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Are there any applications of abstract algebra (group theory, rings, fields etc.) in any branch of either economics or finance?

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    $\begingroup$ Great question! Well, linear algebra is extremely useful. From regressions to asset payoffs in discrete time financial market models, matrices are everywhere. Eigenvectors pop up in principal component analysis. Riesz representation theorem links linear pricing functionals to risk-neutral expectations but that theorem may be classified as functional analysis instead. I heard that algebraic geometry is used in game theory but I'm not too sure on this. $\endgroup$ – KeSchn Apr 29 at 17:57
  • $\begingroup$ Indeed, I use a lot of linear algebra in my researches and work. However are there any applications of more general obejcts such groups, rings and fields? $\endgroup$ – SlavicDoomer Apr 29 at 18:01
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    $\begingroup$ I know there are some papers on Lie algebras and option pricing (just google the two terms) and on algebraic geometry in game theory. But I don't know of any major, mainstream applications. I remember watching an interview with Peter Carr last year who seems to be working on applications of abstracr algebra to QF. I've not checked out his most recent papers to know whether he found any applications. $\endgroup$ – KeSchn Apr 29 at 18:12
  • $\begingroup$ Have a look here for quant.stackexchange.com/questions/3297/… and quant.stackexchange.com/questions/7112/… $\endgroup$ – KeSchn Apr 29 at 18:18
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Yes, I've seen some interesting papers that improve one's insight into how things work, even if it is not clearly applicable to practice.

Belal Ehsan Baaquie published several books on applications of quantum mechanics and quantum field theory to finance, particularly interest rates. They're definitely fun to read. The most recent one is Quantum Field Theory for Economics and Finance (2018).

There was a related question here (not answered) a few years ago about using Lie Groups for interest rates. The paper Park, Chun, Han Webber, Interest rate models on Lie groups is an example of this.

There have been attemps to view credit from the category theory viewpoint, for example Joseph Tanega. Default Invariance, A Naïve Category Theory of Law and Finance.

Not quite finance, but Algebraic Models for Accounting Systems by Nehmer, Perez, Robinson, Rambaud explores a group-theoretic view of double-entry accounting.

Not quite modern algrebra, but related:

Fractional Calculus and Fractional Processes with Applications to Financial Economics by Fallahgoul, Focardi, Fabozzi.

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