All Questions
Tagged with symmetric-spaces lie-algebras
20 questions
3
votes
1
answer
162
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Compact symmetric spaces and sub-root systems
Given two semisimple complex Lie algebras $\frak{g}$ and $\frak{n}$ such that the root system of $\frak{n}$ arises as a sub-root system of the root system of $\frak{g}$, does this then imply that $\...
4
votes
0
answers
158
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Relation between two Harish-Chandra homomorphisms
Let $\mathfrak{g}$ be a Lie algebra admitting a triangular decomposition $\frak g = N^-\oplus h\oplus N^+$ and $\gamma$ the classic Harish-Chandra isomorphism defined on the center $\frak Z$ of the ...
- 141
5
votes
1
answer
395
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Buildings as generalizations of symmetric spaces
In almost every introductory notes on Tits buildings these are motivated
as structures capturing/ sharing several features of symmetric spaces. Could somebody elaborate what are precisely the main ...
- 5,998
1
vote
0
answers
137
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A question about Araki's 1962 paper on classification of irreducible symmetric spaces
I am looking at Sôhô Araki's 1962 paper for the classification of real semisimple lie algebras. Here's the link to the paper: On root systems and an infinitesimal classification of irreducible ...
- 251
3
votes
0
answers
142
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Conjugacy classes of Cartan subspaces in parahermitian symmetric spaces
Are there any good tables of the numbers of conjugacy classes of Cartan subspaces in pseudo-Riemannian symmetric spaces? Or a good method to count them? In particular, I am interested in the ...
- 954
1
vote
1
answer
197
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Semisimple Lie algebra and convexity
There is any relation between semisimple lie algebras and symmetric cones? I'm saying this because the classification of the euclidean Jordan algebras, dual by Koecher-Vinberg theorem to homogeneous ...
4
votes
0
answers
464
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Isometry groups of symmetric spaces
Let $M=G/K$ be a symmetric space where $G=\mathrm{Isom}(M)$ and $K$ is the isotropy at some point $o\in M$. Moreover, let $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ be the Cartan decomposition of $\...
- 201
1
vote
1
answer
449
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Existence of commuting Chevalley involution
Let $\mathfrak{g}$ be a simple finite-dimensional complex Lie algebra, and let $\theta$ be a complex linear involution on $\mathfrak{g}$. Let $\mathfrak{a}$ be a Cartan subspace, and choose a $\theta$...
- 1,077
10
votes
1
answer
590
views
Involutive automorphism of simple Lie algebra
I am sorry if this question is too elementary to be posted here, but no experts answer this question when I post it on Math Stackexchange.
Let $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ be a Cartan ...
- 951
6
votes
1
answer
465
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Holonomy groups of compact Riemannian symmetric spaces
Let $M$ be a compact Riemannian symmetric space. By the classification of Cartan, it belongs to the table of homogeneous spaces given in the Wikipedia page:
https://en.wikipedia.org/wiki/...
- 643
7
votes
0
answers
507
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Explicit formula for the Levi-Civita connection on a non-compact Riemannian symmetric space
Let $G/K$ be a non-compact Riemannian symmetric space, endowed with the Riemannian metric coming from the Killing form on the Lie algebra $\mathfrak{g}$ of the semi-simple Lie group $G$. Here $K$ is ...
- 1,942
1
vote
0
answers
235
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Connectedness of symmetric subgroup of simply connected Lie group
Let $G$ be a connected real simple Lie group, and $\tau$ be an involutive automorphism of $G$. Then $\tau$ defines a symmetric subgroup $G^\tau$ of $G$. In general, $G^\tau$ is not necessarily ...
- 951
3
votes
1
answer
422
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Irreducible Symmetric Pairs
Let $\mathfrak{g}$ be a simple Lie algebra with a compact subalgebra $\mathfrak{k}$ such that $(\mathfrak{g},\mathfrak{k})$ corresponds to an irreducible Riemann symmetric space. Denote by $\sigma$ be ...
- 951
7
votes
0
answers
138
views
Symmetric pairs of holomorphic type
Let $G$ be a real simple Lie group of Hermitian type; that is, $G/K$ carries a structure of a Hermitian symmetric space where $K$ is a maximal compact subgroup of $G$. Equivalently, the center $Z(\...
- 951
6
votes
2
answers
512
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Constructing real forms of the Tits-Freudenthal magic square for (Rosenfeld) projective planes
If $\mathbb{K},\mathbb{L} \in \{\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}\}$ then the Rosenfeld projective ("elliptic"?) plane $\mathbb{P}^2(\mathbb{K}\otimes\mathbb{L})$ is "the" compact Riemannian ...
- 32.4k
2
votes
0
answers
158
views
Kernel of the Weil homomorphism for compact symmetric spaces
Let $X = G/K$ be a Riemannian symmetric space of compact type and consider the "Weil homomorphism" $$w^\bullet: H^\bullet(BK; \mathbb R) \to H^\bullet(X; \mathbb R),$$ i.e. the map in cohomology ...
3
votes
1
answer
931
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Weyl group of a symmetric space
Let $G/K$ be a symmetric space of a non-compact type, i.e. $G$ is a semi-simple connected Lie group, and $K$ is its maximal compact subgroup. Helgason in his book "Differential geometry and symmetric ...
- 21.8k
8
votes
1
answer
650
views
Harish-Chandra isomorphism for compact symmetric spaces
I would be interested to have an explicit description of the algebra of invariant differential operators on functions on a compact symmetric space $G/K$. A reference would be especially useful.
Below ...
- 21.8k
2
votes
0
answers
339
views
volume form in a symmetric space of real rank one
I want to compare the two canonical volume forms on a noncompact symmetric space fo real rank one.
The first one is the volume form induced by the Riemannian structure given by the Killing form ...
- 2,446
22
votes
2
answers
2k
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"isotropic" subspaces of a simple Lie algebra
Let $\bf g$ be a finite-dimensional real simple Lie algebra of compact type and let $\left<-,-\right>$ denote the positive-definite inner product induced from the negative of the Killing form. ...
- 33.3k