Questions tagged [lie-groups]
Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.
3,059 questions
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rank of a Lie group over a non-archimedean local field of positive characteristic
In the case of a Lie algebra over a non-archimedean local field of positive characteristic (I have been led to believe that) it is not necessarily true that all Cartan subalgebras have the same ...
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Restriction map for Lie algebra/Lie group cohomology associated to a complex semisimple Lie algebra and a semisimple Lie-subalgebra
Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra (or the corresponding Lie group). For definiteness, I'll take $\mathfrak{g}$ to be of type $A_n$, that is, $\mathfrak{g} = \...
- 2,160
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On matrices conjugated in a faithful representation
Let $k$ an algebraically closed field.
Let $O=k[[\pi]]$ and $F=k((\pi))$ and $G\rightarrow GL_{n}$ a faithful representation of a semisimple group.
Let $A, B\in G(O)\cap G(F)^{rs}$ (rs for regular ...
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Does the diffeomorphism group preserving a particular section act transitively?
Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$, be the space of homogeneous
degree $d$ polynomials in three variables $[X,Y,Z] \in \mathbb{P}^2$ upto scaling, where
$\delta_d = \frac{d(d+3)}{2}$. ...
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Why "non-linear similarity" is the same as equivalence of representations for connected Lie groups?
Let $G$ be a compact Lie group and $V$ a finite-dimensional orthogonal $G$-representation. Write $S^V$ for the quotient $D(V)/S(V)$, where $D(V)$ and $S(V)$ are the unit disk and sphere in $V$, ...
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density of conjugate of arithmetic subgroup
$K=Q(\sqrt{d} ) , d<0 $, $\Gamma $ an arithmetic subgroup of $G=SU(2,1)(K)$ . Is $\cup_{g\in G}(g^{-1}\Gamma g)$ dense in G for the complex topology?
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Hilbert's Finiteness Theorem for connected semisimple Lie groups over $\mathbb{C}$ in Weyl's "Classical Groups" [duplicate]
In Nagata's "Lectures on the 14th problem of Hilbert" I found a reference to Weyl's "Classical Groups". Nagata writes that Weyl gives a positive answer to the original problem
If $G\subseteq\...
- 339
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Computing equivariant K-theory using the amalgamted product
If I have a Lie group (or a Kac-Moody group) $G$ such that it's the amalgated product of it's proper parabolic subgroups $P_J$, i.e. $G = \text{colim} P_J$, then could I use this to compute the ...
- 257
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Characterization of the weight orbit in the projective space via second order Casimir.
This is the spin-off of the question I previously asked.
First, let me remind you some notation from that question:
$G_0$ - compact, simply connected Lie group giving rise (by complexification) ...
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$\Gamma$ cohomology of principal series
Let $G$ be a noncompact connected real semisimple Lie group with finited center. Let $\Gamma$ be a cocompact discrete subgroup of $G$, and let $P$ be a parabolique subgroup
with Langlands ...
- 1,111
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299
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Intertwining Integral defined on a Weyl group?
Why does the intertwining integral such as the one defined in A. W. Knapp's paper "Intertwining operators for semisimple groups" depend only on an element w of a Weyl group?
http://www.jstor.org/...
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A question on lie groups( Lie algebras)
What is an example of a compact Lie group $G$ which is not isomorphic to a product of two nontrivial lie groups and satisfies the following property:
There are two non zero vector fields $X, Y \in ...
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semisimple conjugacy classes over general bases
Let $k$ be an algebraically closed field, $G$ a connected reductive group, $T$ a maximal torus, $W$ the Weyl group and $\chi:G\rightarrow T/W$ the Steinberg morphism.
We know that if $\gamma,\gamma'\...
- 3,472
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560
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Decomposition of Haar measure other than Hurwitz's
Hurwitz defined a decomposition of the Haar measure on $SO(n)$ based on Given's rotation. So by left multiplication of Givens rotation one can always bring an orthogonal matrix into the identity. The ...
- 4,466
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What is the homology of the real coordinate ring of SO(n,R)? Other compact matrix groups?
As someone whose knowledge of cohomology is patchy and picked up on a need-to-know basis, and whose algebraic geometry is even worse, I wondered if someone could help with this question. (I ran into ...
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The centralizer $Z_G(X)$ of a nilpotent element in a real simple Lie group
I am looking for the description of the centralizer $Z_G(X)$ , where $G$ is a real simple Lie Group and $X\in \ Lie (G) $ such that $X^d=0,\ X^{d-1}\neq 0 $. It is is helpful to me any references or ...
- 116
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Intersections of conjugates of the icosahedral group in SO(3)
(Related question)
Let $I$ be the group of orientation preserving symmetries of a regular icosahedron. This is a $60$ element subgroup of $SO(3)$, isomorphic with the alternating group $A_5$. It is ...
