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## New answers tagged lie-algebra

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There is a real Lie group $\tilde{Diff}(S^1)$ which is a $U(1)$ central extension of the real Lie group $Diff(S^1)$, and the Virasoro algebra is the Lie algebra of this Lie group. The central extension $\tilde{Diff}(S^1)$ can be realized geometrically in two ways. The first is via a Hilbert space embedding (as in the book of Pressley-Segal), and the second ...

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SECTION A : What remains invariant for a complex $\:3\times 3\:$ tensor depends upon its transformation law under $\:U \in SU(3)\:$ CASE 1 : $\:\boldsymbol{3}\boldsymbol{\otimes}\boldsymbol{3}=\boldsymbol{6}\boldsymbol{\oplus}\overline{\boldsymbol{3}}\:$ The transformation law for the complex $\:3\times 3\:$ tensor $\:\mathrm{X}\:$ in this case is ...

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Every cotangent bundle is a symplectic manifold and hence a Poisson manifold. Symplectic manifolds encode unconstrained mechanics. A symplectic manifold has only a single symplectic leaf, namely itself. More general Poisson manifolds (often obtained by a process called symplectic reduction) encode already constrained mechanics: Each Casimir becomes a ...

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First Some Definitions Before I answer, let's first make a subtle nomenclature issue clear to check that I'm understanding your definitions: a set $\mathcal{T}=\{X_1,\,X_2,\,\cdots,\,X_M\}$ of Lie algebra vectors is called a set of topological generators for a group $\mathfrak{G}$ iff the closure of the smallest group containing elements of the form ...

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Yes, there is a relation between both things. Let $$g(t)=\exp(tX), \qquad g(0)=e$$ Be a curve on $G$, such that $\gamma_a(t)=g(t)\cdot a$ Then, by deffinition, for every smooth function $f:G\to\mathbb{R}$ this curve satisfies the differential equation $$\frac{d}{dt}(f\circ g)(t)=X(f\circ g)(t)$$ or directly $$\frac{d}{dt}g(t)=X(g(t))$$ In an analogous way, ...

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Section A : The connection of the transformations of complex $\:3\times 3\:$ antisymmetric tensors and their representative complex $\:3$-vectors. Let $\:U\:$ be a special unitary transformation in $\:SU(3)\:$ represented by the $\:3\times 3\:$ complex matrix U= \begin{bmatrix} u_{11} & u_{12} & u_{13} \\ u_{21} & u_{22} ...

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