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Keywords:
Hall algebra; morphism category; Heisenberg Lie algebra; simple Lie algebra
Hall algebra; morphism category; Heisenberg Lie algebra; simple Lie algebra
Summary:
This paper investigates a universal PBW-basis and a minimal set of generators for the Hall algebra $\mathcal {H}(C_2(\mathcal {P}))$, where $C_2(\mathcal {P})$ is the category of morphisms between projective objects in a finitary hereditary exact category $\mathcal A$. When $\mathcal A$ is the representation category of a Dynkin quiver, we develop multiplication formulas for the degenerate Hall Lie algebra $\mathcal {L}$, which is spanned by isoclasses of indecomposable objects in $C_2(\mathcal {P})$. As applications, we demonstrate that $\mathcal {L}$ contains a Lie subalgebra isomorphic to the central extension of the Heisenberg Lie algebra and construct the Borel subalgebra of the simple Lie algebra associated with $\mathcal A$ as a Lie subquotient algebra of $\mathcal {L}$.
References:
[1] Auslander, M., Reiten, I.: On the representation type of triangular matrix rings. J. Lond. Math. Soc., II. Ser. 12 (1976), 371-382. DOI 10.1112/jlms/s2-12.3.371 | MR 0399174 | Zbl 0316.16034
[2] Bautista, R.: The category of morphisms between projective modules. Commun. Algebra 32 (2004), 4303-4331. DOI 10.1081/AGB-200034145 | MR 2102451 | Zbl 1081.16025
[3] Berenstein, A., Greenstein, J.: Primitively generated Hall algebras. Pac. J. Math. 281 (2016), 287-331. DOI 10.2140/pjm.2016.281.287 | MR 3463039 | Zbl 1338.16016
[4] Birkhoff, G.: Subgroups of Abelian groups. Proc. Lond. Math. Soc. (1935), 385-401. DOI 10.1112/plms/s2-38.1.385 | MR 1576323 | Zbl 0010.34304
[5] Bridgeland, T.: Quantum groups via Hall algebras of complexes. Ann. Math. (2) 177 (2013), 739-759. DOI 10.4007/annals.2013.177.2.9 | MR 3010811 | Zbl 1268.16017
[6] Irelli, G. Cerulli, Feigin, E., Reineke, M.: Quiver Grassmannians and degenerate flag varieties. Algebra Number Theory 6 (2012), 165-194. DOI 10.2140/ant.2012.6.165 | MR 2950163 | Zbl 1282.14083
[7] Chen, Q., Deng, B.: Cyclic complexes, Hall polynomials and simple Lie algebras. J. Algebra 440 (2015), 1-32. DOI 10.1016/j.jalgebra.2015.04.043 | MR 3373385 | Zbl 1328.16007
[8] Deng, B., Du, J., Parshall, B., Wang, J.: Finite Dimensional Algebras and Quantum Groups. Mathematical Surveys and Monographs 150. AMS, Providence (2008). DOI 10.1090/surv/150 | MR 2457938 | Zbl 1154.17003
[9] Ding, M., Xu, F., Zhang, H.: Acyclic quantum cluster algebras via Hall algebras of morphisms. Math. Z. 296 (2020), 945-968. DOI 10.1007/s00209-020-02465-0 | MR 4159816 | Zbl 1509.17010
[10] Eiríksson, "O.: From submodule categories to the stable Auslander algebra. J. Algebra 486 (2017), 98-118. DOI 10.1016/j.jalgebra.2017.05.012 | MR 3666209 | Zbl 1407.16005
[11] Gabriel, P.: Unzerlegbare Darstellungen. I. Manuscr. Math. 6 (1972), 71-103 German. DOI 10.1007/BF01298413 | MR 0332887 | Zbl 0232.08001
[12] Gabriel, P.: Indecomposable representations. II. Symposia Mathematica, Vol. XI Academic Press, London (1973), 81-104. MR 0340377 | Zbl 0276.16001
