Homogeneous irreducible supermanifolds and graded Lie superalgebras

A depth one grading $\mathfrak{g}= \mathfrak{g}^{-1}\oplus \mathfrak{g}^0 \oplus \mathfrak{g}^1 \oplus \cdots \oplus \mathfrak{g}^{\ell}$ of a finite dimensional Lie superalgebra $\mathfrak{g}$ is called nonlinear irreducible if the isotropy representation $\mathrm{ad}_{\mathfrak{g}^0}|_{\mathfrak{g}^{-1}}$ is irreducible and $\mathfrak{g}^1 \neq (0)$. An example is the full prolongation of an irreducible linear Lie superalgebra $\mathfrak{g}^0 \subset \mathfrak{gl}(\mathfrak{g}^{-1})$ of finite type with non-trivial first prolongation. We prove that a complex Lie superalgebra $\mathfrak{g}$ which admits a depth one transitive nonlinear irreducible grading is a semisimple Lie superalgebra with the socle $\mathfrak{s}\otimes \Lambda(\mathbb{C}^n)$, where $\mathfrak{s}$ is a simple Lie superalgebra, and we describe such gradings. The graded Lie superalgebra $\mathfrak{g}$ defines an isotropy irreducible homogeneous supermanifold $M=G/G_0$ where $G$, $G_0$ are Lie supergroups respectively associated with the Lie superalgebras $\mathfrak{g}$ and $\mathfrak{g}_0 := \bigoplus_{p\geq 0} \mathfrak{g}^p$.Comment: 28 pages, 8 Tables (v2: acknowledgments updated, final version to be published in IMRN

The necessity of the second postulate in special relativity

Many authors noted that the principle of relativity together with space-time homogeneity and isotropy restrict the form of the coordinate transformations from one inertial frame to another to being Lorentz-like. The equations contain a free parameter, $k$ (equal to $c^{-2}$ in special relativity), which value is claimed to be merely an empirical matter, so that special relativity does not need the postulate of constancy of the speed of light. I analyze this claim and argue that the distinction between the cases $k = 0$ and $k \neq 0$ is on the level of a postulate and that until we assume one or the other, we have an incomplete structure that leaves many fundamental questions undecided, including basic prerequisites of experimentation. I examine an analogous case in which isotropy is the postulate dropped and use it to illustrate the problem. Finally I analyze two attempts by Sfarti, and Behera and Mukhopadhyay to derive the constancy of the speed of light from the principle of relativity. I show that these attempts make hidden assumptions that are equivalent to the second postulate