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Equivariant Poincare Series of Based Loop Group of SU(2)
Let $\Omega SU(2)$ denote the based loop group of $SU(2)$, and consider the action of $S^1$ on $\Omega SU(2)$ as a maximal torus of $SU(2)$. (This is not the "loop rotation" action.) Is there an ...
- 4,920
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On Eigenvalues of the symmetric linear transformation related to a lie algebra's representation?
Let $\mathfrak{g}$ be a quadratic (finite dimensional) lie algebra and $\rho:\mathfrak{g}\rightarrow \mathfrak{gl}(W)$ be an anti-symmetric representation of $\mathfrak{g}$ on a finite dimensional ...
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Orbits of Product Lie Groups Action
Hi to all,
Let $G$ be a Lie group of linear isometries of $\mathbb{R}^n_{\nu}$ ($\mathbb{R}^n_{\nu}$ is the semi-Euclidean space) and $G_1$ ,$G_2$ two Lie subgroups of $G$. Let $G_1 \times G_2$ as ...
user30435
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Euler number of the complex of basic forms
Let $G$ be a compact Lie group and $\pi:P \to M$ a principal $G$-bundle. I would like to understand the geometry of $M$ through $P$ with the $G$-action.
I am trying to understand the Hopf bundle ($G=...
- 181
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241
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Orbit of the identity matrix under Lie group algebra actions
I would like an explicit description of $\mathbb{R} SO(n) I_n$, i.e., the image of the identity under the action of the group algebra of $SO(n)$ by left multiplication. Equivalently, what is an ...
- 4,466
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279
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Heights in reductive groups
Let $G$ be a reductive group over a local non-archimedean field $F$, and let $B$ a Borel subgroup. For my purposes, the case $G = GL_2(\mathbb{Q}_p)$ will be sufficient with $B$ upper triangular ...
- 11.2k
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272
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minuscule representations and classical groups
Let $G$ a semisimple group over an algebraically closed field $k$.
We assume that $G$ is classical.
We call a $z$-extension, a group $\tilde{G}$ such that $\tilde{G}$ is a central extension of $G$ by ...
- 3,472
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A possible refinement of a theorem of Malliavin
Due to a theorem of Malliavin (an improvement of an erlier theorem of Dixmier and Malliavin, see here) we know that every compactly supported smooth function $f$ on $\mathbb R^n$ can be written as a ...
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Largest Set of Special Unitary Matricies With Invariant Subspace For Adjoint Action
I am trying to solve the following. Given the special unitary group $SU(n)$ and its adjoint action $Ad_{U}: \mathfrak{su}(n) \rightarrow \mathfrak{su}(n)$, what is the largest subset of $SU(n)$ such ...
- 2,099
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Meaning of "Compact" in 1932 Paper by van der Waerden "Continuity Theorem for Semisimple Lie Groups".
I am putting together an exposition on Lie theory; maths research is not my day job, let alone real maths history, so apologies in advance for any ignorance shown by these questions.
I am attempting ...
- 1,161
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547
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Maximal Tori and group structures on spheres
It is known that for any compact Lie group $G$ with maximal torus $T$, that any other maximal torus $T'$ is conjugate to $T$. This might be a bit of a stretch, but I was wondering if it is possible to ...
- 757
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594
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Analogies between orthogonal/unitary groups and their indefinite counterparts
Suppose I have $A\in U(n)$ such that $A^t=A$ (which is a bit un-natural, as usually you'd consider the hermitian transpose, not the transpose).
Well, then $A=X+iY$ say, for $X$ and $Y$ real matrices. ...
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How big is the center of an orthogonal group?
How big is the center of an arbitrary orthogonal group $O(m,n)$?
In the special case of the "circle group" $O(2)$, it's clear that $|\zeta O(2)|$ = 1. In the case of $O(3)$, it seems clear that the ...
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The quotient of a Lie group by the Levi factor of a parabolic subgroup
I am interested in some references on the quotient spaces obtained by quotienting G, a simple Lie group, by L, the group generated by the Levi factor of a parabolic subalgebra.
Presumably the case ...
- 2,123
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A kind of orthogonal subtorus
Here $\mathbb{T}^n := (\mathbb{R} / \mathbb{Z})^n$ is the topological group of the n-dimensional torus and $k \in \mathbb{Z}^n$ is a non-null vector, I'm working about the subgroup
$S = \{x \in \...
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358
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What is the group O(4)/H, where H is the center of O(4)?
What is the group $O(4)/H$?
Here $O(4)$ is the group of orthogonal matrices and H is the center of $O(4)$.
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Which cocompact subgroups of $G$ do contain a cocompact normal subgroup of $G$?
