Several versions of non-linear Lie algebras exist - at least in the physics literature. One version is just an ordinary Lie algebra but where the underlying vector space is a polynomial algebra. Another is a `field dependent' version of a Lie algebra. The quadratic case at least is physically relevant. Any really mathematical references available?
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2$\begingroup$ Can you give some physics references? $\endgroup$– Ben McKayFeb 19, 2019 at 16:53
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13$\begingroup$ Can you also elaborate a bit on what you mean by "non linear"? (Maybe by fleshing out the examples you hinted at) $\endgroup$– QfwfqFeb 19, 2019 at 16:56
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$\begingroup$ To elaborate on @Qfwfq's question: by "non-linear" do you mean "not the Lie algebra of a finite-dimensional linear Lie group"? $\endgroup$– LSpiceFeb 19, 2019 at 17:01
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$\begingroup$ Thanks for pushing me: Ikeda Two-dimensional gravity and nonlinear gauge theory with references to special cases - e.g W-algebras. One version is just a strict Lie algebra with underly8ng vector space a free polynomial algebra. Others seem to be field dependent. $\endgroup$– Jim StasheffFeb 20, 2019 at 17:11
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6$\begingroup$ You've referred twice to field dependence, but it's still not clear what that means. What is an example of a field-dependent Lie algebra, or version of a Lie algebra? What condition on them makes you call them 'non-linear'? $\endgroup$– LSpiceFeb 22, 2019 at 2:28
1 Answer
The main reference I can find is de Sole and Kac's papers https://arxiv.org/abs/math-ph/0312042 and https://arxiv.org/abs/math-ph/0511055.
Definition: [p2 of first paper] A nonlinear Lie algebra is a $\mathbf{Z}_{\ge 0}$-graded vector space $L$ along with a map to the tensor algebra $$L\wedge L\ \to\ T(L)$$ respecting the grading $|[\alpha,\beta]|\le |\alpha|+|\beta|$ (having extended the grading to $T(L)$ in the obvious way) and such that $$ (\text{Jacobi identity for }\alpha,\beta,\gamma)$$ is a sum of elements of the form $a\otimes (b\otimes c-c\otimes b-[b,c])\otimes d$ with total degree $<|\alpha|+|\beta|+|\gamma|$.
You can then define $U(L)$ as the quotient of $T(L)$ by the ideal generated by elements of this form (with any degree).
About "field dependence": there are in fact four different notions: $$\require{AMScd}\begin{CD} \text{Lie algebra} @. \text{non-linear Lie algebra} \\ @. @. \\ \text{Lie conformal algebra} @. \text{non-linear Lie conformal algebra} \end{CD}$$ and the bottom row are "vertex algebra" sort of objects, i.e. depend on a parameter $z\in \mathbf{A}^1$.
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4$\begingroup$ It's terrible to make a definition of "non-P" which is something else than the negation of some given Property P. I hope subsequent papers will opt for a more reasonable terminology. $\endgroup$– YCorSep 16, 2023 at 15:18
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2$\begingroup$ Reminds me of the definition of noncommutative spaces (which includes commutative spaces). $\endgroup$ Oct 16, 2023 at 14:52