# Questions tagged [lie-superalgebras]

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11 questions
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### Coordinate superalgebra

Lie algebra: Let $G$ be a semisimple, simply connected linear algebraic group with a fixed Borel subgroup $B$. Let $P$ be a parabolic subgroup containing $B$. Let $\lambda$ be a dominant integral ...
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### super commutation of matrices

Let $M_{p|q}(\mathbb{C}) = M_{p|q}(\mathbb{C})_0 \oplus M_{p|q}(\mathbb{C})_1$ be the super algebra of all $(p+q) \times (p+q)$ matrices. Let $A, B \in M_{p|q}(\mathbb{C})$ (not necessarily ...
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### General commutators of derivations of the exterior algebra

Let $M$ be a smooth manifold and let $\Omega(M)$ be the exterior algebra of smooth differential forms over $M$. The $\mathbb R$-linear map $D:\Omega(M)\rightarrow\Omega(M)$ is a derivation of the ...
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### Action of quadratic Casimir on cochain/cohomology spaces

Let $\mathfrak g$ be a semisimple Lie algebra and let $M$ be a nontrivial simple $\mathfrak g$-module. Then quadratic Casimir of $\mathfrak g$ acts on $M$ as multiplication by a nonzero scalar $c_M$. ...
### What is the commutator in $gl(m|n)$?
For Lie algebra $gl(m)$, the commutator is \begin{align} [E_{ij}, E_{kl}] = \delta_{jk}E_{il} - \delta_{li}E_{kj}. \end{align} What is the commutator $[E_{ij}, E_{kl}]$ in Lie superalgebra $gl(m|n)$? ...