166436072050006X
6. Ayupov, S. & Kudaybergenov, K. (2020). Infinite dimensional central simple regular algebras with outer derivations. Lobachevskii J. Math., 41, No. 3, pp. 326-332. https://doi.org/10.1134/S1995080220030063
7. Strade, H. (1999). Locally finite dimensional Lie algebras and their derivation algebras. Abh. Math. Sem. Univ. Hamburg, 69, pp. 373-391. https://doi.org/10.1007/BF02940886
8. Willard, S. (2004). General Topology. Mineola, New York: Dover Publications.
9. Köthe, G. (1931). Schiefkörper unendlichen Ranges uber dem Zentrum. Math. Ann., 105, pp. 15-39.
10. Bezushchak, O.O. & Sushchans’kyi, V.I. (2016). Groups of periodically defined linear transformations of an infinite-dimensional vector space. Ukr. Math. J., 67, No. 10, pp. 1457-1468. https://doi.org/10.1007/s11253-016-1165-x
11. Drozd, Yu.A. & Kirichenko, V.V. (1994). Finite dimensional algebras. Berlin, Heidelberg, New York: Springer.
12. Golubchik, I.Z. & Mikhalev, A.V. (1983). Isomorphisms of the general linear group over an associative ring. Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 3, pp. 61-72.
13. Golubchik, I.Z. (1998). Linear groups over associative rings (Unpublished Doctor thesis). Ufa Scientific Center, Bashkir State Pedagogical Institute, Ufa, Russia (in Russian).
14. Zelmanov, E.I. (1985). Isomorphisms of linear groups over associative rings. Sib. Mat. Zh., 26, No. 4, pp. 49-67.