and
A super Lie algebra is the analog of a Lie algebra in superalgebra/supergeometry.
See also supersymmetry.
A super Lie algebra over a field $k$ is a Lie algebra internal to the symmetric monoidal $k$-linear category SVect of super vector spaces.
This means that a super Lie algebra is
a super vector space $\mathfrak{g} = \mathfrak{g}_{even} \oplus \mathfrak{g}_{odd}$;
equipped with a bilinear bracket
that is graded skew-symmetric: is is skew symmetric on $\mathfrak{g}_{even}$ and symmetric on $\mathfrak{g}_{odd}$.
that satisfied the $\mathbb{Z}_2$-graded Jacobi identity:
Equivalently, a super Lie algebra is a “super-representable” Lie algebra internal to the cohesive (∞,1)-topos Super∞Grpd over the site of super points.
See the discussion at superalgebra for details on this.
The super Poincare Lie algebra and various of its polyvector extension are super-extension of the ordinary Poincare Lie algebra. These are the supersymmetry algebras in the strict original sense of the word.
higher super Lie algebras
Just as Lie algebras are categorified to L-infinity algebras and L-infinity algebroids, so super Lie algebras categorifie to super L-infinity algebras. A secretly famous example is the
One of the original references (or the original reference?) is
A review of the classification is in
Surveys:
A useful survey with more pointers to the literature is
Another useful survey is
D. Leites, Lie superalgebras, J. Soviet Math. 30 (1985), 2481–2512 (web)
M. Scheunert, The theory of Lie superalgebras. An introduction, Lect. Notes Math. 716 (1979)
A useful PhD thesis covering Lie superalgebras and superalgebras more generally is