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Wreath products by a Leavitt path algebra and affinizations. (English) Zbl 1306.16002

Starting from an associative algebra \(A\) and a graph \(\Gamma=(V,E)\), the authors construct the wreath product \(B:=A\text{\,wr\,}L(\Gamma)\) of \(A\) and the Leavitt path algebra \(L(\Gamma)\). The algebra \(B\) has an ideal \(I\) consisting of (possibly infinite) matrices over \(A\) and such that \(B/I\cong L(\Gamma)\). Moreover if \(W\) is a hereditary subset of the set of vertices of a graph \(\Gamma\), then the Leavitt path algebra \(L(\Gamma)\) can be written as a wreath product of \(L(W)\) with \(L(\Gamma/W)\) (for a suitable notion of quotient graph). The authors take advantage of these facts to construct new examples of (i) affine algebras with non-nil Jacobson radicals, (ii) affine algebras with non-nilpotent locally nilpotent radicals.

MSC:

16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16G20 Representations of quivers and partially ordered sets
16N40 Nil and nilpotent radicals, sets, ideals, associative rings
16S99 Associative rings and algebras arising under various constructions
46L05 General theory of \(C^*\)-algebras
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References:

[1] DOI: 10.1016/j.jalgebra.2005.07.028 · Zbl 1119.16011
[2] DOI: 10.1016/j.jpaa.2012.03.003 · Zbl 1266.16001
[3] Alahmedi A., J. Algebra Appl. 11 (2012)
[4] DOI: 10.1073/pnas.1311216110 · Zbl 1296.16008
[5] DOI: 10.1007/s10468-006-9044-z · Zbl 1123.16006
[6] Beidar K. I., Uspekhi Mat. Nauk 36 pp 203– (1981)
[7] DOI: 10.1016/S0021-8693(03)00021-8 · Zbl 1028.16012
[8] DOI: 10.1016/S0021-8693(02)00513-6 · Zbl 1020.16014
[9] Herstein I. N., Carus Mathematical Monographs 15, in: Noncommutative Rings (1994) · Zbl 0177.05801
[10] DOI: 10.1007/978-1-4612-9964-6
[11] Rjabuhin Ju. M., Algebra Logika 7 pp 100– (1968)
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