Journal of Lie Theory 34 (2024), No. 4, 957--974
Copyright Heldermann Verlag 2024
Invariants in Varieties of Lie Algebras
Constantinos E. Kofinas
Department of Mathematics, University of the Aegean, Karlovassi, Samos, Greece
kkofinas@aegean.gr
[Abstract-pdf]
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 $R_{2}^{S_{2}}$.
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.
MSC: 17B01, 17B30.
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