Auslander-Reiten theory of small half quantum groups
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Abstract:
For the small half quantum groups $u_\zeta (\mathfrak {b})$ and $u_\zeta (\mathfrak {n})$ we show that the components of the stable Auslander-Reiten quiver containing gradable modules are of the form $\mathbb {Z}[\mathbb {A}_\infty ]$.References
- Maurice Auslander, Idun Reiten, and Sverre O. Smalø, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, Cambridge, 1995. MR 1314422, DOI 10.1017/CBO9780511623608
- Ibrahim Assem, Daniel Simson, and Andrzej Skowroński, Elements of the representation theory of associative algebras. Vol. 1, London Mathematical Society Student Texts, vol. 65, Cambridge University Press, Cambridge, 2006. Techniques of representation theory. MR 2197389, DOI 10.1017/CBO9780511614309
- Petter Andreas Bergh and Karin Erdmann, The stable Auslander-Reiten quiver of a quantum complete intersection, Bull. Lond. Math. Soc. 43 (2011), no. 1, 79–90. MR 2765552, DOI 10.1112/blms/bdq081
- D. J. Benson, Representations and cohomology. II, Cambridge Studies in Advanced Mathematics, vol. 31, Cambridge University Press, Cambridge, 1991. Cohomology of groups and modules. MR 1156302
- Allen D. Bell and Rolf Farnsteiner, On the theory of Frobenius extensions and its application to Lie superalgebras, Trans. Amer. Math. Soc. 335 (1993), no. 1, 407–424. MR 1097163, DOI 10.1090/S0002-9947-1993-1097163-5
- Claude Cibils, Half-quantum groups at roots of unity, path algebras, and representation type, Internat. Math. Res. Notices 12 (1997), 541–553. MR 1456563, DOI 10.1155/S1073792897000366
- Vyjayanthi Chari and Andrew Pressley, A guide to quantum groups, Cambridge University Press, Cambridge, 1994. MR 1300632
- Corrado De Concini and Victor G. Kac, Representations of quantum groups at roots of $1$, Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989) Progr. Math., vol. 92, Birkhäuser Boston, Boston, MA, 1990, pp. 471–506. MR 1103601
- Christopher Martin Drupieski, Cohomology of Frobenius-Lusztig kernels of quantized enveloping algebras, Ph.D. thesis, University of Virginia, 2009.
- Christopher M. Drupieski, On injective modules and support varieties for the small quantum group, Int. Math. Res. Not. IMRN 10 (2011), 2263–2294. MR 2806565, DOI 10.1093/imrn/rnq156
- Christopher M. Drupieski, Representations and cohomology for Frobenius-Lusztig kernels, J. Pure Appl. Algebra 215 (2011), no. 6, 1473–1491. MR 2769244, DOI 10.1016/j.jpaa.2010.09.006
- Karin Erdmann, Miles Holloway, Rachel Taillefer, Nicole Snashall, and Øyvind Solberg, Support varieties for selfinjective algebras, $K$-Theory 33 (2004), no. 1, 67–87. MR 2199789, DOI 10.1007/s10977-004-0838-7
- Karin Erdmann, Blocks of tame representation type and related algebras, Lecture Notes in Mathematics, vol. 1428, Springer-Verlag, Berlin, 1990. MR 1064107, DOI 10.1007/BFb0084003
- Karin Erdmann, On Auslander-Reiten components for group algebras, J. Pure Appl. Algebra 104 (1995), no. 2, 149–160. MR 1360172, DOI 10.1016/0022-4049(94)00133-7
- Karin Erdmann, The Auslander-Reiten quiver of restricted enveloping algebras, Representation theory of algebras (Cocoyoc, 1994) CMS Conf. Proc., vol. 18, Amer. Math. Soc., Providence, RI, 1996, pp. 201–214. MR 1388051
- Karin Erdmann and Andrzej Skowroński, On Auslander-Reiten components of blocks and self-injective biserial algebras, Trans. Amer. Math. Soc. 330 (1992), no. 1, 165–189. MR 1144759, DOI 10.1090/S0002-9947-1992-1144759-7
- Rolf Farnsteiner, Jordan types for indecomposable modules of finite group schemes, preprint, arXiv:0910.3093.