[13] Guo, J. Y., Peng, L.: Universal PBW-basis of Hall-Ringel algebras and Hall polynomials. J. Algebra 198 (1997), 339-351. DOI 10.1006/jabr.1997.7065 | MR 1489901 | Zbl 0893.16005
[14] Hafezi, R., Eshraghi, H.: Determination of some almost split sequences in morphism categories. J. Algebra 633 (2023), 88-113. DOI 10.1016/j.jalgebra.2023.05.045 | MR 4610782 | Zbl 1528.16008
[15] Hubery, A.: From triangulated categories to Lie algebras: A theorem of Peng and Xiao. Trends in Representation Theory of Algebras and Related Topics Contemporary Mathematics 406. AMS, Providence (2006), 51-66. DOI 10.1090/conm/406 | MR 2258041 | Zbl 1107.16021
[16] Kussin, D., Lenzing, H., Meltzer, H.: Nilpotent operators and weighted projective lines. J. Reine. Angew. Math. 685 (2013), 33-71. DOI 10.1515/crelle-2012-0014 | MR 3181563 | Zbl 1293.16008
[17] Lin, Z.: Abelian quotients arising from extriangulated categories via morphism categories. Algebr. Represent. Theory 26 (2023), 117-136. DOI 10.1007/s10468-021-10087-1 | MR 4546135 | Zbl 1509.18014
[18] Luo, X.-H., Zhang, P.: Separated monic representations. I: Gorenstein-projective modules. J. Algebra 479 (2017), 1-34. DOI 10.1016/j.jalgebra.2017.01.038 | MR 3627275 | Zbl 1405.16022
[19] Peng, L.: Some Hall polynomials for representation-finite trivial extension algebras. J. Algebra 197 (1997), 1-13. DOI 10.1006/jabr.1997.7113 | MR 1480775 | Zbl 0891.16010
[20] Peng, L., Xiao, J.: Root categories and simple Lie algebras. J. Algebra 198 (1997), 19-56. DOI 10.1006/jabr.1997.7152 | MR 1482975 | Zbl 0893.16007
[21] Quillen, D.: Higher algebraic $K$-theory. I. Algebr. $K$-Theory. I Lecture Notes in Mathematics 341. Springer, Berlin (1973), 85-147. DOI 10.1007/BFb0067053 | MR 0338129 | Zbl 0292.18004
[22] Riedtmann, C.: Lie algebras generated by indecomposables. J. Algebra 170 (1994), 526-546. DOI 10.1006/jabr.1994.1351 | MR 1302854 | Zbl 0841.16018
[23] Ringel, C. M., Zhang, P.: From submodule categories to preprojective algebras. Math. Z. 278 (2014), 55-73. DOI 10.1007/s00209-014-1305-7 | MR 3267569 | Zbl 1344.16011
[24] Ruan, S., Sheng, J., Zhang, H.: Lie algebras arising from 1-cyclic perfect complexes. J. Algebra 586 (2021), 232-288. DOI 10.1016/j.jalgebra.2021.06.030 | MR 4287774 | Zbl 1477.16022
[25] Sevenhant, B., Bergh, M. Van den: On the double of the Hall algebra of a quiver. J. Algebra 221 (1999), 135-160. DOI 10.1006/jabr.1999.7958 | MR 1722908 | Zbl 0955.16016
[26] Nasab, A. R. Shir Ali, Hosseini, S. N.: Pullback in partial morphism categories. Appl. Categ. Struct. 25 (2017), 197-225. DOI 10.1007/s10485-015-9420-0 | MR 3638360 | Zbl 1397.18006
[27] Wang, G.-J., Li, F.: On minimal horse-shoe lemma. Taiwanese J. Math. 12 (2008), 373-387. DOI 10.11650/twjm/1500574161 | MR 2402122 | Zbl 1143.18012
[28] Xiong, B.-L., Zhang, P., Zhang, Y.-H.: Auslander-Reiten translations in monomorphism categories. Forum Math. 26 (2014), 863-912. DOI 10.1515/forum-2011-0003 | MR 3200353 | Zbl 1319.16017
[29] Zhang, H.: Minimal generators of Hall algebras of 1-cyclic perfect complexes. Int. Math. Res. Not. 2021 (2021), 402-425. DOI 10.1093/imrn/rnz151 | MR 4198500 | Zbl 1508.16023
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