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Cartan's Theorem (other)
Cartan's theorem may refer to several mathematical results by Élie Cartan: *Closed-subgroup theorem, 1930, that any closed subgroup of a Lie group is a Lie subgroup *Theorem of the highest weight, that the irreducible representations of Lie algebras or Lie groups are classified by their highest weights *Lie's third theorem, an equivalence between Lie algebras and simply-connected Lie groups See also * Cartan's theorems A and B, c.1931 results by Henri Cartan concerning a coherent sheaf on a Stein manifold * Cartan's lemma In mathematics, Cartan's lemma refers to a number of results named after either Élie Cartan or his son Henri Cartan: * In exterior algebra: Suppose that ''v''1, ..., ''v'p'' are linearly independent elements of a vector space ''V'' and ''w''1, . ..., several results by Élie or Henri Cartan * Cartan–Dieudonné theorem, a result on orthogonal transformations and reflections Lie groups Theorems in abstract algebra {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed-subgroup Theorem
In mathematics, the closed-subgroup theorem (sometimes referred to as Cartan's theorem) is a theorem in the theory of Lie groups. It states that if is a closed subgroup of a Lie group , then is an embedded Lie group with the smooth structure (and hence the group topology) agreeing with the embedding. One of several results known as Cartan's theorem, it was first published in 1930 by Élie Cartan, who was inspired by John von Neumann's 1929 proof of a special case for groups of linear transformations.; . Overview Let G be a Lie group with Lie algebra \mathfrak. Now let H be an arbitrary closed subgroup of G. It is necessary to show that H is a smooth embedded submanifold of G. The first step is to identify something that could be the Lie algebra of H, that is, the tangent space of H at the identity. The challenge is that H is not assumed to have any smoothness and therefore it is not clear how one may define its tangent space. To proceed, define the "Lie algebra" \mathfrak of H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorem Of The Highest Weight
In representation theory, a branch of mathematics, the theorem of the highest weight classifies the irreducible representations of a complex semisimple Lie algebra \mathfrak g. Theorems 9.4 and 9.5 There is a closely related theorem classifying the irreducible representations of a connected compact Lie group K. Theorem 12.6 The theorem states that there is a bijection :\lambda \mapsto ^\lambda/math> from the set of "dominant integral elements" to the set of equivalence classes of irreducible representations of \mathfrak g or K. The difference between the two results is in the precise notion of "integral" in the definition of a dominant integral element. If K is simply connected, this distinction disappears. The theorem was originally proved by Élie Cartan in his 1913 paper. The version of the theorem for a compact Lie group is due to Hermann Weyl. The theorem is one of the key pieces of representation theory of semisimple Lie algebras. Statement Lie algebra case Let \mathfrak ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Highest Weight
In the mathematical field of representation theory, a weight of an algebra ''A'' over a field F is an algebra homomorphism from ''A'' to F, or equivalently, a one-dimensional representation of ''A'' over F. It is the algebra analogue of a multiplicative character of a group. The importance of the concept, however, stems from its application to representations of Lie algebras and hence also to representations of algebraic and Lie groups. In this context, a weight of a representation is a generalization of the notion of an eigenvalue, and the corresponding eigenspace is called a weight space. Motivation and general concept Given a set ''S'' of n\times n matrices over the same field, each of which is diagonalizable, and any two of which commute, it is always possible to simultaneously diagonalize all of the elements of ''S''.In fact, given a set of commuting matrices over an algebraically closed field, they are simultaneously triangularizable, without needing to assume that they are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie's Third Theorem
In the mathematics of Lie theory, Lie's third theorem states that every finite-dimensional Lie algebra \mathfrak over the real numbers is associated to a Lie group ''G''. The theorem is part of the Lie group–Lie algebra correspondence. Historically, the third theorem referred to a different but related result. The two preceding theorems of Sophus Lie, restated in modern language, relate to the infinitesimal transformations of a group action on a smooth manifold. The third theorem on the list stated the Jacobi identity for the infinitesimal transformations of a local Lie group. Conversely, in the presence of a Lie algebra of vector fields, integration gives a ''local'' Lie group action. The result now known as the third theorem provides an intrinsic and global converse to the original theorem. Historical notes The equivalence between the category of simply connected real Lie groups and finite-dimensional real Lie algebras is usually called (in the literature of the second h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartan's Theorems A And B
In mathematics, Cartan's theorems A and B are two results proved by Henri Cartan around 1951, concerning a coherent sheaf on a Stein manifold . They are significant both as applied to several complex variables, and in the general development of sheaf cohomology. Theorem B is stated in cohomological terms (a formulation that Cartan ( 1953, p. 51) attributes to J.-P. Serre): Analogous properties were established by Serre (1957) for coherent sheaves in algebraic geometry, when is an affine scheme. The analogue of Theorem B in this context is as follows : These theorems have many important applications. For instance, they imply that a holomorphic function on a closed complex submanifold, , of a Stein manifold can be extended to a holomorphic function on all of . At a deeper level, these theorems were used by Jean-Pierre Serre to prove the GAGA theorem. Theorem B is sharp in the sense that if for all coherent sheaves on a complex manifold (resp. quasi-coherent she ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartan's Lemma
In mathematics, Cartan's lemma refers to a number of results named after either Élie Cartan or his son Henri Cartan: * In exterior algebra: Suppose that ''v''1, ..., ''v''''p'' are linearly independent elements of a vector space ''V'' and ''w''1, ..., ''w''''p'' are such that ::v_1\wedge w_1 + \cdots + v_p\wedge w_p = 0 :in Λ''V''. Then there are scalars ''h''''ij'' = ''h''''ji'' such that ::w_i = \sum_^p h_v_j. * In several complex variables: Let and and define rectangles in the complex plane C by ::\begin K_1 &= \ \\ K_1' &= \ \\ K_1'' &= \ \end :so that K_1 = K_1'\cap K_1''. Let ''K''2, ..., ''K''''n'' be simply connected domains in C and let ::\begin K &= K_1\times K_2\times\cdots \times K_n\\ K' &= K_1'\times K_2\times\cdots \times K_n\\ K'' &= K_1''\times K_2\times\cdots \times K_n \end :so that again K = K'\cap K''. Suppose that ''F''(''z'') is a complex analytic matrix-valued function on a rectangle ''K'' in C''n'' such that ''F''(''z'') is an invertibl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartan–Dieudonné Theorem
In mathematics, the Cartan–Dieudonné theorem, named after Élie Cartan and Jean Dieudonné, establishes that every orthogonal transformation in an ''n''-dimensional symmetric bilinear space can be described as the composition of at most ''n'' reflections. The notion of a symmetric bilinear space is a generalization of Euclidean space whose structure is defined by a symmetric bilinear form (which need not be positive definite, so is not necessarily an inner product – for instance, a pseudo-Euclidean space is also a symmetric bilinear space). The orthogonal transformations in the space are those automorphisms which preserve the value of the bilinear form between every pair of vectors; in Euclidean space, this corresponds to preserving distances and angles. These orthogonal transformations form a group under composition, called the orthogonal group. For example, in the two-dimensional Euclidean plane, every orthogonal transformation is either a reflection across a line through ... [...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] |
