Abstract
For a postive integer n and a subset S of \(\mathbb {Z}_n\), let \(\left\langle S\right\rangle =\mathbb {Z}_n\), and \(w:S\rightarrow \mathbb {N}\) be a function. The weighted Cayley graph of the cyclic group \(\mathbb {Z}_n\) with respect to S and w is denoted by \(C_n(S,w)\). We give an explicit description of the Grothendieck group of the Leavitt path algebras of \(C_n(S,w)\). We also give description of Leavitt path algebras of \(C_n(S,w)\) in some special cases.
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References
Abrams G, Ánh P N, Louly A and Pardo E, The classification question for Leavitt path algebras, J. Algebra 320(5) (2008) 1983–2026
Abrams G, Aranda Pino G and Siles Molina M, Locally finite Leavitt path algebras, Israel J. Math. 165 (2008) 329–348
Abrams G, Ara P and Molina M S, Leavitt path algebras, volume 2191 of Lecture Notes in Mathematics (2017) (London: Springer)
Abrams G and Pino G A, Purely infinite simple Leavitt path algebras, J. Pure Appl. Algebra 207(3) (2006) 553–563
Abrams G and Pino G A, The Leavitt path algebras of generalized Cayley graphs, Mediterr. J. Math. 13(1) (2016) 1–27
Abrams G, Erickson S and Gil Canto C, Leavitt path algebras of Cayley graphs \(C_n^j\), Mediterr. J. Math. 15(5) (2018) Art. 197, 23
Abrams G, Louly A, Pardo E and Smith C, Flow invariants in the classification of Leavitt path algebras, J. Algebra 333 (2011) 202–231
Abrams G and Schoonmaker B, Leavitt path algebras of Cayley graphs arising from cyclic groups, in: Noncommutative rings and their applications, volume 634 of Contemp. Math., pages 1–10, Amer. Math. Soc., Providence, RI (2015)
Ara P, Goodearl K R and Pardo E, \(K_0\) of purely infinite simple regular rings, \(K\)-Theory 26(1) (2002) 69–100
Ara P, Moreno M A and Pardo E, Nonstable \(K\)-theory for graph algebras, Algebr. Represent. Theory 10(2) (2007) 157–178
Kra I and Simanca S R, On circulant matrices, Notices Amer. Math. Soc. 59(3) (2012) 368–377
Leavitt W G, The module type of a ring, Trans. Amer. Math. Soc. 103 (1962) 113–130
Nam T G and Phuc N T, The structure of Leavitt path algebras and the invariant basis number property, J. Pure Appl. Algebra 223(11) (2019) 4827–4856
Newman M, Integral matrices, Pure and Applied Mathematics, Vol. 45, (1972) (New York-London: Academic Press)
Acknowledgements
The author would like to thank B. Sury for fruitful discussions during the preparation of this paper. The author is grateful to Aditya Challa for his valuable help with Python Programming Language. The author sincerely thanks Ramesh Sreekantan, Roozbeh Hazrat, Gene Abrams, and Cristóbal Gil Canto for their very useful comments towards improving the paper. The author gratefully acknowledges Department of Atomic Energy (National Board for Higher Mathematics), Government of India for their financial support through Ph.D. scholarship.
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Communicating Editor: Parameswaran Sankaran
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Mohan, R. Leavitt path algebras of weighted Cayley graphs \(\varvec{C_n(S,w)}\). Proc Math Sci 131, 17 (2021). https://doi.org/10.1007/s12044-021-00610-1
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DOI: https://doi.org/10.1007/s12044-021-00610-1