Abstract
Let B be an algebra over a field \(\mathbf {k}\). We define what it means for a subset of \({{\,\mathrm{Der}\,}}_\mathbf {k}(B)\) to be a locally nilpotent set. We prove some basic results about that notion and explore the following questions. Let L be a Lie subalgebra of \({{\,\mathrm{Der}\,}}_\mathbf {k}(B)\); if \(L \subseteq {{\,\mathrm{LND}\,}}(B)\) then does it follow that L is a locally nilpotent set? Does it follow that L is a nilpotent Lie algebra?
Research supported by grant RGPIN/2015-04539 from NSERC Canada.
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Notes
- 1.
- 2.
Recall that \(a_n \cdots a_0 \cdot x = 0\) means \(a_n \cdot ( a_{n-1} \cdots ( a_1 \cdot (a_0 \cdot x)) \dots )=0\), by the right-associativity convention (1.4).
- 3.
A subset H of an associative algebra A is nilpotent if there exists n such that \(h_n \cdots h_1=0\) for all \((h_1, \dots , h_n) \in H^n\). By Corollary 4.9, H is nilpotent if and only if \({\bar{H}}\) is nilpotent, where \({\bar{H}}\) is the subalgebra of A generated by H.
- 4.
See for instance p. 6 of [5].
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Daigle, D. (2020). Locally Nilpotent Sets of Derivations. In: Kuroda, S., Onoda, N., Freudenburg, G. (eds) Polynomial Rings and Affine Algebraic Geometry. PRAAG 2018. Springer Proceedings in Mathematics & Statistics, vol 319. Springer, Cham. https://doi.org/10.1007/978-3-030-42136-6_2
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DOI: https://doi.org/10.1007/978-3-030-42136-6_2
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