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$P(1)$ is not simple: To see why, consider the strange, type I, classical, simple, complex, LS $P(n)$, $n\geq 2$ realized as the set of complex, $(2n+2)\times(2n+2)$ matrices $\mathbf{M}$, with grad …

Akshay Venkatesh has some application-oriented work: On quantum unique ergodicity for locally symmetric spaces, see also: Heat-kernel asymptotics of locally symmetric spaces of rank one and Chern-S …

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 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 …

There are actually lots of applications of solvable Lie algebras, especially in the field of integrable systems where the solvabiltiy of Hamilton's equations of motion is frequently related to the so …

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 $ …

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 \ \ \ \ …

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 …

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 …

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 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 …

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 …

Maybe the following paper might prove helpful to your question: Masatoshi Yamazaki, Branching Diagram for Special Unitary Group SU(n), J. Phys. Soc. Jpn. 21, pp. 1829-1832 (1966)

Classical, Simple, Complex, Lie superalgebras and Complex, Affine, Kac-Moody algebras and Complex, Kac-Moody Lie superalgebras have an associated graph -up to isomorphism- in the sense of a generalize …

I am not sure if this is the qualitative/geometric interpretation -of the integrality of the $l$ parameter- you are looking for, but if the parameter $l$ is a non-negative integer then the Legendre po …