@article{ASENS_2005_4_38_2_193_0, author = {Geiss, Christof and Leclerc, Bernard and Schr\"oer, Jan}, title = {Semicanonical bases and preprojective algebras}, journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure}, pages = {193--253}, publisher = {Elsevier}, volume = {Ser. 4, 38}, number = {2}, year = {2005}, doi = {10.1016/j.ansens.2004.12.001}, mrnumber = {2144987}, zbl = {02211345}, language = {en}, url = {https://www.numdam.org/articles/10.1016/j.ansens.2004.12.001/} }
TY - JOUR AU - Geiss, Christof AU - Leclerc, Bernard AU - Schröer, Jan TI - Semicanonical bases and preprojective algebras JO - Annales scientifiques de l'École Normale Supérieure PY - 2005 SP - 193 EP - 253 VL - 38 IS - 2 PB - Elsevier UR - https://www.numdam.org/articles/10.1016/j.ansens.2004.12.001/ DO - 10.1016/j.ansens.2004.12.001 LA - en ID - ASENS_2005_4_38_2_193_0 ER -
%0 Journal Article %A Geiss, Christof %A Leclerc, Bernard %A Schröer, Jan %T Semicanonical bases and preprojective algebras %J Annales scientifiques de l'École Normale Supérieure %D 2005 %P 193-253 %V 38 %N 2 %I Elsevier %U https://www.numdam.org/articles/10.1016/j.ansens.2004.12.001/ %R 10.1016/j.ansens.2004.12.001 %G en %F ASENS_2005_4_38_2_193_0
Geiss, Christof; Leclerc, Bernard; Schröer, Jan. Semicanonical bases and preprojective algebras. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 38 (2005) no. 2, pp. 193-253. doi : 10.1016/j.ansens.2004.12.001. https://www.numdam.org/articles/10.1016/j.ansens.2004.12.001/
[1] Representation Theory of Artin Algebras, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, Cambridge, 1997, Corrected reprint of the 1995 original. xiv+425pp. | MR | Zbl
, , ,[2] On algebras of strongly unbounded representation type, Comment. Math. Helv. 60 (3) (1985) 392-399. | MR | Zbl
,[3] Cluster algebras III: Upper bounds and double Bruhat cells, Duke Math. J. 126 (1) (2005) 1-52. | MR | Zbl
, , ,
[4] String bases for quantum groups of type
[5] Geometry of modules over tame quasi-tilted algebras, Colloq. Math. 79 (1) (1999) 85-118. | MR | Zbl
, ,[6] Algebras and quadratic forms, J. London Math. Soc. 28 (3) (1983) 461-469. | MR | Zbl
,[7] A geometric version of the Morita equivalence, J. Algebra 139 (1) (1991) 159-171. | MR | Zbl
,[8] Adapted algebras for the Berenstein Zelevinsky conjecture, Transform. Groups 8 (1) (2003) 37-50. | MR | Zbl
,[9] A multiplicative property of quantum flag minors, Representation Theory 7 (2003) 164-176. | MR | Zbl
,[10] A multiplicative property of quantum flag minors II, J. London Math. Soc. 69 (3) (2004) 608-622. | MR | Zbl
, ,[11] Polytopal realizations of generalized associahedra, Canad. Math. Bull. 45 (2002) 537-566. | MR | Zbl
, , ,[12] On the exceptional fibres of Kleinian singularities, Amer. J. Math. 122 (2000) 1027-1037. | MR | Zbl
,[13] Geometry of the moment map for representations of quivers, Compositio Math. 126 (2001) 257-293. | MR | Zbl
,[14] Irreducible components of varieties of modules, J. Reine Angew. Math. 553 (2002) 201-220. | MR | Zbl
, ,[15] The module theoretical approach to quasi-hereditary algebras, in: Representations of Algebras and Related Topics, Kyoto, 1990, Cambridge Univ. Press, Cambridge, 1992, pp. 200-224. | MR | Zbl
, ,[16] On Galois coverings of tame algebras, Arch. Math. 44 (1985) 522-529. | MR | Zbl
, ,[17] Galois coverings of representation-infinite algebras, Comment. Math. Helv. 62 (1987) 311-337. | MR | Zbl
, ,[18] Cluster algebras I: Foundations, J. Amer. Math. Soc. 15 (2002) 497-529. | MR | Zbl
, ,[19] Y-systems and generalized associahedra, Annals of Math. 158 (3) (2003) 977-1018. | MR | Zbl
, ,[20] Cluster algebras II: Finite type classification, Invent. Math. 154 (2003) 63-121. | MR | Zbl
, ,[21] Auslander-Reiten sequences and representation-finite algebras, in: Representation Theory I, Carleton, 1979, Lecture Notes in Math., vol. 831, Springer-Verlag, Berlin, 1980, pp. 1-71. | MR | Zbl