Let $G$ be a locally compact group and let $H$ be a cocompact (or more generally, a cofinite) subgroup of $G$.
Is there any criterion to determine whether $H$ contains a cocompact normal subgroup of ...
user23860
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What matrix groups can be embedded in $Sp_4$?
In a joint paper with Yifan Yang we constructed an "exotic" embedding
of $SL_2(\mathbb R)$ in $Sp_4(\mathbb R)$ (in fact, of $PSL_2(\mathbb R)$ in $PSp_4(\mathbb R)$),
namely,
$$
\iota\colon\begin{...
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$SU(n)$-invariant subring of $\Lambda^{*}\mathbb{R}^{2n}$
I have the following question: Let $R \subset \Lambda^{*}\mathbb{R}^{2n}$ be the sub-ring of forms which are preserved by $SU(n)$. How can one show that this subring is generated by $\Omega_{0}$ and $\...
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Converting Dirichlet boundary conditions for E-L equations on a Lie group into an equivilent condition for EP equaiton
I need to find a specific geodesic of a right invariant Finsler geodesic on a Lie group ($SU(n)$) that connects $I$ to some desired $O$. These are Dirichlet boundary conditions for the E-L equations (...
- 2,099
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394
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Principal Bundle Connection Correspondence for two descriptions of the $\mathbb{CP}^2$
Consider the following pair of principal bundle descriptions of $\mathbb{CP}^2$:
$$
\mathbb{CP}^2 \simeq SU(3)/U(2) \simeq S^5/U(1).
$$
If I have a principal $U(2)$-bundle connection for $\mathbb{CP}^...
- 751
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339
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Invariants of co-diagonalizability in real symmetric matrices
This question has been mentionned to me by U. Frisch. He wanders whether it has ever been considered by algebraists.
In the vector space ${\bf Sym}_n({\mathbb R})$, two elements commutte to each ...
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On homeomorphic compact connected topological groups
I wish to thank Professor Claudio Gorodski for his very helpful
answers to my question on the webcite:
If compact connected Lie groups are homeomorphic as topological space, are they isomorphic as Lie ...
- 491
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An identity for sheaf cohomology of flag varieties
Let $G$ be a connected complex semisimple Lie-group, $T$ a maximal torus and $B$ a Borel subgroup containing it. Let $\phi:G\rightarrow G/B$ denote the projection.
Given a representation ($\theta,V$) ...
user6311
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Dominant weights appear in Discrete Series
If $\lambda$ is a Harish-Chandra paramater. Let $\pi_\lambda$ it's associated discrete series, it's known by the minimal K-type thm that every K-type of $\pi_\lambda\mid_K$ has highest weight of the ...
- 11
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293
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the relation between cohomology and Dynkin graphs of lie groups
I heard it said that the cohomology rings of some Lie groups and Grassmannians can be read from the Dynkin graph. Can someone give me any reference?
- 139
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284
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Packing symmetric matrices in spectral norm, and defining measures on symmetric matrices
I'm trying to upper bound the $\epsilon$-packing number of $\Theta=\{A\in\mathbb{S}^{d}:\; a\preceq A \preceq b\}$ (where $\mathbb{S}$ are symmetric $d\times d$ matrices) for some $a\leq b$ with ...
- 300
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The real group orbits on the flag variety always contains the holomorphic directions?
Let $G$ be a real semisimple Lie group and $\mathfrak{g}$ be its complexified Lie algebra. We have the flag variety $\mathcal{B}$ of $\mathfrak{g}$ which is the set of all Borel subalgebras of $\...
- 9,019
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99
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Unions of orbits of dimension $\leq n$
Let $G$ be a complex linear algebraic group acting on a smooth complex projective variety $X$ with finitely many orbits. Note that each $G$-orbit is a smooth locally closed subvariety of $X$.
For a ...
- 4,920
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110
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Brauer characters of finite simple group $E_8(5)$
I would like to find the irreducible characters of the group $E_8(5)$ (mod 2)?
Can anyone help? (I am elementary in working with Brauer characters)
Many thanks
- 169
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module of sections of the horizontal bundle
Some times ago I posted this question here. There I carelessly assumed that if you have a set of sections of a vector bundle which span every fiber pointwise, they also generate the module of smooth ...
- 1,222
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995
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Lower bound for Jacobian of matrix exponential map near origin
What is a lower bound for the Jacobian of the exponential map from the skew-symmetric matrices to the orthogonal matrices near the origin?
- 145
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2
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1k
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Representations of reductive Lie group
Let $G$ be a reductive algebraic group and $\varrho$ a representation of $G$ in $GL(n)$. Is it true that $\varrho$ is completely reducible? Moreover, how are related the representations of the Lie ...
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