Article
| Title: | Symmetries in connected graded algebras and their PBW-deformations (English) |
| Author: | Xu, Yongjun |
| Author: | Zhang, Xin |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 73 |
| Issue: | 4 |
| Year: | 2023 |
| Pages: | 1255-1272 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | We focus on connected graded algebras and their PBW-deformations endowed with additional symmetric structures. Many well-known algebras such as negative parts of Drinfeld-Jimbo's quantum groups, cubic Artin-Schelter algebras and three-dimensional Sklyanin algebras appear in our research framework. As an application, we investigate a $\mathcal {K}_2$ algebra $\mathcal {A}$ which was introduced to compute the cohomology ring of the Fomin-Kirillov algebra $\mathcal {FK}_3$, and explicitly construct all the (self-)symmetric and sign-(self-)symmetric PBW-deformations of $\mathcal {A}$. (English) |
| Keyword: | connected graded algebra |
| Keyword: | PBW-deformation |
| Keyword: | self-symmetry |
| Keyword: | sign-symmetry |
| Keyword: | $\mathcal {K}_2$ algebra |
| MSC: | 16S80 |
| DOI: | 10.21136/CMJ.2023.0511-22 |
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| Date available: | 2023-11-23T12:27:40Z |
| Last updated: | 2023-11-27 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151958 |
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| Reference: | [1] Berger, R.: Koszulity for nonquadratic algebras.J. Algebra 239 (2001), 705-734. Zbl 1035.16023, MR 1832913, 10.1006/jabr.2000.8703 |
| Reference: | [2] Berger, R., Ginzburg, V.: Higher symplectic reflection algebras and non-homogeneous $N$-Koszul property.J. Algebra 304 (2006), 577-601. Zbl 1151.16026, MR 2256407, 10.1016/j.jalgebra.2006.03.011 |
| Reference: | [3] Berger, R., Taillefer, R.: Poincaré-Birkhoff-Witt deformations of Calabi-Yau algebras.J. Noncommut. Geom. 1 (2007), 241-270. Zbl 1161.16022, MR 2308306, 10.4171/JNCG/6 |
| Reference: | [4] Braverman, A., Gaitsgory, D.: Poincaré-Birkhoff-Witt theorem for quadratic algebras of Koszul type.J. Algebra 181 (1996), 315-328. Zbl 0860.17002, MR 1383469, 10.1006/jabr.1996.0122 |
| Reference: | [5] Cassidy, T., Shelton, B.: PBW-deformation theory and regular central extensions.J. Reine Angew. Math. 610 (2007), 1-12. Zbl 1147.16022, MR 2359848, 10.1515/CRELLE.2007.065 |
| Reference: | [6] Cassidy, T., Shelton, B.: Generalizing the notion of Koszul algebra.Math. Z. 260 (2008), 93-114. Zbl 1149.16026, MR 2413345, 10.1007/s00209-007-0263-8 |
| Reference: | [7] Etingof, P., Ginzburg, V.: Noncommutative del Pezzo surfaces and Calabi-Yau algebras.J. Eur. Math. Soc. (JEMS) 12 (2010), 1371-1416. Zbl 1204.14004, MR 2734346, 10.4171/JEMS/235 |
| Reference: | [8] Fløystad, G., Vatne, J. E.: PBW-deformations of $N$-Koszul algebras.J. Algebra 302 (2006), 116-155. Zbl 1159.16026, MR 2236596, 10.1016/j.jalgebra.2005.08.032 |
| Reference: | [9] Fuchs, J., Schellekens, B., Schweigert, C.: From Dynkin diagram symmetries to fixed point structures.Commun. Math. Phys. 180 (1996), 39-97. Zbl 0863.17020, MR 1403859, 10.1007/BF02101182 |
| Reference: | [10] Gavrilik, A. M., Klimyk, A. U.: $q$-deformed orthogonal and pseudo-orthogonal algebras and their representations.Lett. Math. Phys. 21 (1991), 215-220. Zbl 0735.17020, MR 1102131, 10.1007/BF00420371 |
