A class of simple Lie algebras attached to unit forms

doi:10.1007/s11464-016-0616-x

Front. Math. China

2017, Vol. 12

Issue (4) : 787-803 https://doi.org/10.1007/s11464-016-0616-x

RESEARCH ARTICLE

A class of simple Lie algebras attached to unit forms

Jinjing CHEN¹, Zhengxin CHEN²(

)

¹. School of Mathematical Sciences, Xiamen University, Xiamen 361000, China
². School of Mathematics and Computer Science & FJKLMAA, Fujian Normal University, Fuzhou 350117, China

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Abstract

Let n≥3.The complex Lie algebra, which is attached to a unit form $q (x 1, x 2, …, x n) = ∑ i = 1 n x i 2 + ∑ 1 ≤ i ≤ j ≤ n (− 1) j − i x i x j$ and defined by generators and generalized Serre relations, is proved to be a finite-dimensional simple Lie algebra of type $A n$ ,and realized by the Ringel-Hall Lie algebra of a Nakayama algebra of radical square zero. As its application of the realization, we give the roots and a Chevalley basis of the simple Lie algebra.

Keywords Nakayama algebras finite-dimensional simple Lie algebras Ringel-Hall Lie algebras

Corresponding Author(s): Zhengxin CHEN

Issue Date: 06 July 2017

Cite this article:

Jinjing CHEN,Zhengxin CHEN. A class of simple Lie algebras attached to unit forms[J]. Front. Math. China, 2017, 12(4): 787-803.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-016-0616-x
https://academic.hep.com.cn/fmc/EN/Y2017/V12/I4/787

1	AsashibaH. Realization of simple Lie algebras via Hall algebras of tame hereditary algebras. J Math Soc Japan, 2004, 56: 889–905 https://doi.org/10.2969/jmsj/1191334090
2	AsashibaH. Domestic canonical and simple Lie algebras. Math Z, 2008, 259: 713–754 https://doi.org/10.1007/s00209-007-0246-9
3	BarotM, KussinD, LenzingH. The Lie algebra associated to a unit form. J Algebra, 2006, 296: 1–17 https://doi.org/10.1016/j.jalgebra.2005.11.017
4	ChenZ. Canonical algebras of type (n1, n2, n3) and Kac-Moody algebras. Comm Algebra, 2014, 42(8): 3297–3324 https://doi.org/10.1080/00927872.2013.781613
5	ChenZ, LinY. Tubular algebras and affine Kac-Moody algebras. Sci China Ser A, 2007, 50(4): 521–532 https://doi.org/10.1007/s11425-007-0020-9
6	DingM, XiaoJ, XuF. Realizing enveloping algebras via varieties of modules. Acta Math Sin (Engl Ser), 2010, 26: 29–48 https://doi.org/10.1007/s10114-010-9070-y
7	FrenkelI, MalkinA, VybornovM. Affine Lie algebras and tame quiver. Selecta Math, 2001, 7: 1–56 https://doi.org/10.1007/PL00001397
8	HappelD. Triangulated Categories in the Representation Theory of Finite Dimensional Algebras. London Math Soc Lecture Note Ser, Vol 119. Cambridge: Cambridge Univ Press, 1988 https://doi.org/10.1017/CBO9780511629228
9	HappelD, SeidelU. Piecewise hereditary Nakayama algebras. Algebr Represent Theory, 2010, 13: 693–704 https://doi.org/10.1007/s10468-009-9169-y
10	HappelD, ZachariaD. A homological characterization of piecewise hereditary algebras. Math Z, 2008, 260: 177–185 https://doi.org/10.1007/s00209-007-0268-3
11	LinY, PengL. Elliptic Lie algebras and tubular algebras. Adv Math, 2005, 196: 487–530 https://doi.org/10.1016/j.aim.2004.09.006
12	PengL. Intersection matrix Lie algebras and Ringel-Hall Lie algebras of tilted algebras. In: Proc 9th Inter Conf Representations of Algebra. Beijing: Beijing Normal Univ Press, 2002, 98–108
13	PengL, XiaoJ. Root categories and simple Lie algebras. J Algebra, 1997, 198: 19–56 https://doi.org/10.1006/jabr.1997.7152
14	PengL, XiaoJ. Triangulated categories and Kac-Moody algebras. Invent Math, 2000, 140: 563–603 https://doi.org/10.1007/s002220000062
15	PengL, XuM. Symmetrizable intersection matrices and their root systems. arXiv: 0912.1024v1
16	RiedtmannC. Lie algebras generated by indecomposables. J Algebra, 1997, 198: 19–56
17	RingelC M. Hall algebras. Topics in Algebra, Banach Center Publ, 1990, 26: 433–447
18	RingelC M. Hall polynominals for the representation-finite hereditary algebras. Adv Math, 1990, 84(2): 137–178 https://doi.org/10.1016/0001-8708(90)90043-M
19	RingelC M. Lie algebras arising in representation theory. In: Representations of Algebras and Related Topics (Kyoto, 1990). London Math Soc Lecture Note Ser, Vol 168. Cambridge: Cambridge Univ Press, 1992, 284–291 https://doi.org/10.1017/CBO9780511661853.010
20	SerreJ P. Complex Semisimple Lie Algebras. New York: Spinger-Verlag, 1987 https://doi.org/10.1007/978-1-4757-3910-7
21	SlodowyP. Beyond Kac-Moody algebras and inside. In: Britten D J, Lemire F W, Moody R V, eds. Lie Algebras and Related Topics. Canad Math Soc Conf Proc, Vol 5. Providence: Amer Math Soc, 1986, 361–371
22	XiaoJ, XuF, ZhangG. Derived categories and Lie algebras. arXiv: math/0604564v1
23	XuM, PengL. Symmetrizable intersection matrix Lie algebras. Algebra Colloq, 2011, 18(4): 639–646 https://doi.org/10.1142/S1005386711000484

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