Publications by Steffen Koenig

A book: Derived equivalences for group rings (joint with Alexander Zimmermann) with contributions by Bernhard Keller, Markus Linckelmann, Jeremy Rickard and Raphael Rouquier. Appeared as Springer Lecture Notes in Mathematics, Volume 1685.

Articles and preprints:

Preprints / articles to appear:

Double centraliser property and morphism categories (joint with Nan Gao). To appear in Proc. AMS.

Grade, dominant dimension and Gorenstein algebras (joint with Nan Gao). To appear in Journal of Algebra.

Gendo-symmetric algebras, canonical comultiplication, bar cocomplex and dominant dimension. (joint with Ming Fang). pdf-file of preprint (225 k). To appear in Transactions AMS.

Derived simple algebras and restrictions of recollements of derived module categories. (joint with Lidia Angeleri Hügel, Qunhua Liu and Dong Yang). Preprint arXiv:1310.3479.

Simple-minded systems, configurations and mutations for representation-finite self-injective algebras. (joint with Aaron Chan and Yuming Liu). Preprint arXiv:1305.2576. To appear in Journal Pure and Appl. Algebra.

On tilting complexes providing derived equivalences that send simple-minded objects to simple objects (joint with Dong Yang); preprint, arXiv:1011.3938 (now included in arXiv:1203.5657).

Characteristic tilting modules over quasi-hereditary algebras (joint with Michael Klucznik), third version (January 4, 2000), 64 pages. Course notes (from the compact course given in March 1998 at Bielefeld).

Appeared in 2014:

Quasi-hereditary algebras, exact Borel subalgebras, A-infinity-categories and boxes. (joint with Julian Külshammer and Sergiy Ovsienko). Advances in Math. 262 (2014), 546-592. Preprint arXiv:1305.2315.

Silting objects, simple-minded collections, t-structures and co-t-structures for finite-dimensional algebras. (joint with Dong Yang). Doc.Math. 19 (2014), 403-438. Preprint arXiv:1203.5657.

Schur algebras of Brauer algebras, II (joint with Anne Henke). Math. Z. 276 (2014), 1077-1099. pdf-file of preprint (309 k).

Appeared in 2013:

Derived equivalences from cohomological approximations, and mutations of Yoneda algebras (joint with Changchang Xi and Wei Hu); Proc. Roy. Soc. Edinburgh Sect. A 143 (2013), no. 3, 589-629. pdf-file of preprint (262 k).

Appeared in 2012:

Schur algebras of Brauer algebras I. (joint with Anne Henke); Math. Z. 272 (2012), no. 3-4, 729-759. pdf-file of preprint (368 k).

Simple-minded systems in stable module categories. (joint with Yuming Liu); Q. J. Math. 63 (2012), no. 3, 653-674.

On the uniqueness of stratifications of derived module categories (joint with Lidia Angeleri Hügel and Qunhua Liu); J. of Alg., Volume 359 (2012), 120-137.

Jordan-Hölder theorems for derived module categories of piecewise hereditary algebras (joint with Lidia Angeleri Hügel and Qunhua Liu); J. of Alg., Volume 352 (2012), 361-381.

On the socle of an endomorphism algebra (joint with Gerhard Hiss and Natalie Nährig); J. Pure Appl. Alg. Volume 216 (2012), 1288-1294.

Affine cellular algebras (joint with Changchang Xi); Adv. in Math. 229 (2012), 139-182. pdf-file of preprint (444 k).

Transfer maps in Hochschild (co)homology and applications to stable and derived invariants and to the Auslander-Reiten conjecture (joint with Yuming Liu and Guodong Zhou); Trans. Amer. Math. Soc. 364 (2012), 195-232.

Appeared in 2011:

Stratifying derived module categories (joint with Lidia Angeleri Hügel, Qunhua Liu and Dong Yang), Comptes R. Math. Volume 349, Issues 21-22, November 2011, Pages 1139-1144.

Endomorphism algebras of generators over symmetric algebras. (joint with Ming Fang); J. Algebra 332 (2011), 428-433.

Recollements and tilting objects (joint with Lidia Angeleri Hügel and Qunhua Liu); J. of Pure and Appl. Algebra Volume 215, Issue 4, April 2011, Pages 420-438.

Schur functors and dominant dimension (joint with Ming Fang); Transactions A.M.S. 363 (2011), 1555-1576. pdf-file of preprint (268 k)

Appeared in 2010:

Dominant dimension and almost relatively true versions of Schur's theorem; Milan J. of Math. 78 (2010), 457-479.

Cohomological stratification of diagram algebras (joint with Robert Hartmann, Anne Henke and Rowena Paget); Math. Annalen. 347 (2010), no. 4, 765-804.

