Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
solvable Lie algebra; nilpotent residual; Frattini ideal
solvable Lie algebra; nilpotent residual; Frattini ideal
Summary:
Let $\mathcal {N}$ denote the class of nilpotent Lie algebras. For any finite-dimensional Lie algebra $L$ over an arbitrary field $\mathbb {F}$, there exists a smallest ideal $I$ of $L$ such that $L/I\in \mathcal {N}$. This uniquely determined ideal of $L$ is called the nilpotent residual of $L$ and is denoted by $L^{\mathcal {N}}$. In this paper, we define the subalgebra $S(L)=\bigcap \nolimits _{H\leq L}I_L(H^{\mathcal {N}})$. Set $S_0(L) = 0$. Define $S_{i+1}(L)/S_i (L) =S(L/S_i (L))$ for $i \geq 1$. By $S_{\infty }(L)$ denote the terminal term of the ascending series. It is proved that $L= S_{\infty }(L)$ if and only if $L^{\mathcal {N}}$ is nilpotent. In addition, we investigate the basic properties of a Lie algebra $L$ with $S(L)=L$.
References:
[1] Barnes, D. W.: Nilpotency of Lie algebras. Math. Z. 79 (1962), 237-238. DOI 10.1007/BF01193118 | MR 0150177 | Zbl 0122.04001
[2] Barnes, D. W.: On the cohomology of soluble Lie algebras. Math. Z. 101 (1967), 343-349. DOI 10.1007/BF01109799 | MR 0220784 | Zbl 0166.04102
[3] Barnes, D. W.: The Frattini argument for Lie algebras. Math. Z. 133 (1973), 277-283. DOI 10.1007/BF01177868 | MR 0330244 | Zbl 0253.17003
[4] Barnes, D. W., Gastineau-Hills, H. M.: On the theory of soluble Lie algebras. Math. Z. 106 (1968), 343-354. DOI 10.1007/BF01115083 | MR 0232807 | Zbl 0164.03701
[5] Barnes, D. W., Newell, M. L.: Some theorems on saturated homomorphs of solvable Lie algebras. Math. Z. 115 (1970), 179-187. DOI 10.1007/BF01109856 | MR 0266969 | Zbl 0197.03003
[6] Chen, L., Meng, D.: On the intersection of maximal subalgebras in a Lie superalgebra. Algebra Colloq. 16 (2009), 503-516. DOI 10.1142/S1005386709000479 | MR 2536774 | Zbl 1235.17009
[7] Gong, L., Guo, X.: On the intersection of the normalizers of the nilpotent residuals of all subgroups of a finite group. Algebra Colloq. 20 (2013), 349-360. DOI 10.1142/S1005386713000321 | MR 3043320 | Zbl 1281.20020
[8] Gong, L., Guo, X.: On normalizers of the nilpotent residuals of subgroups of a finite group. Bull. Malays. Math. Sci. Soc. (2) 39 (2016), 957-970. DOI 10.1007/s40840-016-0338-y | MR 3515061 | Zbl 06625446
[9] Humphreys, J. E.: Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics 9, Springer, New York (1972). DOI 10.1007/978-1-4612-6398-2 | MR 0323842 | Zbl 0254.17004
[10] Marshall, E. I.: The Frattini subalgebra of a Lie algebra. J. Lond. Math. Soc. 42 (1967), 416-422. DOI 10.1112/jlms/s1-42.1.416 | MR 0217132 | Zbl 0166.04101
[11] Schwarck, F.: Die Frattini-Algebra einer Lie-Algebra. Dissertation, Universität Kiel, Kiel German (1963).
[12] Shen, Z., Shi, W., Qian, G.: On the norm of the nilpotent residuals of all subgroups of a finite group. J. Algebra 352 (2012), 290-298. DOI 10.1016/j.jalgebra.2011.11.018 | MR 2862187 | Zbl 1255.20019
[13] Stitzinger, E. L.: On the Frattini subalgebra of a Lie algebra. J. Lond. Math. Soc., II. Ser. 2 (1970), 429-438. DOI 10.1112/jlms/2.Part_3.429 | MR 0263885 | Zbl 0201.03603
[14] Stitzinger, E. L.: Covering-avoidance for saturated formations of solvable Lie algebras. Math. Z. 124 (1982), 237-249. DOI 10.1007/BF01110802 | MR 0297829 | Zbl 0215.38601
[15] Su, N., Wang, Y.: On the normalizers of $\frak{F}$-residuals of all subgroups of a finite group. J. Algebra 392 (2013), 185-198. DOI 10.1016/j.jalgebra.2013.06.037 | MR 3085030 | Zbl 1312.20015
[16] Towers, D. A.: A Frattini theory for algebras. Proc. Lond. Math. Soc., III. Ser. 27 (1973), 440-462. DOI 10.1112/plms/s3-27.3.440 | MR 0427393 | Zbl 0267.17004
[17] Towers, D. A.: Elementary Lie algebras. J. Lond. Math. Soc., II. Ser. 7 (1973), 295-302. DOI 10.1112/jlms/s2-7.2.295 | MR 0376782 | Zbl 0267.17006
[18] Towers, D. A.: $c$-ideals of Lie algebras. Commun. Algebra 37 (2009), 4366-4373. DOI 10.1080/00927870902829023 | MR 2588856 | Zbl 1239.17006
[19] Towers, D. A.: The index complex of a maximal subalgebra of a Lie algebra. Proc. Edinb. Math. Soc., II. Ser. 54 (2011), 531-542. DOI 10.1017/S0013091509001035 | MR 2794670 | Zbl 1228.17007
[20] Towers, D. A.: Solvable complemented Lie algebras. Proc. Am. Math. Soc. 140 (2012), 3823-3830. DOI 10.1090/S0002-9939-2012-11244-4 | MR 2944723 | Zbl 1317.17010
Search
Partner of
