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Kostant Convexity Theorem
In mathematics, Kostant's convexity theorem, introduced by , states that the projection of every coadjoint orbit of a connected compact Lie group into the dual of a Cartan subalgebra is a convex set. It is a special case of a more general result for symmetric spaces. Kostant's theorem is a generalization of a result of , and for hermitian matrices. They proved that the projection onto the diagonal matrices of the space of all ''n'' by ''n'' complex self-adjoint matrices with given eigenvalues Λ = (λ1, ..., λ''n'') is the convex polytope with vertices all permutations of the coordinates of Λ. Kostant used this to generalize the Golden–Thompson inequality to all compact groups. Compact Lie groups Let ''K'' be a connected compact Lie group with maximal torus ''T'' and Weyl group ''W'' = ''N''''K''(''T'')/''T''. Let their Lie algebras be \mathfrak and \mathfrak. Let ''P'' be the orthogonal projection of \mathfrak onto \mathfrak for some Ad-invariant inner product on \mathfrak ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coadjoint Orbit
In mathematics, the coadjoint representation K of a Lie group G is the dual of the adjoint representation. If \mathfrak denotes the Lie algebra of G, the corresponding action of G on \mathfrak^*, the dual space to \mathfrak, is called the coadjoint action. A geometrical interpretation is as the action by left-translation on the space of right-invariant 1-forms on G. The importance of the coadjoint representation was emphasised by work of Alexandre Kirillov, who showed that for nilpotent Lie groups G a basic role in their representation theory is played by coadjoint orbits. In the Kirillov method of orbits, representations of G are constructed geometrically starting from the coadjoint orbits. In some sense those play a substitute role for the conjugacy classes of G, which again may be complicated, while the orbits are relatively tractable. Formal definition Let G be a Lie group and \mathfrak be its Lie algebra. Let \mathrm : G \rightarrow \mathrm(\mathfrak) denote the adjoint repres ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi Eigenvalue Algorithm
In numerical linear algebra, the Jacobi eigenvalue algorithm is an iterative method for the calculation of the eigenvalues and eigenvectors of a real symmetric matrix (a process known as diagonalization). It is named after Carl Gustav Jacob Jacobi, who first proposed the method in 1846, but only became widely used in the 1950s with the advent of computers. Description Let S be a symmetric matrix, and G=G(i,j,\theta) be a Givens rotation matrix. Then: :S'=G S G^\top \, is symmetric and similar to S. Furthermore, S^\prime has entries: :\begin S'_ &= c^2\, S_ - 2\, s c \,S_ + s^2\, S_ \\ S'_ &= s^2 \,S_ + 2 s c\, S_ + c^2 \, S_ \\ S'_ &= S'_ = (c^2 - s^2 ) \, S_ + s c \, (S_ - S_ ) \\ S'_ &= S'_ = c \, S_ - s \, S_ & k \ne i,j \\ S'_ &= S'_ = s \, S_ + c \, S_ & k \ne i,j \\ S'_ &= S_ &k,l \ne i,j \end where s=\sin(\theta) and c=\cos(\theta). Since G is orthogonal, S and S^\prime have the same Frobenius norm , , \cdot, , _F (the square-root sum of squares ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebras
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identity. The Lie bracket of two vectors x and y is denoted ,y/math>. The vector space \mathfrak g together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative. Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to finite coverings (Lie's third theorem). This correspondence allows one to study the structure and classification of Lie groups in terms of Lie algebras. In physics, Lie groups appear as symmetry groups of ph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Groups
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (division), or equivalently, the concept of addition and the taking of inverses (subtraction). Combining these two ideas, one obtains a continuous group where multiplying points and their inverses are continuous. If the multiplication and taking of inverses are smooth (differentiable) as well, one obtains a Lie group. Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the rotational symmetry in three dimensions (given by the special orthogonal group \text(3)). Lie groups are widely used in many parts of modern mathematics and physics. Lie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complexification
In mathematics, the complexification of a vector space over the field of real numbers (a "real vector space") yields a vector space over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include their scaling ("multiplication") by complex numbers. Any basis for (a space over the real numbers) may also serve as a basis for over the complex numbers. Formal definition Let V be a real vector space. The of is defined by taking the tensor product of V with the complex numbers (thought of as a 2-dimensional vector space over the reals): :V^ = V\otimes_ \Complex\,. The subscript, \R, on the tensor product indicates that the tensor product is taken over the real numbers (since V is a real vector space this is the only sensible option anyway, so the subscript can safely be omitted). As it stands, V^ is only a real vector space. However, we can make V^ into a complex vector space by defining complex multiplication as follows: :\alpha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iwasawa Decomposition
