Infinite-dimensional necklace Lie algebras and some finite-dimensional important subalgebras

doi:10.3868/s140-DDD-023-0025-x

Front. Math. China

2023, Vol. 18

Issue (5) : 353-365 https://doi.org/10.3868/s140-DDD-023-0025-x

Infinite-dimensional necklace Lie algebras and some finite-dimensional important subalgebras

Demin YU¹(

), Caihui LU²

¹. College of Mathematics, Hunan Institute of Science and Technology, Yueyang 414000, China
². School of Mathematical Sciences, Capital Normal University, Beijing 100037, China

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Abstract

In this paper, a new infinite-dimensional necklace Lie algebra is studied and the left and right index arrays of a necklace word in necklace Lie algebra is first defined. Using the left and right index arrays, we divide the necklace words into 5 classes. We discuss finite-dimensional Lie subalgebras of necklace Lie algebras intensively and prove that some subalgebras are isomorphism to simple Lie algebra sl $(n)$ .

Keywords Necklace Lie algebra left and right index arrays subalgebra

Corresponding Author(s): Demin YU

Online First Date: 27 December 2023 Issue Date: 11 January 2024

Cite this article:

Demin YU,Caihui LU. Infinite-dimensional necklace Lie algebras and some finite-dimensional important subalgebras[J]. Front. Math. China, 2023, 18(5): 353-365.

URL:

https://academic.hep.com.cn/fmc/EN/10.3868/s140-DDD-023-0025-x
https://academic.hep.com.cn/fmc/EN/Y2023/V18/I5/353

Fig.1 Loop

c = α i 1 α i 2 ? α i u

corresponds to the necklace word

Fig.2 Image definition of Lie operation

Fig.3 Q

Fig.4 Index array relation

D)

Fig.5 Index array relation

E)

Fig.6 Arrow diagram

Q 1

Fig.7 Arrow diagram

Q 2

Fig.8 Arrow diagram

Q 3

Fig.9 Arrow diagram

Q 4

Fig.10 Arrow diagram

Q 5

1	R Bocklandt, L Le Bruyn. Necklace Lie algebras and noncommutative symplectic geometry. Math Z 2002; 240(1): 141–167
2	V Ginzburg. Non-commutative symplectic geometry, quiver varieties, and operads. Math Res Lett 2001; 8(3): 377–400
3	J Y Guo, R Martínez-Villa. Algebra pairs associated to McKay quivers. Comm Algebra 2002; 30(2): 1017–1032
4	M Lothaire. Combinations on Words. Encyclopedia Math Appl, Vol 17. Reading: Addison-Wesley Publishing Co, 1983
5	C Q Mei, D M Yu. The structure of Necklace Lie algebras. Math Pract Theory 2012; 42(1): 195–204
6	L G Peng. Lie algebras determined by finite Auslander-Reiten quivers. Comm Algebra 1998; 26(9): 2711–2725
7	G F Post. On the structure of transitively differential algebras. J Lie Theory 2001; 11(1): 111–128
8	C Reutenauer. Free Lie Algebras. London Math Soc Monogr Ser, Vol 7. Oxford: Clarendon Press, 1993
9	D M Yu, B J Li, Q H Wan. The automorphism group and simplicity of the generalized Virasoro-like Lie algebra. Adv Math (China) 2013; 42(5): 620–624
10	D M Yu, C H Lu. Special property of Lie algebra L(Z, f, δ). Adv Math (China) 2006; 35(6): 707–711
11	D M Yu, C Q Mei, J Y Guo. Homomorphisms of some special necklace Lie algebras. Chinese Ann Math Ser A 2009; 30(4): 551–562
12	D M Yu, C Q Mei, J Y Guo. Automorphisms and automorphism groups of Necklace Lie algebras. Chinese Ann Math Ser A 2013; 34(5): 569–578

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