Hochschild and cyclic (co)homology of the Fomin-Kirillov algebra on 3 generators. (English) Zbl 07828317
In [Prog. Math. 172, 147–182 (1999; Zbl 0940.05070)], S. Fomin and A. N. Kirillov introduced a family of quadratic algebras which serves as a model for the Schubert calculus of a flag manifold. These quadratic algebras are now called the Fomin-Kirillov algebras \(FK(n)\), indexed by the positive integers \(n\).
The goal of the present paper is to explicitly compute the Hochschild (co)homology of \(A=FK(3)\), the Fomin-Kirillov algebra on three generators, over a field \(k\) of characteristic different from \(2\) and \(3\). The main tool used is the minimal projective resolution of the standard bimodule \(A\) in the category of bounded below graded bimodules, and this resolution builds upon the minimal projective resolution of the trivial module \(k\) in the category of bounded below graded modules introduced by one of the authors in [E. Herscovich, Homology Homotopy Appl. 22, No. 2, 367–386 (2020; Zbl 1454.16033)]. Using this resolution, the authors explicitly compute bases for the Hochschild homology as well as cohomology groups of \(A\). When the characteristic of \(k\) is \(0\), the present paper also investigates the cyclic homology of \(A\). Lastly, the paper describes the algebra structure of the Hochschild cohomology using techniques from Gröbner bases.
The goal of the present paper is to explicitly compute the Hochschild (co)homology of \(A=FK(3)\), the Fomin-Kirillov algebra on three generators, over a field \(k\) of characteristic different from \(2\) and \(3\). The main tool used is the minimal projective resolution of the standard bimodule \(A\) in the category of bounded below graded bimodules, and this resolution builds upon the minimal projective resolution of the trivial module \(k\) in the category of bounded below graded modules introduced by one of the authors in [E. Herscovich, Homology Homotopy Appl. 22, No. 2, 367–386 (2020; Zbl 1454.16033)]. Using this resolution, the authors explicitly compute bases for the Hochschild homology as well as cohomology groups of \(A\). When the characteristic of \(k\) is \(0\), the present paper also investigates the cyclic homology of \(A\). Lastly, the paper describes the algebra structure of the Hochschild cohomology using techniques from Gröbner bases.
Reviewer: Yining Zhang (Singapura)
MSC:
| 16E40 | (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) |
| 16S37 | Quadratic and Koszul algebras |
| 16T05 | Hopf algebras and their applications |
| 18G10 | Resolutions; derived functors (category-theoretic aspects) |
Software:
GBNPReferences:
| [1] | N. Andruskiewitsch and H.-J. Schneider, On the classification of finite-dimensional pointed Hopf algebras. Ann. of Math. (2) 171 (2010), no. 1, 375-417 Zbl 1208.16028 MR 2630042 · Zbl 1208.16028 · doi:10.4007/annals.2010.171.375 |
| [2] | R. Berger and N. Marconnet, Koszul and Gorenstein properties for homogeneous algebras. Algebr. Represent. Theory 9 (2006), no. 1, 67-97 Zbl 1125.16017 MR 2233117 · Zbl 1125.16017 · doi:10.1007/s10468-005-9002-1 |
| [3] | A. M. Cohen and J. W. Knopper, Gbnp -a gap package Version 1.0.3 (2016) |
| [4] | S. Fomin and A. N. Kirillov, Quadratic algebras, Dunkl elements, and Schubert calculus. In Advances in geometry, pp. 147-182, Progr. Math. 172, Birkhäuser Boston, Boston, MA, 1999 Zbl 0940.05070 MR 1667680 · Zbl 0940.05070 |
