Malcev’s paper “On a class of homogeneous spaces” in English

I am struggling to find the English translation of Malcev’s paper “On a class of homogenous spaces” providing foundational material for nil-manifolds. To be precise this paper: Malcev, A. I. On a class of homogeneous spaces. Amer. Math. Soc. Translation 1951, (1951). no. 39, 33 pp. (mathscinet link) . It would be really important, for a project I am doing, to find this paper and I did not succeed neither on the website of the AMS nor by standard googling, which gives tons of papers referring to it.

Can anyone provide a reference to a place where to download the paper? I am at an institution with free access virtually everywhere, I just need a place with the actual paper in English (yeah in Russian I could find it).

Volume of balls in homogeneous manifolds

Let $ X=G/H$ be a homogeneous manifold, where $ G$ and $ H$ are connected Lie groups and assume there is given a $ G$ -invariant Riemannian metric on $ X$ . Let $ B(R)$ be the closed ball of radius $ R>0$ around the base point $ eH$ and let $ b(R)$ denote its volume. Is it rue that $ $ \lim_{\varepsilon\to 0}\ \limsup_{R\to\infty}\ \frac{b(R+\varepsilon)}{b(R)}=1? $ $ The idea somehow being that volume growth is largest with constant negative curvature in which case it is exponential and thus satisfies our claim.

Homogeneous space for intersection of subgroups

Suppose we have a Lie group $ G$ and two subgroups $ P_1$ and $ P_2$ . We can then study the homogeneous spaces $ M_1=G/P_1$ and $ M_2=G/P_2$ , and bundles on these spaces associated to representations of $ P_1$ and $ P_2$ . For example, let us take representations $ \rho_1$ and $ \rho_2$ of the respective $ P_i$ and imagine that we studied the associated bundles and maybe found some interesting structures.

I now want to consider the homogeneous space $ M_{12}=G/(P_1\cap P_2)$ . Now $ \rho_1\otimes \rho_2$ is naturally a representation of $ P_1\cap P_2$ , and therefore there is an associated bundle on $ M_{12}$ . Furthermore, there are natural subspaces of $ T M_{12}$ corresponding to $ \mathfrak{p}_i/(\mathfrak{p}_1\cap \mathfrak{p}_2)$ . In the case when $ P_1$ is conjugate to $ P_2$ , one can interpret $ M_{12}$ as a $ G$ -orbit in the configuration space of pairs of points in $ M_1$ . If there is only one orbit in this configuration space, then the subspaces $ \mathfrak{p}_i/(\mathfrak{p}_1\cap \mathfrak{p}_2)$ span $ TM_{12}$ .

I am working on a problem in this general setup (in my case I have three $ P$ ‘s and they are conjugate parabolic subgroups), but I imagine that this is some fairly standard construction. My question is whether it is the case, and if yes, what is a good reference or some keywords to look for? Many thanks in advance.

Finding the linear mapping between homogeneous coordinates of affine camera

If I have an affine camera with a projection relationship governed by:

\begin{equation} \begin{bmatrix} x & y \end{bmatrix}^T = A \begin{bmatrix} X & Y & Z \end{bmatrix}^T + b \end{equation} where A is a 2×3 matrix and b is a 2×1 vector. How can I form a matrix representing the linear mapping between the world point $ (X,Y,Z)$ and image point $ (x,y)$ if they are represented by homogeneous vectors?

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.