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This article is cited in 4 scientific papers (total in 4 papers)
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.
Received: 09.11.2007 and 25.07.2008
Citation:
V. V. Gorbatsevich, “Some properties of the space of $n$-dimensional Lie algebras”, Sb. Math., 200:2 (2009), 185–213
Linking options:
https://www.mathnet.ru/eng/sm4032https://doi.org/10.1070/SM2009v200n02ABEH003991 https://www.mathnet.ru/eng/sm/v200/i2/p31
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