,[22] The universal cover of a representation-finite algebra, in: Representations of Algebras, Puebla, 1980, Lecture Notes in Math., vol. 903, Springer-Verlag, Berlin, 1981, pp. 68-105. | MR | Zbl
,[23] A class of weighted projective curves arising in representation theory of finite-dimensional algebras, in: Singularities, Representation of Algebras, and Vector Bundles, Lambrecht, 1985, Lecture Notes in Math., vol. 1273, Springer-Verlag, Berlin, 1987, pp. 265-297. | MR | Zbl
, ,[24] Varieties of modules over tubular algebras, Colloq. Math. 95 (2003) 163-183. | MR | Zbl
, ,[25] Extension-orthogonal components of nilpotent varieties, Trans. Amer. Math. Soc. 357 (2005) 1953-1962. | MR | Zbl
, ,[26] Triangulated Categories in the Representation Theory of Finite-Dimensional Algebras, London Mathematical Society Lecture Note Series, vol. 119, Cambridge University Press, Cambridge, 1988, x+208pp. | MR | Zbl
,[27] The derived category of a tubular algebra, in: Representation Theory, I, Ottawa, Ont., 1984, Lecture Notes in Math., vol. 1177, Springer-Verlag, Berlin, 1986, pp. 156-180. | MR | Zbl
, ,[28] Minimal algebras of infinite representation type with preprojective component, Manuscripta Math. 42 (1983) 221-243. | MR | Zbl
, ,[29] On crystal bases of the q-analogue of universal enveloping algebras, Duke Math. J. 63 (1991) 465-516. | MR | Zbl
,[30] Geometric construction of crystal bases, Duke Math. J. 89 (1997) 9-36. | MR | Zbl
, ,
[31] Imaginary vectors in the dual canonical basis of
[32] Dual canonical bases, quantum shuffles and q-characters, Math. Z. 246 (2004) 691-732. | MR | Zbl
,[33] Induced representations of affine Hecke algebras and canonical bases of quantum groups, in: Studies in Memory of Issai Schur, Progress in Mathematics, vol. 210, Birkhäuser, Basel, 2003. | MR | Zbl
, , ,[34] A K-theoretic study of canonical algebras, in: CMS Conf. Proc., vol. 18, 1996, pp. 433-454. | MR | Zbl
,[35] Coxeter transformations associated with finite dimensional algebras, in: Computational Methods for Representations of Groups and Algebras, Essen, 1997, Progress in Math., vol. 173, Birkhäuser, Basel, 1999. | MR | Zbl
,[36] Sheaves on a weighted projective line of genus one, and representations of a tubular algebra, in: Representations of Algebras, Ottawa, ON, 1992, CMS Conf. Proc., vol. 14, 1993, pp. 313-337. | Zbl
, ,[37] Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990) 447-498. | MR | Zbl
,[38] Quivers, perverse sheaves and quantized enveloping algebras, J. Amer. Math. Soc. 4 (1991) 365-421. | MR | Zbl
,[39] Affine quivers and canonical bases, Publ. Math. IHES 76 (1994) 365-416. | Numdam | Zbl
,[40] Constructible functions on the Steinberg variety, Adv. Math. 130 (1997) 365-421. | MR | Zbl
,[41] Semicanonical bases arising from enveloping algebras, Adv. Math. 151 (2000) 129-139. | MR | Zbl
,[42] Marsh R., Reineke M., Personal communication.
[43] Multiplicative properties of dual canonical bases of quantum groups, J. Algebra 211 (1999) 134-149. | MR | Zbl
,[44] Dynkin diagrams and the representation theory of algebras, Notices AMS 44 (1997) 546-556. | MR | Zbl
,[45] Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math., vol. 1099, Springer-Verlag, Berlin, 1984, xiii+376pp. | MR | Zbl
,[46] The preprojective algebra of a quiver, in: Algebras and Modules II, Geiranger, 1966, CMS Conf. Proc., vol. 24, AMS, 1998, pp. 467-480. | MR | Zbl
,[47] Ringel C.M., The multisegment duality and the preprojective algebras of type A, Algebra Montpellier Announcements 1.1 (1999) (6 pages). | MR | Zbl
[48] Representation theory of finite-dimensional algebras, in: Representations of Algebras, Proceedings of the Durham Symposium 1985, Lecture Note Series, vol. 116, LMS, 1986, pp. 9-79. | MR | Zbl
,[49] Schröer J., Module theoretic interpretation of quantum minors, in preparation.
