## 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:

$$$$\begin{bmatrix} x & y \end{bmatrix}^T = A \begin{bmatrix} X & Y & Z \end{bmatrix}^T + b$$$$ where A is a 2×3 matrix and b is a 2×1 vector. How can I form a matrix representing the linear mapping between the world point $$(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.

## Is $x f(x,y)+y g(x,y)$ homogeneous if f and g homogeneous?

If $$f_{1}(x,y)$$ and $$f_{2}(x,y)$$ are two homogeneous polynomials then prove that : $$\frac{f_{1}(x,y)+f_{2}(x,y)}{xf_{1}(x,y)+yf_{2}(x,y)}$$ is homogeneous.

## Connected and homogeneous $T_2$-space not homeomorphic to a subset of $\mathbb{R}^n$

What is an example of an connected and homogeneous $$T_2$$-space $$(X,\tau)$$ with $$|X|=2^{\aleph_0}$$ such that for no $$n\in\mathbb{R}$$ the space $$(X,\tau)$$ is homeomorphic to a subspace of $$\mathbb{R}^n$$?