The Homogeneous Complex Monge-Ampère Equation and the Infinite Dimensional Versions of Classic Symmetric Spaces

Abstract The notion of a valuation on convex bodies is very classical; valuations on a class of functions have been introduced and studied by M. Ludwig and others. We study an explicit relation between continuous valuations on convex functions which are invariant under adding arbitrary linear functionals, and translation invariant continuous valuations on convex bodies. More precisely, we construct a natural linear map from the former space to the latter and prove that it has dense image and infinite-dimensional kernel. The proof uses the author’s irreducibility theorem and properties of the real Monge–Ampère operators due to A.D. Alexandrov and Z. Blocki. Furthermore we show how to use complex, quaternionic, and octonionic Monge–Ampère operators to construct more examples of continuous valuations on convex functions in an analogous way.

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TOWARD SYMMETRIC SPACES OF AFFINE KAC–MOODY TYPE

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887806001648 ◽

2006 ◽

Vol 03 (05n06) ◽

pp. 881-898 ◽

Cited By ~ 4

Author(s):

ERNST HEINTZE

Keyword(s):

Fixed Point ◽

Lie Groups ◽

Symmetric Spaces ◽

Point Group ◽

Infinite Dimensional ◽

Simple Lie Groups ◽

Finite Dimensional ◽

Expository Article

In this expository article we discuss some ideas and results which might lead to a theory of infinite dimensional symmetric spaces [Formula: see text] where [Formula: see text] is an affine Kac–Moody group and [Formula: see text] the fixed point group of an involution (of the second kind). We point out several striking similarities of these spaces with their finite dimensional counterparts and discuss their geometry. Furthermore we sketch a classification and show that they are essentially in 1 : 1 correspondence with hyperpolar actions on compact simple Lie groups.

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JORDAN SYMMETRIC SPACES

Asian-European Journal of Mathematics ◽

10.1142/s1793557109000339 ◽

2009 ◽

Vol 02 (03) ◽

pp. 407-415

Author(s):

Cho-Ho Chu

Keyword(s):

Symmetric Spaces ◽

Hermitian Symmetric Spaces ◽

Triple Systems ◽

Infinite Dimensional ◽

Jordan Triple Systems ◽

Riemannian Symmetric Spaces

We introduce a class of Riemannian symmetric spaces, called Jordan symmetric spaces, which correspond to real Jordan triple systems and may be infinite dimensional. This class includes the symmetric R-spaces as well as the Hermitian symmetric spaces.

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