- Rolf Farnsteiner, On Auslander-Reiten quivers of enveloping algebras of restricted Lie algebras, Math. Nachr. 202 (1999), 43–66. MR 1694715, DOI 10.1002/mana.19992020105
- Rolf Farnsteiner, Auslander-Reiten components for Lie algebras of reductive groups, Adv. Math. 155 (2000), no. 1, 49–83. MR 1789848, DOI 10.1006/aima.2000.1925
- Rolf Farnsteiner, Auslander-Reiten components for $G_1T$-modules, J. Algebra Appl. 4 (2005), no. 6, 739–759. MR 2192153, DOI 10.1142/S0219498805001502
- Jörg Feldvoss and Sarah Witherspoon, Support varieties and representation type of small quantum groups, Int. Math. Res. Not. IMRN 7 (2010), 1346–1362. MR 2609022, DOI 10.1093/imrn/rnp189
- Jörg Feldvoss and Sarah Witherspoon, Support varieties and representation type of self-injective algebras, Homology Homotopy Appl. 13 (2011), no. 2, 197–215. MR 2854335, DOI 10.4310/HHA.2011.v13.n2.a13
- Jörg Feldvoss and Sarah Witherspoon, Erratum to “Support varieties and representation type of small quantum groups”, arXiv: 1308.4928, August 2013.
- Lillian E. Peters Hupert, Homological characteristics of pro-uniserial rings, J. Algebra 69 (1981), no. 1, 43–66. MR 613856, DOI 10.1016/0021-8693(81)90126-5
- Otto Kerner and Dan Zacharia, Auslander-Reiten theory for modules of finite complexity over self-injective algebras, Bull. Lond. Math. Soc. 43 (2011), no. 1, 44–56. MR 2765548, DOI 10.1112/blms/bdq082
- Julian Külshammer, Auslander-Reiten theory of Frobenius-Lusztig kernels, Bull. London Math. Soc. 45 (2013), 1285–1300, DOI: 10.1112/blms/bdt064.
- —, Representation type and Auslander-Reiten theory of Frobenius-Lusztig kernels, Ph.D. thesis, University of Kiel, 2012.
- Idun Reiten and Christine Riedtmann, Skew group algebras in the representation theory of Artin algebras, J. Algebra 92 (1985), no. 1, 224–282. MR 772481, DOI 10.1016/0021-8693(85)90156-5
- Sarah Scherotzke, Euclidean components for a class of self-injective algebras, Colloq. Math. 115 (2009), no. 2, 219–245. MR 2491746, DOI 10.4064/cm115-2-7
- Mariano Suarez-Alvarez, The Hilton-Heckmann argument for the anti-commutativity of cup products, Proc. Amer. Math. Soc. 132 (2004), no. 8, 2241–2246. MR 2052399, DOI 10.1090/S0002-9939-04-07409-X
- Maurice Auslander, Idun Reiten, and Sverre O. Smalø, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, Cambridge, 1995. MR 1314422, DOI 10.1017/CBO9780511623608
- Ibrahim Assem, Daniel Simson, and Andrzej Skowroński, Elements of the representation theory of associative algebras. Vol. 1, London Mathematical Society Student Texts, vol. 65, Cambridge University Press, Cambridge, 2006. Techniques of representation theory. MR 2197389, DOI 10.1017/CBO9780511614309
- Petter Andreas Bergh and Karin Erdmann, The stable Auslander-Reiten quiver of a quantum complete intersection, Bull. Lond. Math. Soc. 43 (2011), no. 1, 79–90. MR 2765552, DOI 10.1112/blms/bdq081
- D. J. Benson, Representations and cohomology. II, Cambridge Studies in Advanced Mathematics, vol. 31, Cambridge University Press, Cambridge, 1991. Cohomology of groups and modules. MR 1156302
- Allen D. Bell and Rolf Farnsteiner, On the theory of Frobenius extensions and its application to Lie superalgebras, Trans. Amer. Math. Soc. 335 (1993), no. 1, 407–424. MR 1097163, DOI 10.1090/S0002-9947-1993-1097163-5
- Claude Cibils, Half-quantum groups at roots of unity, path algebras, and representation type, Internat. Math. Res. Notices 12 (1997), 541–553. MR 1456563, DOI 10.1155/S1073792897000366
- Vyjayanthi Chari and Andrew Pressley, A guide to quantum groups, Cambridge University Press, Cambridge, 1994. MR 1300632
- Corrado De Concini and Victor G. Kac, Representations of quantum groups at roots of $1$, Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989) Progr. Math., vol. 92, Birkhäuser Boston, Boston, MA, 1990, pp. 471–506. MR 1103601
- Christopher Martin Drupieski, Cohomology of Frobenius-Lusztig kernels of quantized enveloping algebras, Ph.D. thesis, University of Virginia, 2009.
