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Lie's Theorem
In mathematics, specifically the theory of Lie algebras, Lie's theorem states that, over an algebraically closed field of characteristic zero, if \pi: \mathfrak \to \mathfrak(V) is a finite-dimensional representation of a solvable Lie algebra, then there's a flag V = V_0 \supset V_1 \supset \cdots \supset V_n = 0 of invariant subspaces of \pi(\mathfrak) with \operatorname V_i = i, meaning that \pi(X)(V_i) \subseteq V_i for each X \in \mathfrak and ''i''. Put in another way, the theorem says there is a basis for ''V'' such that all linear transformations in \pi(\mathfrak) are represented by upper triangular matrices. This is a generalization of the result of Frobenius that commuting matrices are simultaneously upper triangularizable, as commuting matrices generate an abelian Lie algebra, which is a fortiori solvable. A consequence of Lie's theorem is that any finite dimensional solvable Lie algebra over a field of characteristic 0 has a nilpotent derived algebra (see #Consequen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, 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 [x,y]. The vector space \mathfrak g together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative property, associative. Lie algebras are closely related to Lie groups, which are group (mathematics), 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 space, connected Lie group unique up to finite coverings (Lie's third theorem). This Lie group–Lie algebra correspondence, correspondence allows one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Engel's Theorem
In representation theory, a branch of mathematics, Engel's theorem states that a finite-dimensional Lie algebra \mathfrak g is a nilpotent Lie algebra_if_and_only_if_for_each_X_\in_\mathfrak_g,_the_adjoint_representation_of_a_Lie_algebra.html" "title="mathfrak,\mathfrak ... if and only if for each X \in \mathfrak g, the adjoint representation of a Lie algebra">adjoint map :\operatorname(X)\colon \mathfrak \to \mathfrak, given by \operatorname(X)(Y) = [X, Y], is a nilpotent endomorphism on \mathfrak; i.e., \operatorname(X)^k = 0 for some ''k''. It is a consequence of the theorem, also called Engel's theorem, which says that if a Lie algebra of matrices consists of nilpotent matrices, then the matrices can all be simultaneously brought to a strictly upper triangular form. Note that if we merely have a Lie algebra of matrices which is nilpotent ''as a Lie algebra'', then this conclusion does ''not'' follow (i.e. the naïve replacement in Lie's theorem of "solvable" with "nilpotent", ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie–Kolchin Theorem
In mathematics, the Lie–Kolchin theorem is a theorem in the representation theory of linear algebraic groups; Lie's theorem is the analog for linear Lie algebras. It states that if ''G'' is a connected and solvable linear algebraic group defined over an algebraically closed field and :\rho\colon G \to GL(V) a representation on a nonzero finite-dimensional vector space ''V'', then there is a one-dimensional linear subspace ''L'' of ''V'' such that : \rho(G)(L) = L. That is, ρ(''G'') has an invariant line ''L'', on which ''G'' therefore acts through a one-dimensional representation. This is equivalent to the statement that ''V'' contains a nonzero vector ''v'' that is a common (simultaneous) eigenvector for all \rho(g), \,\, g \in G . It follows directly that every irreducible finite-dimensional representation of a connected and solvable linear algebraic group ''G'' has dimension one. In fact, this is another way to state the Lie–Kolchin theorem. The result for Lie alg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor Product Of Representations
In mathematics, the tensor product of representations is a tensor product of vector spaces underlying representations together with the factor-wise group action on the product. This construction, together with the Clebsch–Gordan procedure, can be used to generate additional irreducible representations if one already knows a few. Definition Group representations If V_1, V_2 are linear representations of a group G, then their tensor product is the tensor product of vector spaces V_1 \otimes V_2 with the linear action of G uniquely determined by the condition that :g \cdot (v_1 \otimes v_2) = (g\cdot v_1) \otimes (g\cdot v_2) for all v_1\in V_1 and v_2\in V_2. Although not every element of V_1\otimes V_2 is expressible in the form v_1\otimes v_2, the universal property of the tensor product operation guarantees that this action is well defined. In the language of homomorphisms, if the actions of G on V_1 and V_2 are given by homomorphisms \Pi_1:G\rightarrow\operatorname(V_1) and \P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radical Of A Lie Algebra
