Invariants in Varieties of Lie Algebras
Journal of Lie Theory, Volume 34 (2024) no. 4, pp. 957-974
For a positive integer $n$, with $n \geq 2$, let $L_{n}$ be the free Lie algebra over a field $K$ of characteristic 0 and let $P_{n} = L_{n}/V_{1}(L_{n})$ and $Q_{n} = L_{n}/V_{2}(L_{n})$ be relatively free Lie algebras, with $V_{1}(L_{n}) \subseteq V_{2}(L_{n})$. For a non-trivial finite subgroup $G$ of ${\rm GL}_{n}(K)$, let $P_{n}^{G}$ and $Q_{n}^{G}$ be the Lie subalgebras of invariants in $P_{n}$ and $Q_{n}$, respectively. We give connections between $P_{n}^{G}$ and $Q_{n}^{G}$. For $G = S_{2}$, we apply our methods to $L_{2}/L_{2}^{\prime\prime}$ and $R_{2} = L_{2}/(\gamma_{3}(L_{2}^{\prime}) + (\gamma_{3}(L_{2}))^{\prime})$ (i.e., $R_{2}$ is a free (nilpotent of class 2)-by-abelian and abelian-by-(nilpotent of class 2) Lie algebra of rank 2). We give a basis and a minimal infinite generating set for $R_{2}^{S_{2}}$ and we find a presentation of $R_{2}^{S_{2}}$.
DOI:
10.5802/jolt.1367
Classification:
17B01, 17B30
Keywords: Varieties of Lie algebras, relatively free Lie algebras, algebra of invariants, symmetric polynomials, free (nilpotent of class 2)-by-abelian and abelian-by-(nilpotent of class 2) Lie algebra
Keywords: Varieties of Lie algebras, relatively free Lie algebras, algebra of invariants, symmetric polynomials, free (nilpotent of class 2)-by-abelian and abelian-by-(nilpotent of class 2) Lie algebra
@article{JOLT_2024_34_4_a8,
author = {C. E. Kofinas},
title = {Invariants in {Varieties} of {Lie} {Algebras}},
journal = {Journal of Lie Theory},
pages = {957--974},
year = {2024},
volume = {34},
number = {4},
doi = {10.5802/jolt.1367},
zbl = {1567.17006},
url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1367/}
}
C. E. Kofinas. Invariants in Varieties of Lie Algebras. Journal of Lie Theory, Volume 34 (2024) no. 4, pp. 957-974. doi: 10.5802/jolt.1367
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