- Christopher M. Drupieski, On injective modules and support varieties for the small quantum group, Int. Math. Res. Not. IMRN 10 (2011), 2263–2294. MR 2806565, DOI 10.1093/imrn/rnq156
- Christopher M. Drupieski, Representations and cohomology for Frobenius-Lusztig kernels, J. Pure Appl. Algebra 215 (2011), no. 6, 1473–1491. MR 2769244, DOI 10.1016/j.jpaa.2010.09.006
- Karin Erdmann, Miles Holloway, Rachel Taillefer, Nicole Snashall, and Øyvind Solberg, Support varieties for selfinjective algebras, $K$-Theory 33 (2004), no. 1, 67–87. MR 2199789, DOI 10.1007/s10977-004-0838-7
- Karin Erdmann, Blocks of tame representation type and related algebras, Lecture Notes in Mathematics, vol. 1428, Springer-Verlag, Berlin, 1990. MR 1064107, DOI 10.1007/BFb0084003
- Karin Erdmann, On Auslander-Reiten components for group algebras, J. Pure Appl. Algebra 104 (1995), no. 2, 149–160. MR 1360172, DOI 10.1016/0022-4049(94)00133-7
- Karin Erdmann, The Auslander-Reiten quiver of restricted enveloping algebras, Representation theory of algebras (Cocoyoc, 1994), CMS Conf. Proc., vol. 18, American Mathematical Society, Providence, RI, 1996, pp. 201–214.
- Karin Erdmann and Andrzej Skowroński, On Auslander-Reiten components of blocks and self-injective biserial algebras, Trans. Amer. Math. Soc. 330 (1992), no. 1, 165–189. MR 1144759, DOI 10.1090/S0002-9947-1992-1144759-7
- Rolf Farnsteiner, Jordan types for indecomposable modules of finite group schemes, preprint, arXiv:0910.3093.
- Rolf Farnsteiner, On Auslander-Reiten quivers of enveloping algebras of restricted Lie algebras, Math. Nachr. 202 (1999), 43–66. MR 1694715, DOI 10.1002/mana.19992020105
- Rolf Farnsteiner, Auslander-Reiten components for Lie algebras of reductive groups, Adv. Math. 155 (2000), no. 1, 49–83. MR 1789848, DOI 10.1006/aima.2000.1925
- Rolf Farnsteiner, Auslander-Reiten components for $G_1T$-modules, J. Algebra Appl. 4 (2005), no. 6, 739–759. MR 2192153, DOI 10.1142/S0219498805001502
- Jörg Feldvoss and Sarah Witherspoon, Support varieties and representation type of small quantum groups, Int. Math. Res. Not. IMRN 7 (2010), 1346–1362. MR 2609022, DOI 10.1093/imrn/rnp189
- Jörg Feldvoss and Sarah Witherspoon, Support varieties and representation type of self-injective algebras, Homology Homotopy Appl. 13 (2011), no. 2, 197–215. MR 2854335, DOI 10.4310/HHA.2011.v13.n2.a13
- Lillian E. Peters Hupert, Homological characteristics of pro-uniserial rings, J. Algebra 69 (1981), no. 1, 43–66. MR 613856, DOI 10.1016/0021-8693(81)90126-5
- Otto Kerner and Dan Zacharia, Auslander-Reiten theory for modules of finite complexity over self-injective algebras, Bull. Lond. Math. Soc. 43 (2011), no. 1, 44–56. MR 2765548, DOI 10.1112/blms/bdq082
- Julian Külshammer, Auslander-Reiten theory of Frobenius-Lusztig kernels, preprint, arXiV: 1201.5303, January 2012.
- Idun Reiten and Christine Riedtmann, Skew group algebras in the representation theory of Artin algebras, J. Algebra 92 (1985), no. 1, 224–282. MR 772481, DOI 10.1016/0021-8693(85)90156-5
- Sarah Scherotzke, Euclidean components for a class of self-injective algebras, Colloq. Math. 115 (2009), no. 2, 219–245. MR 2491746, DOI 10.4064/cm115-2-7
Additional Information
- Julian Külshammer
- Affiliation: Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, 24098 Kiel, Germany
- Address at time of publication: Institut für Algebra und Zahlentheorie, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
- MR Author ID: 928718
- Email: julian.kuelshammer@mathematik.uni-stuttgart.de
- Received by editor(s): February 8, 2012
- Received by editor(s) in revised form: May 3, 2012
- Published electronically: January 10, 2014
- Additional Notes: This work was supported by the D.F.G. priority program SPP1388 “Darstellungstheorie”
- Communicated by: Kailash C. Misra
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 142 (2014), 1109-1119
- MSC (2010): Primary 16G20, 16G70, 17B37
- DOI: https://doi.org/10.1090/S0002-9939-2014-11848-X
- MathSciNet review: 3162234