Abstract
We develop a new approach to highest weight categories \(\cal{C}\) with good (and cogood) posets of weights via pseudocompact algebras by introducing ascending (and descending) quasi-hereditary pseudocompact algebras. For \(\cal{C}\) admitting a Chevalley duality, we define and investigate tilting modules and Ringel duals of the corresponding pseudocompact algebras. Finally, we illustrate all these concepts on an explicit example of the general linear supergroup GL(1|1).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anderson, F.W. Fuller, K.R.: Rings and Categories of Modules, 2nd edn. Graduate Texts in Mathematics, vol. 13, Springer, New York (1992)
Auslander, M., Reiten, I. Smalø, S.: Representation theory of Artin algebras. In: Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, Cambridge (1997)
Boe, B.D., Kujawa, J.R. Nakano, D.K.: Complexity and module varieties for classical Lie superalgebras. Int. Math. Res. Not. IMRN 3, 696–724 (2011)
Brenner, S., Butler, M.C.R.: Generalizations of the Bernstein-Gel’fand-Ponomarev reflection functors. Representation theory, II (Proc. second internat. conf., Carleton Univ., Ottawa, Ont., 1979). In: Lecture Notes in Math., vol. 832, pp. 103–169. Springer, Berlin (1980)
Brumer, A.: Pseudocompact algebras, profinite groups and class formations. J. Algebra 4, 442–470 (1966)
Brundan, J., Kujawa, J.: A new proof of the Mullineux conjecture. J. Algebr. Comb. 18, 13–39 (2003)
Brundan, J.: Kazhdan–Lusztig polynomials and character formulae for the Lie superalgebra q(n). Adv. Math. 182(1), 28–77 (2004)
Brundan, J.: Kazhdan–Lusztig polynomials and character formulae for the Lie superalgebra gl(m|n). J. Am. Math. Soc. 16(1), 185–231 (2003)
Brundan, J.: Tilting modules for Lie superalgebras. Commun. Algebra 32(6), 2251–2268 (2004)
Brundan, J., Stroppel, C.: Highest weight categories arising from Khovanov’s diagram algebra I: cellularity. Mosc. Math. J. 11, 685–722 (2011)
Brundan, J., Stroppel, C.: Highest weight categories arising from Khovanov’s diagram algebra IV: the general linear supergroup. arXiv:0907.2543v2 [math.RT]. To appear in JEMS
Bucur, I., Deleanu, A.: Introduction to the theory of categories and functors. In: Pure and Applied Mathematics, vol. XIX. Wiley, New York (1968)
Cheng, S.-J., Wang, W., Zhang, R.B.: Super duality and Kazhdan-Lusztig polynomials. Trans. Am. Math. Soc. 360(11), 5883–5924 (2008)
Chuang, J., Turner, W.: Cubist algebras. Adv. Math. 217(4), 1614–1670 (2008)
Cline, E., Parshall, B., Scott, L.: Finite-dimensional algebras and highest weight categories. J. Reine Angew. Math. 391, 85–99 (1988)
Dlab, V., Ringel, C.M.: The module theoretical approach to quasi-hereditary algebras. Representations of algebras and related topics (Kyoto, 1990). In: London Math. Soc. Lecture Note Ser., vol. 168, pp. 200–224. Cambridge Univ. Press, Cambridge (1992)
Donkin, S.: A filtration for rational modules. Math. Z. 177(1), 1–8 (1981)
Donkin, S.: On Schur algebras and related algebras, I. J. Algebra 104(2), 310–328 (1986)
Donkin, S.: On projective modules for algebraic groups. J. Lond. Math. Soc. (2) 54(1), 75–88 (1996)
Donkin, S.: The q-Schur algebra. In: London Mathematical Society Lecture Note Series, vol. 253. Cambridge University Press, Cambridge (1998)
Gabriel, P.: Des catégories abéliennes (French). Bull. Soc. Math. Fr. 90, 323–448 (1962)
Green, J.A.: Locally finite representations. J. Algebra 41, 137–171 (1976)
Germoni, J.: Indecomposable representations of osp(3, 2), D(2, 1; α) and G(3). Colloquium on Homology and Representation Theory (Spanish) (Vaquerias, 1998). Bol. Acad. Nac. Cienc. (Cordoba) 65, 147–163 (2000)
Jantzen, J.C.: Representations of algebraic groups. In: Pure and Applied Mathematics, vol. 131. Academic, Boston (1987)
Lang, S.: Algebra, 3rd rev. edn. Springer, New York (2002)
Marko, F., Zubkov, A.N.: Schur superalgebras in characteristic p. Algebr. Represent. Theory 9(1), 1–12 (2006)
Mazorchuk, V.: Koszul duality for stratified algebras II. Standardly stratified algebras. J. Aust. Math. Soc. 89(1), 23–49 (2010)
Mazorchuk, V., Miemietz, V.: Idempotent subquotients of symmetric quasi-hereditary algebras. Ill. J. Math. 53(3), 737–756 (2009)
Mazorchuk, V., Miemietz, V.: Serre functors for Lie algebras and superalgebras. arXiv:1008.1166v2 [math.RT]. To appear in Annales de l’Institut Fourier
Miemietz, V., Turner, W.: Rational representations of GL2. Glasg. Math. J. 53(2), 257–275 (2011)
Ringel, C.M.: The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences. Math. Z. 208(2), 209–223 (1991)
Simson, D.: Coalgebras, comodules, pseudocompact algebras and tame comodule type. Colloq. Math. 90(1), 101–150 (2001)
Soergel, W.: Charakterformeln für Kipp–Moduln über Kac–Moody-Algebren [Character formulas for tilting modules over Kac-Moody algebras]. Represent. Theory 1, 115–132 (electronic) (1997)
Takeuchi, M.: Morita theorems for categories of comodules. J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 24(3), 629–644 (1977)
Van den Bergh, M.: Blowing up of non-commutative smooth surfaces. Mem. Am. Math. Soc. 154(734) (2001)
Van Gastel, M., Van den Bergh, M.: Graded modules of Gelfand-Kirillov dimension one over three-dimensional Artin–Schelter regular algebras. J. Algebra 196(1), 251–282 (1997)
Zubkov, A.N: Some properties of general linear supergroups and Schur superalgebras. Algebra Logic (Algebra i Logika, Russian) 45, 257–299 (2006)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Marko, F., Zubkov, A.N. Pseudocompact Algebras and Highest Weight Categories. Algebr Represent Theor 16, 689–728 (2013). https://doi.org/10.1007/s10468-011-9326-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-011-9326-y