| [5] | M. Gerstenhaber, The cohomology structure of an associative ring. Ann. of Math. (2) 78 (1963), 267-288 Zbl 0131.27302 MR 161898 · Zbl 0131.27302 · doi:10.2307/1970343 |
| [6] | M. Graña, Nichols algebras of non-abelian group type: zoo examples. 2016, httpW//mate.dm. uba.ar/ lvendram/zoo/, visited on Aug 31, 2023 |
| [7] | E. Herscovich, An elementary computation of the cohomology of the Fomin-Kirillov algebra with 3 generators. Homology Homotopy Appl. 22 (2020), no. 2, 367-386 Zbl 1454.16033 MR 4102553 · Zbl 1454.16033 · doi:10.4310/hha.2020.v22.n2.a22 |
| [8] | A. N. Kirillov, On some quadratic algebras. In L. D. Faddeev’s Seminar on Mathematical Physics, pp. 91-113, Amer. Math. Soc. Transl. Ser. 2 201, American Mathematical Society, Providence, RI, 2000 Zbl 0966.05080 MR 1772287 · Zbl 0966.05080 · doi:10.1090/trans2/201/07 |
| [9] | A. N. Kirillov, On some quadratic algebras I 1 2 : combinatorics of Dunkl and Gaudin elements, Schubert, Grothendieck, Fuss-Catalan, universal Tutte and reduced polynomials. SIGMA Sym-metry Integrability Geom. Methods Appl. 12 (2016), article no. 002, 172 Zbl 1348.05213 MR 3439199 · Zbl 1348.05213 · doi:10.3842/SIGMA.2016.002 |
| [10] | J.-L. Loday, Cyclic homology. Second edn., Grundlehren Math. Wiss. 301, Springer, Berlin, 1998 Zbl 0885.18007 MR 1600246 · Zbl 0885.18007 · doi:10.1007/978-3-662-11389-9 |
| [11] | A. Milinski and H.-J. Schneider, Pointed indecomposable Hopf algebras over Coxeter groups. In New trends in Hopf algebra theory (La Falda, 1999), pp. 215-236, Contemp. Math. 267, American Mathematical Society, Providence, RI, 2000 Zbl 1093.16504 MR 1800714 · Zbl 1093.16504 · doi:10.1090/conm/267/04272 |
| [12] | D. Ştefan and C. Vay, The cohomology ring of the 12-dimensional Fomin-Kirillov algebra. Adv. Math. 291 (2016), 584-620 Zbl 1366.18015 MR 3459024 · Zbl 1366.18015 · doi:10.1016/j.aim.2016.01.001 |
| [13] | V. A. Ufnarovskij, Combinatorial and asymptotic methods in algebra. In Algebra, VI, pp. 1-196, Encyclopaedia Math. Sci. 57, Springer, Berlin, 1995 MR 1360005 · Zbl 0826.16001 · doi:10.1007/978-3-662-06292-0_1 |
| [14] | V. S. Varadarajan, Supersymmetry for mathematicians: an introduction. Courant Lect. Notes Math. 11, New York University, Courant Institute of Mathematical Sciences, New York; Amer-ican Mathematical Society, Providence, RI, 2004 Zbl 1142.58009 MR 2069561 · Zbl 1142.58009 · doi:10.1090/cln/011 |
| [15] | C. A. Weibel, An introduction to homological algebra. Cambridge Stud. Adv. Math. 38, Cam-bridge University Press, Cambridge, 1994 Zbl 0797.18001 MR 1269324 · Zbl 0797.18001 · doi:10.1017/CBO9781139644136 |
| [16] | S. J. Witherspoon, Hochschild cohomology for algebras. Grad. Stud. Math. 204, American Mathematical Society, Providence, RI, 2019 Zbl 1462.16002 MR 3971234 Received 06 January 2022. Estanislao Herscovich Institut Fourier, UMR 5582, Laboratoire de Mathématiques, Université Grenoble Alpes, 38058 · Zbl 1462.16002 · doi:10.1090/gsm/204 |
| [17] | Grenoble, France; estanislao.herscovich@univ-grenoble-alpes.fr Ziling Li Institut Fourier, UMR 5582, Laboratoire de Mathématiques, Université Grenoble Alpes, 38058 |
| [18] | Grenoble, France; ziling.li@univ-grenoble-alpes.fr |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
