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There has been quite a lot of literature on the applications of $q$-numbers, $q$-derivatives, $q$-deformations, etc, of various algebraic models of physics. Such applications range from $q$-deformatio …

I think that the answer lies in the "educational culture" of physicists. Physicists are often used -well at least at the undergraduate level- to learn and perform complicated computations with abstra …

The three formalisms of classical mechanics, i.e. the Newtonian, the Lagrangian (analytical mechanics) and the Hamiltonian (canonical formalism) are generally not equivalent to each other -at least no …

About your Q1: I think that the simplest—and most obvious—way to think mathematically about the physical meaning of Planck's constant $h$ is that it is a kind of quantitative measure of the departure …

This is a very interesting question. I have also made some search but i have not found this result explicitly mentioned somewhere in the literature. However, i remember i have heard such a claim in th …

I am not sure if this is exactly what you are looking for, but there have been some classic works, developing general methods for such topics: In Dynkin, Semisimple subalgebras of semisimple Lie al …

Some further references, that might be of interest for your purposes: You can see at this article and the book Supermanifolds Theory and Applications by Alice Rogers. The article discusses -among o …

I do not think that it makes sense to say that the gauge-fixing is a special case of BRST quantization: $\rightarrow$ The gauge-fixing procedure is actually a normalization technique and it is utili …

The Heisenberg-Weyl algebra or the Weyl algebra or the algebra of the Canonical Commutation Relations (CCR) is generated by the $p,q$ generators subject to the relation $$ [q, p] = i \hbar I \ \ \ \ …

The Serre relations (some authors also call them Serre-Chevalley relations) for the finite dimensional, complex, basic, classical, simple Lie superalgebras -in analogy with the Lie algebra case- rea …

I understand that the OP's original focus is classical statistical mechanics. However, i think that the question is of interest from a more general viewpoint including the dynamical systems/integrabil …

Here's what I have done: $\bullet$ Let the Lagrangian $L(q_{i},\dot{q}_{i},t)$, which under the point transformations $$ \{q_{i}\}\leftrightsquigarrow\{Q_{i}\} $$ given by the invertible relations $ …

Regarding your second question, on other possible deformations of $sl(2)$: There have been various studies on (multi-parametric) deformations of Lie algebras -as has already been mentioned in the co …

I think that -apart from the applications in CFT and TFT already mentioned in previous answers- one of the most fundamental applications of braided Hopf algebras (with both non-trivial and "calculable …

If you are looking for geometric-algebraic interpretations of supersymmetric field theories, then non-commutative geometry -in the sense of A. Connes- seems to be the natural playground. There has bee …