Appeared in 2009:

Hochschild cohomology and stratifying ideals (joint with Hiroshi Nagase); J. Pure Appl. Algebra 213 (2009), no. 5, 886--891.

Appeared in 2008:

A panorama of diagram algebras; Trends in representation theory of algebras and related topics, 491--540, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2008.

Gluing of idempotents, radical embeddings and two classes of stable equivalences (joint with Yuming Liu); Journal of Algebra 319, no. 12, 5144-5164.

Cyclotomic extensions of diagram algebras (joint with Junchang Wang); Communications in Algebra 36 (2008), no.5, 1739--1757.

Comparing GL_n-representations by characteristic free isomorphisms between generalized Schur algebras (joint with Ming Fang and Anne Henke, with an appendix by Stephen Donkin); Forum Mathematicum 20 (2008), no. 1, 45--79.

Cohomological reduction by split pairs (joint with Luca Diracca); Journal of Pure and Applied Algebra 212 (2008), no. 3, 471--485.

From triangulated categories to abelian categories - cluster tilting in a general framework (joint with Bin Zhu); Mathematische Zeitschrift 258 (2008), no. 1, 143--160.

Appeared in 2006:

Comparing Lusztig's algebra and Hall algebras at v=-1 (joint with Libin Li); Journal of Algebra 305 (2006), 775-788.

Appeared in 2005:

Tilting modules and Ringel duality (joint with Ronghua Tan); Communications in Algebra 33 (2005), 3749--3769.

Appeared in 2004:

Finitistic dimension and tilting modules for stratified algebras (joint with Oleksandr Khomenko and Volodymyr Mazorchuk); Journal of Algebra 86, 456-475 (2005). For quasi-hereditary or standardly stratified algebras there are well-known upper bounds for the global or finitistic dimensions, but the precise values of these dimensions are usually not known. Recently, the projective and injective dimensions of tilting modules have been related to the global or finitistic dimensions, in particular through a conjecture of Mazorchuk and Parker. This paper contributes new techniques and classes of examples, where the conjecture is true.

Filtrations, stratifications and applications, Proceedings of ICRA 10 (Toronto 2002), Fields Institute Communications 40, AMS, 65-111 (2004) . Survey article on cellular, stratified and quasi-hereditary algebras, focussing on applications to representation theory of algebraic groups and Lie algebras.

Appeared in 2003:

On Hochschild cohomology for orders (joint with Katsunori Sanada and Nicole Snashall); Archiv der Mathematik 81, 627-635 (2003). Certain tiled orders are shown to have periodic resolutions. This is used for determining their Hochschild cohomology, thus generalizing earlier results of Larsen and of Sanada.

Appeared in 2002:

Relating polynomial GL(n)-representations in different degrees (joint with Anne Henke); Journal fuer die reine und angewandte Mathematik 551, 219-235 (2002). Explicit isomorphisms are constructed between (generalized) Schur algebras in different degrees. This establishes and explains repeating patterns in decomposition matrices of general linear and symmetric groups.

Enright's completions and injectively copresented modules (joint with Volodymyr Mazorchuk); Transactions of the American Mathematical Society 354, 2725-2743 (2002). It is shown that the category of (absolutely or relatively) complete modules in the sense of Enright is equivalent to a category of injectively copresented modules, and therefore to the category of eAe-modules for some idempotent e in the algebra A associated with the given block of O. This leads to an easy proof of Deodhar's and Bouaziz's theorem (Enright's conjecture) that completion functors satisfy braid relations. Moreover, the same category is equivalent to some category of Harish-Chandra bimodules (studied by Bernstein and Gelfand) and to some parabolic category O. In the second part of the paper, it is shown that the algebra eAe is projectively (=standardly) stratified. It satisfies a double centralizer property similar to Soergel's results for A itself.

An equivalence of two categories of sl(n,C)-modules (joint with Volodymyr Mazorchuk); Algebras and Representation Theory 5, 319-329 (2002). An equivalence is explicitly constructed between 1) a category of (Enright-)complete modules filtered by submodules of Verma modules and 2) a category constructed from a generic Gelfand-Zetlin module by tensoring with finite dimensional modules.

Ringel duality and Kazhdan-Lusztig theory; Pacific Journal of Mathematics 203, 415-428 (2002). Kazhdan-Lusztig conjecture (for category O) and Lusztig conjecture (for Schur algebras) are formulated in terms of the structure of tilting modules.

Categories of induced modules and standardly stratified algebras (joint with Vyacheslav Futorny and Volodymyr Mazorchuk); Algebras and Representation Theory 5, 259-276 (2002). Generalizations of the BGG-category O are constructed, where certain infinite dimensional modules are the input of parabolic induction. We prove reciprocity formulae and relate the situation to finite dimensional algebras.

