On solvability of the first Hochschild cohomology of a finite-dimensional algebra

Eisele, Florian; Raedschelders, Theo

doi:10.1090/tran/8064

On solvability of the first Hochschild cohomology of a finite-dimensional algebra
HTML articles powered by AMS MathViewer

by Florian Eisele and Theo Raedschelders PDF

Trans. Amer. Math. Soc. 373 (2020), 7607-7638 Request permission

Abstract:

For an arbitrary finite-dimensional algebra $A$, we introduce a general approach to determining when its first Hochschild cohomology $\mathrm {HH}^1(A)$, considered as a Lie algebra, is solvable. If $A$ is, moreover, of tame or finite representation type, we are able to describe $\mathrm {HH}^1(A)$ as the direct sum of a solvable Lie algebra and a sum of copies of $\mathfrak {sl}_2$. We proceed to determine the exact number of such copies, and give an explicit formula for this number in terms of certain chains of Kronecker subquivers of the quiver of $A$. As a corollary, we obtain a precise answer to a question posed by Chaparro, Schroll, and Solotar.

References

Dalia Artenstein, Marcelo Lanzilotta, and Andrea Solotar, Gerstenhaber structure on Hochschild cohomology of toupie algebras, Algebr. Represent. Theory 23 (2020), no. 2, 421–456. MR 4097322, DOI 10.1007/s10468-019-09854-y
Maurice Auslander, Idun Reiten, and Sverre O. Smalø, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, Cambridge, 1997. Corrected reprint of the 1995 original. MR 1476671
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, DOI 10.1017/CBO9780511614309
Michael J. Bardzell, The alternating syzygy behavior of monomial algebras, J. Algebra 188 (1997), no. 1, 69–89. MR 1432347, DOI 10.1006/jabr.1996.6813
Thomas Brüstle and Yang Han, Tame two-point algebras without loops, Comm. Algebra 29 (2001), no. 10, 4683–4692., DOI 10.1081/AGB-100106780
Ragnar-Olaf Buchweitz and Shiping Liu, Hochschild cohomology and representation-finite algebras, Proc. London Math. Soc. (3) 88 (2004), no. 2, 355–380. MR 2032511, DOI 10.1112/S0024611503014394
Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1999. With an appendix by David A. Buchsbaum; Reprint of the 1956 original. MR 1731415
Cristian Chaparro, Sibylle Schroll, and Andrea Solotar, On the Lie algebra structure of the first Hochschild cohomology of gentle algebras and Brauer graph algebras, J. Algebra 558 (2020), 293–326. MR 4102135, DOI 10.1016/j.jalgebra.2020.02.003
Peter Donovan and Mary Ruth Freislich, The representation theory of finite graphs and associated algebras, Carleton Mathematical Lecture Notes, No. 5, Carleton University, Ottawa, Ont., 1973. MR 0357233
Karin Erdmann and Thorsten Holm, Twisted bimodules and Hochschild cohomology for self-injective algebras of class $A_n$, Forum Math. 11 (1999), no. 2, 177–201. MR 1680594, DOI 10.1515/form.1999.002
Karin Erdmann, Blocks of tame representation type and related algebras, Lecture Notes in Mathematics, vol. 1428, Springer-Verlag, Berlin, 1990. MR 1064107, DOI 10.1007/BFb0084003
Daniel R. Farkas, Christof Geiss, and Eduardo N. Marcos, Smooth automorphism group schemes, Representations of algebras (São Paulo, 1999), 2002, pp. 71–89.
Peter Gabriel, Unzerlegbare Darstellungen. I, Manuscripta Math. 6 (1972), 71–103; correction, ibid. 6 (1972), 309 (German, with English summary). MR 332887, DOI 10.1007/BF01298413
