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New Algebras from Old

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Quiver Representations

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Abstract

In this chapter, we present several popular constructions for algebras, each one in a separate section. We introduce tilted algebras, trivial extensions, cluster-tilted algebras, triangular matrix algebras, and one-point extensions.

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References

  1. Lidia Angeleri Hügel, Dieter Happel, and Henning Krause (eds.), Handbook of tilting theory, London Mathematical Society Lecture Note Series, vol. 332, Cambridge University Press, Cambridge, 2007. MR 2385175 (2008i:16001)

    Google Scholar 

  2. Ibrahim Assem, Thomas Brüstle, and Ralf Schiffler, Cluster-tilted algebras and slices, J. Algebra 319 (2008), no. 8, 3464–3479. MR 2408327 (2009f:16021)

    Google Scholar 

  3. Ibrahim Assem, Thomas Brüstle, and Ralf Schiffler, Cluster-tilted algebras as trivial extensions, Bull. Lond. Math. Soc. 40 (2008), no. 1, 151–162. MR 2409188 (2009c:16086)

    Google Scholar 

  4. Ibrahim Assem, Thomas Brüstle, and Ralf Schiffler, Cluster-tilted algebras without clusters, J. Algebra 324 (2010), no. 9, 2475–2502. MR 2684150

    Google Scholar 

  5. Ibrahim Assem, Dieter Happel, and Oscar Roldán, Representation-finite trivial extension algebras, J. Pure Appl. Algebra 33 (1984), no. 3, 235–242. MR 761629 (85m:16009)

    Google Scholar 

  6. Ibrahim Assem, Daniel Simson, and Andrzej Skowroński, Elements of the representation theory of associative algebras. Vol. 1, London Mathematical Society Student Texts, vol. 65, Cambridge University Press, Cambridge, 2006, Techniques of representation theory. MR 2197389 (2006j:16020)

    Google Scholar 

  7. Maurice Auslander, María Inés Platzeck, and Idun Reiten, Coxeter functors without diagrams, Trans. Amer. Math. Soc. 250 (1979), 1–46. MR 530043 (80c:16027)

    Google Scholar 

  8. Michael Barot, Elsa Fernández, María Inés Platzeck, Nilda Isabel Pratti, and Sonia Trepode, From iterated tilted algebras to cluster-tilted algebras, Adv. Math. 223 (2010), no. 4, 1468–1494. MR 2581376

    Google Scholar 

  9. Marco Angel Bertani-Økland, Steffen Oppermann, and Anette Wrålsen, Constructing tilted algebras from cluster-tilted algebras, J. Algebra 323 (2010), no. 9, 2408–2428. MR 2602387

    Google Scholar 

  10. Grzegorz Bobiński and Aslak Bakke Buan, The algebras derived equivalent to gentle cluster tilted algebras, J. Algebra Appl. 11 (2012), no. 1, 1250012, 26. MR 2900882

    Google Scholar 

  11. Sheila Brenner and M. C. R. Butler, Generalizations of the Bernstein-Gel’fand-Ponomarev reflection functors, Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979), Lecture Notes in Math., vol. 832, Springer, Berlin, 1980, pp. 103–169. MR 607151 (83e:16031)

    Google Scholar 

  12. Aslak Bakke Buan, Robert Marsh, Markus Reineke, Idun Reiten, and Gordana Todorov, Tilting theory and cluster combinatorics, Adv. Math. 204 (2006), no. 2, 572–618. MR 2249625 (2007f:16033)

    Google Scholar 

  13. Aslak Bakke Buan, Robert J. Marsh, and Idun Reiten, Cluster-tilted algebras of finite representation type, J. Algebra 306 (2006), no. 2, 412–431. MR 2271343 (2008f:16032)

    Google Scholar 

  14. Aslak Bakke Buan, Robert J. Marsh, and Idun Reiten, Cluster-tilted algebras, Trans. Amer. Math. Soc. 359 (2007), no. 1, 323–332 (electronic). MR 2247893 (2007f:16035)

    Google Scholar 

  15. Aslak Bakke Buan, Robert J. Marsh, and Idun Reiten, Cluster mutation via quiver representations, Comment. Math. Helv. 83 (2008), no. 1, 143–177. MR 2365411 (2008k:16026)