Appeared in 2001:

The coinvariant algebra and representation types of blocks of category O (joint with Thomas Bruestle and Volodymyr Mazorchuk); Bulletin of the London Mathematical Society 33, 669-681 (2001). Using an embedding from the module category of a certain subalgebra of the coinvariant algebra, we classify both the blocks of O and the subcategories of modules with Verma flags into finite, tame and wild representation type.

Blocks of category O, double centralizer properties, and Enright's completions; Proceedings of NATO-ASI (Constanta, 2000), Algebras - Representation Theory, 113-134, Kluwer (2001). Survey.

Categories of induced modules for Lie algebras with triangular decomposition (joint with Vyacheslav Futorny and Volodymyr Mazorchuk); Forum Math. 13, 641-661 (2001). Some of our previous results for parabolic category O are generalized to Lie algebras with triangular decomposition, in particular to affine Kac-Moody algebras. Again, projectively stratified algebras appear.

Double centralizer properties, dominant dimension and tilting modules (joint with Inger Heidi Slungard and Changchang Xi); Journal of Algebra 240, 393-412 (2001). A machine is developped for establishing double centralizer properties from structures in ring theory and representation theory. As applications, new and easy proofs are obtained for both classical and quantized Schur-Weyl duality and for Soergel's double centralizer property for category O.

A characteristic free approach to Brauer algebras (joint with Changchang Xi); Transactions of the American Mathematical Society 353, 1489-1505 (2001). Brauer algebras (which arise in representation theory of orthogonal or symplectic groups) are shown to be inflations of group algebras of symmetric groups, in particular they are cellular (which had been proven before by Graham and Lehrer). In some cases we find block decompositions of these algebras.

Appeared in 2000:

S-subcategories in O (joint with Vyacheslav Futorny and Volodymyr Mazorchuk); manuscripta mathematica 102, 487-503 (2000). A combinatorial description is obtained for certain subcategories of O consisting of complete modules having a quasi-Verma flag. These subcategories have an abelian structure which is different from the one in O.

A combinatorial description of blocks in O(P,Lambda) associated with sl(2)-induction (joint with Vyacheslav Futorny and Volodymyr Mazorchuk); Journal of Algebra 231, 86-103 (2000). Analogues of Soergel's results are demonstrated for a generalized BGG-category O. This includes a description of the endomorphism ring of the big projective module as coinvariant algebra, a double centralizer property, and a character formula for tilting modules.

A self-injective cellular algebra is weakly symmetric (joint with Changchang Xi); Journal of Algebra 228, 51-59 (2000). Not surprisingly, the main result of this note is the following: A self-injective cellular algebra is weakly symmetric.

Cyclotomic Schur algebras and blocks of cyclic defect ; Canadian Mathematical Bulletin 43, 79-86 (2000). Generalizing and reproving a result of C.C.Xi, the algebras in the title are classified. This uses the classification of blocks of cyclic defect of finite groups together with a double centralizer property.

Older papers

On cellular algebras, in particular Hecke algebras and Brauer algebras:

When is a cellular algebra quasi-hereditary? (joint with Changchang Xi) : Mathematische Annalen 315 (1999), 281-293. Quasi-hereditary algebras are characterized among the cellular ones by having finite global dimension or equivalently having Cartan determinant one. As a consequence we determine precisely when a Brauer or a partition algebra is quasi-hereditary. Here we use C.C.Xi's recent construction of a cell chain for the partition algebra. His paper Partition algebras are cellular appeared in Compositio Mathematica 119, 107-118 (1999).
Cellular algebras and quasi-hereditary algebras: a comparison (joint with Changchang Xi): ERA 5, 71-75 (1999). Survey on some recent results on cellular algebras.
Cellular algebras: inflations and Morita equivalences (joint with Changchang Xi): Journal of the London Mathematical Society 60, 700-722 (1999). A linear algebra construction of cellular algebras is given, which has been used in other papers to prove that Brauer or partition algebras are cellular. As a consequence we prove that cellular structures can be transported by Morita equivalences if and only if the ground field has characteristic different from two.
On the number of cells of a cellular algebra (joint with Changchang Xi): Communications in Algebra 27, 5463-5470 (1999). An example is given of a cellular algebra having two different cellular structures given by cell chains of two different lengths.
On cellular algebras (joint with Changchang Xi): appeared in: Algebras and Modules II, Proceedings of ICRA VIII (Geiranger), CMS Conference Proceedings. By a mistake of the publisher, the printed version is not the final one. The latter one is given here and contains less misprints. Cellular algebras (defined by Graham and Lehrer) are related to quasi-hereditary ones, and some of Graham and Lehrer's results are given new and different proofs.