[50] Extended affine root systems I, Publ. Res. Inst. Math. Sci. 21 (1985) 75-179. | MR | Zbl
,[51] Zelevinsky A., Personal communication.
[52] The multisegment duality, in: Documenta Mathematica, Extra Volume, ICM, Berlin, 1998, III, pp. 409-417. | MR | Zbl
,- On Mirković–Vilonen Polytopes, Algebras and Representation Theory (2025) | DOI:10.1007/s10468-025-10317-w
- Geometric realisations of the unipotent enveloping algebra of a quiver, Advances in Mathematics, Volume 441 (2024), p. 109536 | DOI:10.1016/j.aim.2024.109536
- Hopf PBW-deformations of a new type quantum group Uq(𝔰𝔩2∗) and deformed preprojective algebras, International Journal of Mathematics, Volume 35 (2024) no. 03 | DOI:10.1142/s0129167x24500034
- Locally free representations of quivers over commutative Frobenius algebras, Selecta Mathematica, Volume 30 (2024) no. 2 | DOI:10.1007/s00029-023-00914-2
- Equivariant multiplicities via representations of quantum affine algebras, Selecta Mathematica, Volume 29 (2023) no. 1 | DOI:10.1007/s00029-022-00805-y
- PBW parametrizations and generalized preprojective algebras, Advances in Mathematics, Volume 395 (2022), p. 108144 | DOI:10.1016/j.aim.2021.108144
- The characteristic cycles and semi-canonical bases on type A quiver variety, Journal of Algebra, Volume 598 (2022), p. 392 | DOI:10.1016/j.jalgebra.2022.01.019
- Twist Automorphisms on Quantum Unipotent Cells and Dual Canonical Bases, International Mathematics Research Notices, Volume 2021 (2021) no. 9, p. 6772 | DOI:10.1093/imrn/rnz040
- Cluster tilting modules for mesh algebras, Linear Algebra and its Applications, Volume 630 (2021), p. 112 | DOI:10.1016/j.laa.2021.07.021
- Mirković–Vilonen basis in type 𝐴₁, Representation Theory of the American Mathematical Society, Volume 25 (2021) no. 27, p. 780 | DOI:10.1090/ert/582
- Equivariant multiplicities of simply-laced type flag minors, Representation Theory of the American Mathematical Society, Volume 25 (2021) no. 37, p. 1049 | DOI:10.1090/ert/589
- Conjectures and results about parabolic induction of representations of
, Inventiones mathematicae, Volume 222 (2020) no. 3, p. 695 | DOI:10.1007/s00222-020-00982-7 - Tilting Modules and Support τ-Tilting Modules over Preprojective Algebras Associated with Symmetrizable Cartan Matrices, Algebras and Representation Theory, Volume 22 (2019) no. 5, p. 1239 | DOI:10.1007/s10468-018-9819-z
- Maximal Green Sequences for Preprojective Algebras, Algebras and Representation Theory, Volume 20 (2017) no. 1, p. 163 | DOI:10.1007/s10468-016-9635-2
- Separable equivalence, complexity and representation type, Journal of Algebra, Volume 490 (2017), p. 219 | DOI:10.1016/j.jalgebra.2017.06.039
- The greedy basis equals the theta basis: A rank two haiku, Journal of Combinatorial Theory, Series A, Volume 145 (2017), p. 150 | DOI:10.1016/j.jcta.2016.08.004
- Monoidal categorification of cluster algebras, Journal of the American Mathematical Society, Volume 31 (2017) no. 2, p. 349 | DOI:10.1090/jams/895
- Tensor diagrams and cluster algebras, Advances in Mathematics, Volume 300 (2016), p. 717 | DOI:10.1016/j.aim.2016.03.030
- Lifting preprojective algebras to orders and categorifying partial flag varieties, Algebra Number Theory, Volume 10 (2016) no. 7, p. 1527 | DOI:10.2140/ant.2016.10.1527
- A categorification of Grassmannian cluster algebras, Proceedings of the London Mathematical Society, Volume 113 (2016) no. 2, p. 185 | DOI:10.1112/plms/pdw028
- Graded quantum cluster algebras and an application to quantum Grassmannians, Proceedings of the London Mathematical Society, Volume 109 (2014) no. 3, p. 697 | DOI:10.1112/plms/pdu018
