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Let $K$ be a commutative ring with unity. Let $A$ be a unital algebra over $K$. We write $[x,y]= xy-yx$ for every $x,y \in A$ and we call it Lie product (or Lie bracket). A linear map $L: A \to A$ is called a Lie derivation if $L([x,y]) = [L(x),y]+[x,L(y)]$ for every $x,y \in A$.

This kind of derivations is studied in many papers but none of them provides a real-world application (to the best of my knowledge). I want to know some applications of this kind of derivations in the real world.

Thanks.

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  • $\begingroup$ Most results in abstract algebra lack direct real-world applications, because algebraic structures arise in pure rather than applied mathematics. Why would you expect real-world applications of Lie derivations? $\endgroup$
    – Marc van Leeuwen
    Commented Apr 8, 2020 at 11:04

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They appear in control theory. For example, one can use them to show that a car which can drive forwards/backwards and turn the front wheels, can in fact also move sideways, if infinitesimal motion is allowed.

http://planning.cs.uiuc.edu/node829.html

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