6. Hochschild Cohomology of Polynomial Representations of $\GL_2$

  • Vanessa Miemietz School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, UK
  • Will Turner Department of Mathematics, University of Aberdeen, Fraser Noble Building, King's College, Aberdeen AB24 3UE, UK


We compute the Hochschild cohomology algebras of Ringel-self-dual blocks of polynomial representations of $\GL_2$ over an algebraically closed field of
characteristic $p>2$, that is, of any block whose number of simple modules is a power of $p$. These algebras are finite-dimensional and we provide an explicit description of their bases and multiplications.


\bibitem{BGS} A. Beilinson, V. Ginzburg, W. Soergel, \emph{Koszul duality patterns in representation theory}, J. Amer. Math. Soc. 9 (1996), no. 2, 473--527.
DOI 10.1090/S0894-0347-96-00192-0;
arxiv math/0407108

\bibitem{CPS} E. Cline, B. Parshall and L. Scott, \emph{Finite-dimensional algebras and highest weight categories.}
J. Reine Angew. Math. 391 (1988), 85--99.
zbl 0657.18005;

\bibitem{DR} V. Dlab, C.M. Ringel, \emph{A construction for quasi-hereditary algebras}, Compositio Math., 70 (1989) no.~2, 155--175.
zbl 0677.16007;

\bibitem{Ger} M. Gerstenhaber, \emph{The cohomology structure of an associative ring}. Ann. of Math. (2) 78 (1963) 267--288.
DOI 10.2307/1970343;
zbl 0131.27302;

\bibitem{Green} J. A. Green, \emph{Polynomial representations of ${\rm
GL}_{n}$}, Lecture Notes in Mathematics, 830. Springer, Berlin,
zbl 0451.20037;

\bibitem{EH} K. Erdmann and A. Henke, \emph{On Ringel duality for Schur algebras}, Math. Proc.
Cambridge Philos. Soc. 132(1) (2002), 97--116.
DOI 10.1017/S0305004101005485;
zbl 0998.20016;

\bibitem{Ha} D. Happel, \emph{Triangulated categories in the representation theory of finite-dimensional algebras}, London Mathematical Society Lecture Notes 119, Cambridge University Press,1988.
zbl 0635.16017;

B. Keller, \emph{Deriving dg categories}, Ann. Sci. \'Ecole Norm. Sup. (4) 27 (1994), no.~1, 63--102.
DOI 10.24033/asens.1689;
zbl 0799.18007;

\bibitem{Ke2} B. Keller, \emph{$A_\infty$-algebras, modules and functor categories}, Trends in representation theory of algebras and related topics, Contemp. Math. 406, 67--93. Amer. Math. Soc., Providence, RI, 2006.
zbl 1121.18008;
arxiv math/0510508

B. Keller, \emph{On differential graded categories}, International Congress of
Mathematicians. Vol. II, Eur. Math. Soc., Z\"urich, 2006, pp.~151--190.
zbl 1140.18008;
arxiv math/0601185

\bibitem{Kr} U. Kr\"ahmer, \emph{Notes on Koszul algebras}, http://www.maths.gla.ac.uk/~ukraehmer/connected.pdf.

\bibitem{LH} K. Lef\`evre-Hasegawa, Sur les $A_\infty$ cat\'egories, PhD thesis, Universit\'e Paris 7 - Denis Diderot, 2003.

\bibitem{Maz} V. Mazorchuk, \emph{Koszul duality for stratified algebras, I}, Balanced quasi-hereditary algebras. Manuscripta Math. 131 (2010), no. 1--2, 1--10.
DOI 10.1007/s00229-009-0313-0;
zbl 1207.16030;
arxiv 0810.3479

\bibitem{MT2} V. Miemietz, W. Turner, \emph{Homotopy, Homology and $GL_2$}, Proc. London Math. Soc. (3) 100 (2010), no.~2, 585--606.
DOI 10.1112/plms/pdp040;
zbl 1214.16010;
arxiv 0809.0988

\bibitem{MT3} V. Miemietz, W. Turner, \emph{Koszul dual $2$-functors and extension algebras of simple modules for $GL_2$},
Selecta Math. (N.S.) 21 (2015), no.~2, 605--648.
DOI 10.1007/s00029-014-0164-8;
zbl 1332.20048;
arxiv 1106.5411

\bibitem{MT4} V. Miemietz, W. Turner, \emph{The Weyl extension algebra of $GL_2(\bar{\mathbb{F}}_p)$}, Adv. Math. 246 (2013), 144--197.
DOI 10.1016/j.aim.2013.07.003;
zbl 1301.20036;
arxiv 1106.5665

\bibitem{Ne} C. Negron, \emph{The cup product on Hochschil cohomology via twisting cochains and applications to Koszul rings}, J. of Pure and Applied Algebra 221 (2017), 1112--1133.
DOI 10.1016/j.jpaa.2016.09.003;
zbl 1376.16010;

\bibitem{Rickard} J. Rickard, \emph{Derived equivalences as derived functors}, J. London Math. Soc. (2) 43 (1991), no. 1, 37--48.
DOI 10.1112/jlms/s2-43.1.37;
zbl 0683.16030;

\bibitem{RR} R. Rouquier, \emph{Derived equivalences and finite
dimensional algebras}, Proceedings of the International Congress of
Mathematicians (Madrid, 2006), vol II, pp. 191-221, EMS Publishing
House, 2006.
zbl 1108.20006;

\bibitem{Salfelder} F. Salfelder, \emph{Hochschild cohomology of category $\mathcal{O}$}, felix.salfelder.org/misc/HH.ps
\bibitem{Snashall} N. Snashall,\emph{Support varieties and the Hochschild cohomology ring modulo nilpotence}, Proceedings of the 41st Symposium on Ring Theory and Representation Theory, 68--82, Symp. Ring Theory Represent. Theory Organ. Comm., Tsukuba, 2009.
arxiv 0811.4506

\bibitem{Xu} F. Xu, \emph{Hochschild and ordinary cohomology rings of small categories}, Adv. Math. 219 (2008), 1872--1893.
DOI 10.1016/j.aim.2008.07.014;
zbl 1156.18007;
arxiv 0805.3295
How to Cite
MIEMIETZ, Vanessa; TURNER, Will. 6. Hochschild Cohomology of Polynomial Representations of $\GL_2$. DOCUMENTA MATHEMATICA, [S.l.], v. 23, p. 117-170, may 2018. ISSN 1431-0643. Available at: <https://ojs.elibm.org/index.php/dm/article/view/361>. Date accessed: 23 may 2018. doi: https://doi.org/10.25537/dm.2018v23.117-170.
Unassigned Articles