Murray Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math. (2) 78 (1963), 267–288. MR 161898, DOI 10.2307/1970343
Murray Gerstenhaber, On the deformation of rings and algebras, Ann. of Math. (2) 79 (1964), 59–103. MR 171807, DOI 10.2307/1970484
Ezra Getzler and John DS Jones, Operads, homotopy algebra and iterated integrals for double loop spaces, arXiv preprint hep-th (1994).
Murray Gerstenhaber and Samuel D. Schack, Algebraic cohomology and deformation theory, Deformation theory of algebras and structures and applications (Il Ciocco, 1986) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 247, Kluwer Acad. Publ., Dordrecht, 1988, pp. 11–264. MR 981619, DOI 10.1007/978-94-009-3057-5_{2}
Dieter Happel, Hochschild cohomology of finite-dimensional algebras, Séminaire d’Algèbre Paul Dubreil et Marie-Paul Malliavin, 39ème Année (Paris, 1987/1988) Lecture Notes in Math., vol. 1404, Springer, Berlin, 1989, pp. 108–126. MR 1035222, DOI 10.1007/BFb0084073
Dieter Happel, Hochschild cohomology of Auslander algebras, Topics in algebra, Part 1 (Warsaw, 1988) Banach Center Publ., vol. 26, PWN, Warsaw, 1990, pp. 303–310. MR 1171239
G. Hochschild, Semi-simple algebras and generalized derivations, Amer. J. Math. 64 (1942), 677–694. MR 7009, DOI 10.2307/2371713
Thorsten Holm, Hochschild cohomology of tame blocks, J. Algebra 271 (2004), no. 2, 798–826. MR 2025551, DOI 10.1016/j.jalgebra.2003.09.030
Nathan Jacobson, Lie algebras, Dover Publications, Inc., New York, 1979. Republication of the 1962 original. MR 559927
Bernhard Keller, Derived invariance of higher structures on the Hochschild complex, preprint (2003).
Henning Krause, Stable equivalence preserves representation type, Comment. Math. Helv. 72 (1997), no. 2, 266–284. MR 1470092, DOI 10.1007/s000140050016
T. Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics, vol. 189, Springer-Verlag, New York, 1999., DOI 10.1007/978-1-4612-0525-8
Jean-Louis Loday, Cyclic homology, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 301, Springer-Verlag, Berlin, 1998. Appendix E by María O. Ronco; Chapter 13 by the author in collaboration with Teimuraz Pirashvili. MR 1600246, DOI 10.1007/978-3-662-11389-9
J. P. Jans, Verification of a conjecture of Gerstenhaber, Proc. Amer. Math. Soc. 11 (1960), 335–336. MR 114836, DOI 10.1090/S0002-9939-1960-0114836-2
Claus Michael Ringel, The representation type of local algebras, Proceedings of the International Conference on Representations of Algebras (Carleton Univ., Ottawa, Ont., 1974) Carleton Math. Lecture Notes, No. 9, Carleton Univ., Ottawa, Ont., 1974, pp. 24. MR 0382356
Markus Linckelmann and Lleonard Rubio y Degrassi, On the Lie algebra structure of $HH^1(A)$ of a finite-dimensional algebra $A$, Proc. Amer. Math. Soc. 148 (2020), no. 5, 1879–1890. MR 4078074, DOI 10.1090/proc/14875
Lleonard Rubio y Degrassi, Sibylle Schroll, and Andrea Solotar, The first Hochschild cohomology as a Lie algebra, arXiv preprint arXiv:1903.12145 (2019).
Daniel Simson and Andrzej Skowroński, Elements of the representation theory of associative algebras. Vol. 3, London Mathematical Society Student Texts, vol. 72, Cambridge University Press, Cambridge, 2007. Representation-infinite tilted algebras. MR 2382332
Gonçalo Tabuada, Noncommutative motives, University Lecture Series, vol. 63, American Mathematical Society, Providence, RI, 2015. With a preface by Yuri I. Manin. MR 3379910, DOI 10.1090/ulect/063