On quasi-hereditary algebras, in particular Schur algebras and blocks of O:

Strong symmetry defined by twisting modules, applied to quasi-hereditary algebras with triangular decomposition and vanishing radical cube (joint with Changchang Xi); Communications in Mathematical Physics 197, 427-441 (1998). A new symmetry of certain quasi-hereditary algebras with triangular decomposition is studied. As applications we get a new proof of the structure of blocks of Temperley-Lieb algebras and we generalize some previous results by Deng and Xi on characteristic tilting modules over certain quasi-hereditary algebras.
A criterion for quasi-hereditary, and an abstract straightening formula ; Inventiones Mathematicae 127, 481-488 (1997).
Exact Borel subalgebras of quasi-hereditary algebras, I (with an appendix by Leonard Scott); Mathematische Zeitschrift 220, 399-426 (1995). (Note: Theorems E and F are wrong. The problem whether a Schur algebra has an exact Borel subalgebra at present is open.)
Exact Borel subalgebras of quasi-hereditary algebras, II; Communications in Algebra 23, 2331-2344 (1995).
Strong exact Borel subalgebras of quasi-hereditary algebras and abstract Kazhdan-Lusztig theory; accepted (in 1994) for publication in Advances in Mathematics. Advances in Mathematics 147, 110-137 (1999).
Exact Borel subalgebras of quasi-hereditary algebras and Kazhdan-Lusztig theory; Comptes Rendus Acad.Sci.Paris 318, 601-606 (1994).
Cartan decompositions and BGG-resolutions; Manuscripta Mathematica 86, 103-111 (1995).
On the global dimension of quasi-hereditary algebras with triangular decomposition; Proceedings of the American Mathematical Society 124, 1993-1999 (1996).
Strong exact Borel subalgebras and global dimensions of quasi-hereditary algebras; Proceedings of ICRA VII (Mexico), Canadian Math.Soc. Conference Proceedings 18, 399-417 (1996).
A guide to exact Borel subalgebras of quasi-hereditary algebras; in: V. Dlab, H. Lenzing (Editors), Representations of Algebras, Sixth International Conference, Ottawa 1992 (Canadian Mathematical Society Conference Proceedings, CMS Vol. 14), 291-308 (1993).
Projective resolutions and quasi-heredity; Archiv der Mathematik 58 (1992), 7-13.
Exakte Borel-Teilalgebren von quasi-erblichen Algebren und Kazhdan-Lusztig-Theorie; Habilitationsschrift, Fakultaet Mathematik der Universitaet Stuttgart (1993), 155 pp.

On derived categories and derived equivalences:

Tilting selfinjective algebras and Gorenstein orders (joint with A.Zimmermann); Quarterly Journal of Mathematics (Oxford) 48, 351-361 (1997).
Tilting hereditary orders (joint with A.Zimmermann); Communications in Algebra 24, 1897-1913 (1996).
Tilting complexes, perpendicular categories and recollements of derived module categories of rings; Journal of Pure and Applied Algebra 73 (1991), 211-232.

On quasi-hereditary orders:

Every order is the endomorphism ring of a projective module over a quasi-hereditary order; Communications in Algebra 19 (1991), 2395-2401.
Global dimension two orders are quasi-hereditary (joint with A. Wiedemann); Manuscripta Mathematica 66 (1989), 17-23.
Quasi-hereditary orders; Manuscripta Mathematica 68 (1990), 417-433.

On Auslander Reiten quivers of orders:

Tame and wild socle-projective categories and generalized Baeckstroem orders; Communications in Algebra 18 (1990), 889-925.
A classification theorem for generalized Baeckstroem orders; Communications in Algebra 17 (1989), 11-32.
Zahme und wilde verallgemeinerte Baeckstroemordnungen und ihre Auslander-Reiten-Koecher; Dissertation, Universitaet Stuttgart (1988), 90 pp.

Editorial work

Since 2002 Editorial advisor of the London Mathematical Society (Bulletin, Journal and Proceedings of the London Mathematical Society).

Since 2008 Member of editorial board, Journal of Algebra and its Applications.

Editor (jointly with Michael Butler and Jan-Erik Roos) of the proceedings of 'Representation Theory and its Applications' (Uppsala 2004), Journal of Algebra and its Applications, volume 4 (2005), issues 5 and 6.

Editor (jointly with Alexander Zimmermann) of two issues of 'Algebras and Representation Theory' dedicated to Klaus Roggenkamp on the occasion of his sixtieth birthday, volume 3, issue 4 (2000) and volume 4, issue 1 (2001).