In mathematics, the Iwasawa decomposition (aka KAN from its expression) of a semisimple Lie group generalises the way a square real matrix can be written as a product of an orthogonal matrix and an upper triangular matrix (QR decomposition, a consequence of Gram–Schmidt orthogonalization). It is named after Kenkichi Iwasawa, the Japanese mathematician who developed this method. Definition *''G'' is a connected semisimple real Lie group. * \mathfrak_0 is the Lie algebra of ''G'' * \mathfrak is the complexification of \mathfrak_0 . *θ is a Cartan involution of \mathfrak_0 * \mathfrak_0 = \mathfrak_0 \oplus \mathfrak_0 is the corresponding Cartan decomposition * \mathfrak_0 is a maximal abelian subalgebra of \mathfrak_0 *Σ is the set of restricted roots of \mathfrak_0 , corresponding to eigenvalues of \mathfrak_0 acting on \mathfrak_0 . *Σ+ is a choice of positive roots of Σ * \mathfrak_0 is a nilpotent Lie algebra given as the sum of the root spaces of Σ+ *''K'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moment Map
In mathematics, specifically in symplectic geometry, the momentum map (or, by false etymology, moment map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. The momentum map generalizes the classical notions of linear and angular momentum. It is an essential ingredient in various constructions of symplectic manifolds, including symplectic (Marsden–Weinstein) quotients, discussed below, and symplectic cuts and sums. Formal definition Let ''M'' be a manifold with symplectic form ω. Suppose that a Lie group ''G'' acts on ''M'' via symplectomorphisms (that is, the action of each ''g'' in ''G'' preserves ω). Let \mathfrak be the Lie algebra of ''G'', \mathfrak^* its dual, and :\langle, \rangle : \mathfrak^* \times \mathfrak \to \mathbf the pairing between the two. Any ξ in \mathfrak induces a vector field ρ(ξ) on ''M'' describing the infinitesimal action of ξ. To be precise, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Manifold
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds. For example, in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the phase space of the system. Motivation Symplectic manifolds arise from classical mechanics; in particular, they are a generalization of the phase space of a closed system. In the same way the Hamilton equations allow one to derive the time evolution of a system from a set of differential equations, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Restricted Weyl Group
In mathematics, restricted root systems, sometimes called relative root systems, are the root systems associated with a symmetric space. The associated finite reflection group is called the restricted Weyl group. The restricted root system of a symmetric space and its dual can be identified. For symmetric spaces of noncompact type arising as homogeneous spaces of a semisimple Lie group, the restricted root system and its Weyl group are related to the Iwasawa decomposition of the Lie group. See also *Satake diagram In the mathematics, mathematical study of Lie algebras and Lie groups, a Satake diagram is a generalization of a Dynkin diagram introduced by whose configurations classify semisimple Lie algebra, simple Lie algebras over the field (mathematics), fi ... References * * * * * Lie groups Lie algebras {{mathematics-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Lie Group
In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural generalization of finite groups with the discrete topology and have properties that carry over in significant fashion. Compact groups have a well-understood theory, in relation to group actions and representation theory. In the following we will assume all groups are Hausdorff spaces. Compact Lie groups Lie groups form a class of topological groups, and the compact Lie groups have a particularly well-developed theory. Basic examples of compact Lie groups include * the circle group T and the torus groups T''n'', * the orthogonal group O(''n''), the special orthogonal group SO(''n'') and its covering spin group Spin(''n''), * the unitary group U(''n'') and the special unitary group SU(''n''), * the compact forms of the exceptional Lie groups ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity Component
In mathematics, specifically group theory, the identity component of a group ''G'' refers to several closely related notions of the largest connected subgroup of ''G'' containing the identity element. In point set topology, the identity component of a topological group ''G'' is the connected component ''G''0 of ''G'' that contains the identity element of the group. The identity path component of a topological group ''G'' is the path component of ''G'' that contains the identity element of the group. In algebraic geometry, the identity component of an algebraic group ''G'' over a field ''k'' is the identity component of the underlying topological space. The identity component of a group scheme ''G'' over a base scheme ''S'' is, roughly speaking, the group scheme ''G''0 whose fiber over the point ''s'' of ''S'' is the connected component ''(Gs)0'' of the fiber ''Gs'', an algebraic group.SGA 3, v. 1, Exposé VI, Définition 3.1 Properties The identity component ''G''0 of a topo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weyl Group
In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection group. In fact it turns out that ''most'' finite reflection groups are Weyl groups. Abstractly, Weyl groups are finite Coxeter groups, and are important examples of these. The Weyl group of a semisimple Lie group, a semisimple Lie algebra, a semisimple linear algebraic group, etc. is the Weyl group of the root system of that group or algebra. Definition and examples Let \Phi be a root system in a Euclidean space V. For each root \alpha\in\Phi, let s_\alpha denote the reflection about the hyperplane perpendicular to \alpha, which is given explicitly as :s_\alpha(v)=v-2\frac\alpha, where (\cdot,\cdot) is the inner product on V. The Weyl group W of \Phi is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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