| Reference: | [11] Heckenberger, I., Vendramin, L.: PBW deformations of a Fomin-Kirillov algebra and other examples.Algebr. Represent. Theory 22 (2019), 1513-1532. Zbl 1454.16037, MR 4034793, 10.1007/s10468-018-9830-4 |
| Reference: | [12] Humphreys, J. E.: Introduction to Lie Algebras and Representation Theory.Graduate Texts in Mathematics 9. Springer, New York (2006). Zbl 0447.17001, MR 0499562, 10.1007/978-1-4612-6398-2 |
| Reference: | [13] Iorgov, N. Z., Klimyk, A. U.: The nonstandard deformation $U'_{q}(so_n)$ for $q$ a root of unity.Methods Funct. Anal. Topol. 6 (2000), 56-71. Zbl 0980.17009, MR 1903121 |
| Reference: | [14] Iorgov, N. Z., Klimyk, A. U.: Classification theorem on irreducible representations of the $q$-deformed algebra $U'_{q}(so_n)$.Int. J. Math. Math. Sci. 2005 (2005), 225-262. Zbl 1127.17016, MR 2143754, 10.1155/IJMMS.2005.225 |
| Reference: | [15] Kolb, S., Pellegrini, J.: Braid group actions on coideal subalgebras of quantized enveloping algebras.J. Algebra 336 (2011), 395-416. Zbl 1266.17011, MR 2802552, 10.1016/j.jalgebra.2011.04.001 |
| Reference: | [16] Letzter, G.: Subalgebras which appear in quantum Iwasawa decompositions.Can. J. Math. 49 (1997), 1206-1223. Zbl 0898.17005, MR 1611652, 10.4153/CJM-1997-059-4 |
| Reference: | [17] Letzter, G.: Symmetric pairs for quantized enveloping algebras.J. Algebra 220 (1999), 729-767. Zbl 0956.17007, MR 1717368, 10.1006/jabr.1999.8015 |
| Reference: | [18] Letzter, G.: Coideal subalgebras and quantum symmetric pairs.New Directions in Hopf Algebras Mathematical Sciences Research Institute Publications 43. Cambridge University Press, Cambridge (2002), 117-166. Zbl 1025.17005, MR 1913438 |
| Reference: | [19] Letzter, G.: Quantum symmetric pairs and their zonal spherical functions.Transform. Groups 8 (2003), 261-292. Zbl 1107.17010, MR 1996417, 10.1007/s00031-003-0719-9 |
| Reference: | [20] Polishchuk, A., Positselski, L.: Quadratic Algebras.University Lecture Series 37. AMS, Providence (2005). Zbl 1145.16009, MR 2177131, 10.1090/ulect/037 |
| Reference: | [21] Ştefan, D., Vay, C.: The cohomology ring of the 12-dimensional Fomin-Kirillov algebra.Adv. Math. 291 (2016), 584-620. Zbl 1366.18015, MR 3459024, 10.1016/j.aim.2016.01.001 |
| Reference: | [22] Walton, C. M.: On Degenerations and Deformations of Sklyanin Algebras: Ph.D. Thesis.University of Michigan, Ann Arbor (2011). MR 2942216 |
| Reference: | [23] Xu, Y., Huang, H.-L., Wang, D.: Realization of PBW-deformations of type $\Bbb A_n$ quantum groups via multiple Ore extensions.J. Pure Appl. Algebra 223 (2019), 1531-1547. Zbl 1439.17019, MR 3906516, 10.1016/j.jpaa.2018.06.017 |
| Reference: | [24] Xu, Y., Wang, D., Chen, J.: Analogues of quantum Schubert cell algebras in PBW-deformations of quantum groups.J. Algebra. Appl. 15 (2016), Article ID 1650179, 13 pages. Zbl 1366.17016, MR 3575969, 10.1142/S0219498816501796 |
| Reference: | [25] Xu, Y., Yang, S.: PBW-deformations of quantum groups.J. Algebra 408 (2014), 222-249. Zbl 1366.17017, MR 3197182, 10.1016/j.jalgebra.2013.08.011 |
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