# Tagged Questions

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### equation for geodesics of a right invairant Finsler metric on $SU(n)$ which are parallel to a linear affine distribution

I am looking for an equation analogous to the Euler-Poincare equations for a right invariant Finlser metric except I want the geodesics which are parallel to a linear affine distribution on $SU(n)$. ...
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### Construction of the Lie functor: left vs. right invariant vector fields on Lie groups and Lie groupoids

When constructing the Lie algebra $L(G)$ of a Lie group $G$, one usually uses the identification of the tangent space $T_1 G$ with left invariant vector fields $\mathcal{V}^l(G)$ to construct the Lie ...
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### Laplace-Beltrami operator on a Lie group

For an arbitrary Lie group, is it always possible to chose a left-invariant Riemannian metric such that the Laplace-Beltrami operator $\Delta$ is given by $$\Delta f = \delta^{i j} X_i X_j f$$ for ...
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It is well known that the stationary curves (say $\Xi(t)$) of a regular Lagrangian $\mathcal{L}$ on a compact, semi-simple Lie group $G$ have the property that $\xi(t) = \frac{d \Xi(t)}{dt} ... 2answers 427 views ### Geodesics on$SU(4)$Are the geodesics of the following metrics on$SU(4)$known or easy (in a way not known to me!) to find? In the adjoint representation, one can express the Killing form as a matrix and consider it as ... 3answers 568 views ### nth term in the Baker-Campbell-Hausdorff formula I am trying to prove a result for which I need the nth term of the Baker-Campbell-Hausdorff formula. I came at this particular result (which is not of significance for the question, but mentioning for ... 1answer 289 views ### The surjectivity of the exponential map for the isometry group Little is known on general conditions guaranteeing that the exponential map between a Lie algebra and an associated Lie group is surjective. Let$M$be a noncompact connected Riemann manifold, and ... 0answers 37 views ### Locus maximizing the holomorphic sectional curvature in a non-compact Hermitian symmetric space Is there a quick way to prove the following statement, if possible without resorting to the classification of simple Lie groups? Let$G$be a simple Lie group of non-compact Hermitian type of rank ... 0answers 302 views ### Solving$T^2 = -\kappa\, \mathrm{Tr}\, (\log(e^{i T \hat{H}_0} \hat{O}) )^2$equation Is there a way to solve the equation:$T^2 = -\kappa\, \mathrm{Tr}\, (\log(e^{i T \hat{H}_0} \hat{O}) )^2$for$T$? Here$\kappa$is an arbitrary positive constant,$\hat{H}_0 \in \mathfrak{su}(N)$... 1answer 250 views ### Special Riemannian connections? Assume that$E$is a bundle of Lie Algebras. Let$g$be an invariant metric on$E$, that is for all$p\in M$, $$g_p([x,y],z)+g_p(y,[x,z])=0,$$ where$x,y,z\in E_p$are arbitrary. Is there a ... 0answers 43 views ### Special family of Metrics on Transitive Lie Algebroids? Let$\rho:E\longrightarrow TM$is a transitive Lie Algebroid, then$L=ker\rho$is bundle of lie algebras. Suppose$\Gamma:TM\longrightarrow E$be a linear splitting. Define $$\nabla_X ... 2answers 457 views ### A question about Marsden-Weinstein reduction theory Let G ba a compat Lie group and \frak g be its Lie algebra, then by Marsden-Weinstein reduction theory we know that if we take M=T^*G and J:M\to \frak g^* be its moment map then the reduced ... 1answer 152 views ### The set of leaves of the distribution D on coadjoint orbit O_{\mu} Let G be a compact connected Lie group and O_{\mu} be a coadjoint orbit where \mu\in \mathfrak{g}^* and \mathfrak{g}^* is the dual of the Lie algebra of \mathfrak{g}=\mathrm{Lie}(G). Let ... 1answer 141 views ### Each coadjoint orbit of a compact connected Lie group G admits a G-invariant generalized complex structure I am looking for a proof or counterexample for following assertion Each coadjoint orbit of a compact connected Lie group G admits a G-invariant generalized complex structure (In sense of ... 0answers 245 views ### Injectivity of Lie group exponential function If G is a (finite-dimensional) Lie group, then the exponential function \exp\colon\mathfrak{g}\to G is injective on some identity neighbourhood. If, moreover, \mathfrak{g} is semi-simple and ... 2answers 104 views ### Derivations of algebra of smooth g-valued function? Let M is a smooth n-manifold and g is a Z_2-graded Lie algebra, we denote the algebra of smooth g-valued function on M by C^{\infty} (M,g). I wanna find all graded derivation of ... 1answer 421 views ### How was this Lie algebra found? In a paper the author lists, without justification, generators for a Lie algebra. I would be grateful if someone could justify these choices and perhaps suggest how I might have found them for myself. ... 1answer 143 views ### Right Invariant Randers metrics I'm hoping to determine the geodesic equation for a right invariant Randers metric F(x) = \sqrt{a(x,x)} + b(x) on SU(N). In my special case the navigation data (h,W) for the Randers metric are ... 1answer 206 views ### Determining the Lie algebra elements exponentiating to the center of a Lie group For a semi-simple compact Lie group G with center Z(G), one can characterize the preimage of Z(G) in the Cartan subalgebra under the exponential map as the nodes of the Stiefel diagram (see for ... 0answers 121 views ### About the Lie algebra of polyvector fields I would like to know if someone already did some computations of the group of Lie algebra automorphisms of the algebra of polyvector fields on \mathbb{R}^n equiped with the Schouten bracket (or ... 1answer 131 views ### Split real form of SL(2,\mathbb{C}) description of the two sphere? If we denote the parabolic subgroup of SL(2,\mathbb{C}) by P, then we have the well known isomorphism SL(2,\mathbb{C})/P \simeq S^2, where S^2 is the two sphere. Now the compact real form of ... 1answer 422 views ### Para-Complexification of Lie Groups Let G be a real Lie group. Then the complexification G_\mathbb{C} of G is the unique complex Lie group equipped with a map Ï†:G\to G_\mathbb{C} such that any map G\to H where H is a ... 1answer 124 views ### Casimir of a three dimensional solvable lie algebra Good morning everyone. I've encountered recently during my computations the following lie algebra$$\mathfrak g=\text{span}(f_0,f_1,f_2),$$with$$\begin{eqnarray}[f_2,f_1]&=&f_0+a f_2,\\ ... 0answers 55 views ### compute the determinant of a conjugacy map Let$k$be an algebraically closed field. Let$F=k((\pi))$and$\mathcal{O}$the ring of integers, let$\gamma\in T(F)$regular semisimple for a connected reductive group$G$. We consider the map ... 1answer 521 views ### Kirillov-Kostant-Souriau Theorem on$\mathfrak{g}\oplus \mathfrak{g^*} $My question is about the extention of kirillov's symplectic structure on coadjoint orbits. The most remarkable feature of the coadjoint representation is the fact that all coadjoint orbits possess a ... 1answer 397 views ### fiber bundle on an orbit of$\mathfrak{g}\oplus\mathfrak{g^*}$Let$G$, be a Lie Group and$\mathfrak{g}$be its Lie algebra ,i.e,$Lie(G)=\mathfrak{g}$. Let$\zeta=(\ X,F)\ \in \mathfrak{g}\oplus\mathfrak{g^*}$. Here$X\in \mathfrak{g} $and$F\in ...
I am looking for a clear reason for following fact:Is there any reference ? Why a $G$-invariant differential form $\omega$ on a homogeneous $G$-manifold $M=G/H$ is uniquely determined by its value at ...