Is every indecomposable homogeneous continuum unicoherent?

  • Continuum = compact connected metrizable space

  • Indecomposable = not the union of any two proper subcontinua.

  • Homogeneous = for every two points $ x$ and $ y$ there is a homeomorphism of the space onto itself which maps $ x$ to $ y$ .

  • Unicoherent = the intersection of every two subcontinua is connected.

Examples of indecomposable homogeneous continua include solenoids, the pseudoarc, and solenoids of pseudoarcs (a solenoid with each point is blown up into a pseudoarc). I think it’s an open problem to determine whether this list is complete, but so far all of the examples are unicoherent. Is there a theorem stating that such continua must be unicoherent?

Holomorphic Poisson structures on $C P^{n-1}$ and homogeneous Poisson structures on $C^n$

Is it correct that any holomorphic Poisson structure on $ C P^{n-1}$ can be lifted to a homogeneous Poisson structure on $ C^n$ ? By homogeneous I mean a quadratic Poisson structure of the form $ \{z_i,z_j\}=q_{ij}^{kl}z_kz_l$ where coefficients $ q_{ij}^{kl}$ are constants.

I suspect that this is correct but do not know any reference nor any idea of possible proof. Could you please help me with these?

Coordinate ring of an equivariant embedding of a homogeneous projective variety

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 weight. It follows from the Borel-Weil-Bott theorem that the homogeneous coordinate ring $ A_\lambda(G/P)$ is a sum of highest weight representations. Namely, \begin{align*} A_\lambda(G/P) = \oplus_{n \in \mathbb{N}} V(n\lambda). \end{align*}

Lie superalgebra: Do we have an analogue result of this in the super case? More precisely, let $ G$ be a simply connected analytic supergroup with $ \mathfrak{g} = Lie(G)$ (a basic classical Lie superalgebra) and a Borel subsupergroup $ B$ . Let $ P$ be a subsupergroup containing $ B$ . I know that there is an analogue of Borel-Weil-Bott theorem for supercase, but I wonder if there is a result for the coordinate ring as above. I guess we might need to assume $ \lambda$ is a typical weight.