- The transition matrix between PBW basis and semicanonical basis of U +(
(ℂ)), Science China Mathematics, Volume 57 (2014) no. 7, p. 1427 | DOI:10.1007/s11425-014-4804-4 - The Hochschild Cohomology Ring of Preprojective Algebras of Type 𝕃nOver a Field of Characteristic 2, Communications in Algebra, Volume 41 (2013) no. 8, p. 3045 | DOI:10.1080/00927872.2012.672602
- CATEGORICAL TINKERTOYS FOR
GAUGE THEORIES, International Journal of Modern Physics A, Volume 28 (2013) no. 05n06, p. 1330006 | DOI:10.1142/s0217751x13300068 - The Hochschild cohomology ring of preprojective algebras of type, Journal of Pure and Applied Algebra, Volume 217 (2013) no. 8, p. 1447 | DOI:10.1016/j.jpaa.2012.11.005
- Cluster algebras in algebraic lie theory, Transformation Groups, Volume 18 (2013) no. 1, p. 149 | DOI:10.1007/s00031-013-9215-z
- Quantum unipotent subgroup and dual canonical basis, Kyoto Journal of Mathematics, Volume 52 (2012) no. 2 | DOI:10.1215/21562261-1550976
- Tubular cluster algebras I: categorification, Mathematische Zeitschrift, Volume 271 (2012) no. 3-4, p. 1091 | DOI:10.1007/s00209-011-0905-8
- Cluster characters for cluster categories with infinite-dimensional morphism spaces, Advances in Mathematics, Volume 227 (2011) no. 1, p. 1 | DOI:10.1016/j.aim.2010.12.010
- Kac–Moody groups and cluster algebras, Advances in Mathematics, Volume 228 (2011) no. 1, p. 329 | DOI:10.1016/j.aim.2011.05.011
- On the cluster category of a marked surface without punctures, Algebra Number Theory, Volume 5 (2011) no. 4, p. 529 | DOI:10.2140/ant.2011.5.529
- Categorification of Skew-symmetrizable Cluster Algebras, Algebras and Representation Theory, Volume 14 (2011) no. 6, p. 1087 | DOI:10.1007/s10468-010-9228-4
- Categorification of Acyclic Cluster Algebras: An Introduction, Higher Structures in Geometry and Physics, Volume 287 (2011), p. 227 | DOI:10.1007/978-0-8176-4735-3_11
- Generic variables in acyclic cluster algebras, Journal of Pure and Applied Algebra, Volume 215 (2011) no. 4, p. 628 | DOI:10.1016/j.jpaa.2010.06.012
- Generic bases for cluster algebras and the Chamber Ansatz, Journal of the American Mathematical Society, Volume 25 (2011) no. 1, p. 21 | DOI:10.1090/s0894-0347-2011-00715-7
- Cluster algebras and quantum affine algebras, Duke Mathematical Journal, Volume 154 (2010) no. 2 | DOI:10.1215/00127094-2010-040
- On the Combinatorics of Rigid Objects in 2-Calabi-Yau Categories, International Mathematics Research Notices (2010) | DOI:10.1093/imrn/rnn029
- Cluster structures for 2-Calabi–Yau categories and unipotent groups, Compositio Mathematica, Volume 145 (2009) no. 4, p. 1035 | DOI:10.1112/s0010437x09003960
- Cluster algebras and triangulated surfaces. Part I: Cluster complexes, Acta Mathematica, Volume 201 (2008) no. 1, p. 83 | DOI:10.1007/s11511-008-0030-7
- Algèbres amassées et algèbres préprojectives : le cas non simplement lacé, Comptes Rendus. Mathématique, Volume 346 (2008) no. 7-8, p. 379 | DOI:10.1016/j.crma.2008.02.007
- From triangulated categories to cluster algebras, Inventiones mathematicae, Volume 172 (2008) no. 1, p. 169 | DOI:10.1007/s00222-008-0111-4
- An approach to non-simply laced cluster algebras, Journal of Algebra, Volume 320 (2008) no. 4, p. 1626 | DOI:10.1016/j.jalgebra.2008.03.018
- Auslander correspondence, Advances in Mathematics, Volume 210 (2007) no. 1, p. 51 | DOI:10.1016/j.aim.2006.06.003
- Rigid modules over preprojective algebras, Inventiones mathematicae, Volume 165 (2006) no. 3, p. 589 | DOI:10.1007/s00222-006-0507-y
- Verma Modules and Preprojective Algebras, Nagoya Mathematical Journal, Volume 182 (2006), p. 241 | DOI:10.1017/s002776300002688x
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