    Google Scholar 

  16. Philippe Caldero, Frédéric Chapoton, and Ralf Schiffler, Quivers with relations and cluster tilted algebras, Algebr. Represent. Theory 9 (2006), no. 4, 359–376. MR 2250652 (2007f:16036)

    Google Scholar 

  17. Philippe Caldero, Frédéric Chapoton, and Ralf Schiffler, Quivers with relations arising from clusters (A n case), Trans. Amer. Math. Soc. 358 (2006), no. 3, 1347–1364. MR 2187656 (2007a:16025)

    Google Scholar 

  18. Edward Cline, Brian Parshall, and Leonard Scott, Derived categories and Morita theory, J. Algebra 104 (1986), no. 2, 397–409. MR 866784 (88a:16075)

    Google Scholar 

  19. Robert M. Fossum, Phillip A. Griffith, and Idun Reiten, Trivial extensions of abelian categories, Lecture Notes in Mathematics, Vol. 456, Springer-Verlag, Berlin, 1975, Homological algebra of trivial extensions of abelian categories with applications to ring theory. MR 0389981 (52 #10810)

    Google Scholar 

  20. Peter Gabriel, Unzerlegbare Darstellungen. I, Manuscripta Math. 6 (1972), 71–103; correction, ibid. 6 (1972), 309. MR 0332887 (48 #11212)

    Google Scholar 

  21. Peter Gabriel, Indecomposable representations. II, Symposia Mathematica, Vol. XI (Convegno di Algebra Commutativa, INDAM, Rome, 1971), Academic Press, London, 1973, pp. 81–104. MR 0340377 (49 #5132)

    Google Scholar 

  22. Dieter Happel, On the derived category of a finite-dimensional algebra, Comment. Math. Helv. 62 (1987), no. 3, 339–389. MR 910167 (89c:16029)

    Google Scholar 

  23. Dieter Happel, Idun Reiten, and Smalø Sverre O., Tilting in abelian categories and quasitilted algebras, Mem. Amer. Math. Soc. 120 (1996), no. 575, viii+ 88. MR 1327209 (97j:16009)

    Google Scholar 

  24. Dieter Happel and Claus Michael Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274 (1982), no. 2, 399–443. MR 675063 (84d:16027)

    Google Scholar 

  25. Mitsuo Hoshino, Trivial extensions of tilted algebras, Comm. Algebra 10 (1982), no. 18, 1965–1999. MR 674704 (84j:16019)

    Google Scholar 

  26. Yasuo Iwanaga and Takayoshi Wakamatsu, Trivial extension of Artin algebras, Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979), Lecture Notes in Math., vol. 832, Springer, Berlin, 1980, pp. 295–301. MR 607160 (82c:16024)

    Google Scholar 

  27. Bernhard Keller and Idun Reiten, Cluster-tilted algebras are Gorenstein and stably Calabi-Yau, Adv. Math. 211 (2007), no. 1, 123–151. MR 2313531 (2008b:18018)

    Google Scholar 

  28. Miki Oryu and Ralf Schiffler, On one-point extensions of cluster-tilted algebras, J. Algebra 357 (2012), 168–182. MR 2905247

    Google Scholar 

  29. Marju Purin, τ-complexity of cluster tilted algebras, J. Pure Appl. Algebra 216 (2012), no. 4, 897–904. MR 2864863 (2012k:16033)

    Google Scholar 

  30. Jeremy Rickard, Morita theory for derived categories, J. London Math. Soc. (2) 39 (1989), no. 3, 436–456. MR 1002456 (91b:18012)

    Google Scholar 

  31. Claus Michael Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Mathematics, vol. 1099, Springer-Verlag, Berlin, 1984. MR 774589 (87f:16027)

    Google Scholar 

  32. Hiroyuki Tachikawa, Representations of trivial extensions of hereditary algebras, Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979), Lecture Notes in Math., vol. 832, Springer, Berlin, 1980, pp. 579–599. MR 607173 (82d:16029)

    Google Scholar 

  33. Bin Zhu, Cluster-tilted algebras and their intermediate coverings, Comm. Algebra 39 (2011), no. 7, 2437–2448. MR 2821722 (2012f:16043)

    Google Scholar 

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Schiffler, R. (2014). New Algebras from Old. In: Quiver Representations. CMS Books in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-09204-1_6

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