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 Mat. Sb., 2009, Volume 200, Number 2, Pages 31–60 (Mi msb4032)

Some properties of the space of $n$-dimensional Lie algebras

V. V. Gorbatsevich

Moscow State Aviation Technological University

Abstract: Some general properties of the space $\mathscr L_n$ of $n$-dimensional Lie algebras are studied. This space is defined by the system of Jacobi's quadratic equations. It is proved that these equations are linearly independent and equivalent to each other (more precisely, the quadratic forms defining these equations are affinely equivalent). Moreover, the problem on the closures of some orbits of the natural action of the group $\mathrm{GL}_n$ on $\mathscr L_n$ is considered. Two Lie algebras are indicated whose orbits are closed in the projectivization of the space $\mathscr L_n$. The intersection of all irreducible components of the space $\mathscr L_n$ is also treated. It is proved that this intersection is nontrivial and consists of nilpotent Lie algebras. Two Lie algebras belonging to this intersection are indicated. Some other results concerning arbitrary Lie algebras and the space $\mathscr L_n$ formed by these algebras are presented.
Bibliography: 17 titles.

Keywords: Lie algebra, Jacobi's identity, irreducible component, contraction.

DOI: https://doi.org/10.4213/sm4032

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English version:
Sbornik: Mathematics, 2009, 200:2, 185–213

Bibliographic databases:

UDC: 512.554.3
MSC: Primary 17B05; Secondary 17B30, 17B40

Citation: V. V. Gorbatsevich, “Some properties of the space of $n$-dimensional Lie algebras”, Mat. Sb., 200:2 (2009), 31–60; Sb. Math., 200:2 (2009), 185–213

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb4032
• https://doi.org/10.4213/sm4032
• http://mi.mathnet.ru/eng/msb/v200/i2/p31

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This publication is cited in the following articles:
1. V. V. Gorbatsevich, “On the intersection of irreducible components of the space of finite-dimensional Lie algebras”, Sb. Math., 203:7 (2012), 976–995
2. V. V. Gorbatsevich, “On the frames of spaces of finite-dimensional Lie algebras of dimension at most 6”, Sb. Math., 205:5 (2014), 633–645
3. D. V. Millionshchikov, R. Jimenez, “Geometry of Central Extensions of Nilpotent Lie Algebras”, Proc. Steklov Inst. Math., 305 (2019), 209–231
4. V. V. Gorbatsevich, “Nekotorye svoistva pochti abelevykh algebr Li”, Izv. vuzov. Matem., 2020, no. 4, 26–42
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