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The linear spans of $[a,x]$ and $[x,a]$, in a Lie superalgebra (i.e. a $\mathbb{Z}_2$-graded Lie algebra) are generally not the same (unlike the Lie algebras case): Since $\mathfrak a$ is not a grade …

$P(1)$ 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 …

Although i am not an expert in the topic, i did some studying on the references you provided together with Kac's monograph on Infinite dimensional Lie algebras. I do not have very clear answers to …

In: Kazhdan-Lustzig polynomials and character formulae for the Lie superalgebra $gl(m|n)$, J. Amer. Math. Soc. 16 (2002), 185–231, J. Brundan develops a conjecture on the characters for the irreducibl …

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 …

Regarding the "what is happening in the super case"; yes i agree that in some sense, it has to do with the odd simple roots but i think it is deeper than that: In the case of semisimple, complex, Lie …

Yes there is a complete classification of finite dimensional, simple Lie superalgebras (over $\mathbb{C}$), which -up to a certain extent- goes very much in parallel with the corresponding case of Lie …

I agree with the suggestion in the comments for searching the front of the math arXiv (as an entry point), because this is a quite broad and active topic (and i am not sure it can be fully covered wit …

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 did not know these identities but after a small search, i think that some relations are missing from your post: In: http://mathworld.wolfram.com/KaehlerIdentities.html some additional relations (see …

If i have correctly understood your question, i think that the answer can be found at M. D. Gould, R. B. Zhang, Classification of all star and grade star irreps of gl(n|1), J. of Math. Phys., 31, 15 …

The notion of hopf algebras has slowly emerged from the work of topologists in the late '30's and '40's on the cohomology of compact Lie groups and their homogeneous spaces. Initially the term had bee …

$\DeclareMathOperator\chr{char}$Yes this is true: Under your assumptions $\mathcal{P}(U(g))=g$. Also, since for any primitive element $x$ we have $\epsilon(x)=0$, for any grouplike element $y$ we have …