In the mathematical field of Lie theory, the radical of a Lie algebra \mathfrak is the largest solvable ideal of \mathfrak.. The radical, denoted by (\mathfrak), fits into the exact sequence :0 \to (\mathfrak) \to \mathfrak g \to \mathfrak/(\mathfrak) \to 0. where \mathfrak/(\mathfrak) is semisimple. When the ground field has characteristic zero and \mathfrak g has finite dimension, Levi's theorem states that this exact sequence splits; i.e., there exists a (necessarily semisimple) subalgebra of \mathfrak g that is isomorphic to the semisimple quotient \mathfrak/(\mathfrak) via the restriction of the quotient map \mathfrak g \to \mathfrak/(\mathfrak). A similar notion is a Borel subalgebra, which is a (not necessarily unique) maximal solvable subalgebra. Definition Let k be a field and let \mathfrak be a finite-dimensional Lie algebra over k. There exists a unique maximal solvable ideal, called the ''radical,'' for the following reason. Firstly let \mathfrak and \mathfrak be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartan's Criterion For Solvability
In mathematics, Cartan's criterion gives conditions for a Lie algebra in characteristic 0 to be solvable, which implies a related criterion for the Lie algebra to be semisimple. It is based on the notion of the Killing form, a symmetric bilinear form on \mathfrak defined by the formula : B(u,v)=\operatorname(\operatorname(u)\operatorname(v)), where tr denotes the trace of a linear operator. The criterion was introduced by .Cartan, Chapitre IV, Théorème 1 Cartan's criterion for solvability Cartan's criterion for solvability states: :''A Lie subalgebra \mathfrak of endomorphisms of a finite-dimensional vector space over a field of characteristic zero is solvable if and only if \operatorname(ab)=0 whenever a\in\mathfrak,b\in mathfrak,\mathfrak'' The fact that \operatorname(ab)=0 in the solvable case follows from Lie's theorem that puts \mathfrak g in the upper triangular form over the algebraic closure of the ground field (the trace can be computed after extending the ground f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nilpotent Lie Algebra
In mathematics, a Lie algebra \mathfrak is nilpotent if its lower central series terminates in the zero subalgebra. The ''lower central series'' is the sequence of subalgebras : \mathfrak \geq mathfrak,\mathfrak\geq mathfrak,[\mathfrak,\mathfrak \geq [\mathfrak, mathfrak,[\mathfrak,\mathfrak] \geq ... We write \mathfrak_0 = \mathfrak, and \mathfrak_n = [\mathfrak,\mathfrak_] for all n > 0. If the lower central series eventually arrives at the zero subalgebra, then the Lie algebra is called nilpotent. The lower central series for Lie algebras is analogous to the lower central series in group theory, and nilpotent Lie algebras are analogs of nilpotent groups. The nilpotent Lie algebras are precisely those that can be obtained from abelian Lie algebras, by successive central extensions. Note that the definition means that, viewed as a non-associative non-unital algebra, a Lie algebra \mathfrak is nilpotent if it is nilpotent as an ideal. Definition Let \mathfrak be a Lie algeb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjoint Representation Of A Lie Algebra
In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is GL(n, \mathbb), the Lie group of real ''n''-by-''n'' invertible matrices, then the adjoint representation is the group homomorphism that sends an invertible ''n''-by-''n'' matrix g to an endomorphism of the vector space of all linear transformations of \mathbb^n defined by: x \mapsto g x g^ . For any Lie group, this natural representation is obtained by linearizing (i.e. taking the differential of) the action of ''G'' on itself by conjugation. The adjoint representation can be defined for linear algebraic groups over arbitrary fields. Definition Let ''G'' be a Lie group, and let :\Psi: G \to \operatorname(G) be the mapping , with Aut(''G'') the automorphism group of ''G'' and given by the inner automorphism (conjugation) :\Psi_g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borel Subalgebra
In mathematics, specifically in representation theory, a Borel subalgebra of a Lie algebra \mathfrak is a maximal solvable subalgebra. The notion is named after Armand Borel. If the Lie algebra \mathfrak is the Lie algebra of a complex Lie group, then a Borel subalgebra is the Lie algebra of a Borel subgroup. Borel subalgebra associated to a flag Let \mathfrak g = \mathfrak(V) be the Lie algebra of the endomorphisms of a finite-dimensional vector space ''V'' over the complex numbers. Then to specify a Borel subalgebra of \mathfrak g amounts to specify a flag of ''V''; given a flag V = V_0 \supset V_1 \supset \cdots \supset V_n = 0, the subspace \mathfrak b = \ is a Borel subalgebra, and conversely, each Borel subalgebra is of that form by Lie's theorem. Hence, the Borel subalgebras are classified by the flag variety of ''V''. Borel subalgebra relative to a base of a root system Let \mathfrak g be a complex semisimple Lie algebra, \mathfrak h a Cartan subalgebra and ''R'' the r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representation Of A Lie Algebra
In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is given by the commutator. In the language of physics, one looks for a vector space V together with a collection of operators on V satisfying some fixed set of commutation relations, such as the relations satisfied by the angular momentum operators. The notion is closely related to that of a representation of a Lie group. Roughly speaking, the representations of Lie algebras are the differentiated form of representations of Lie groups, while the representations of the universal cover of a Lie group are the integrated form of the representations of its Lie algebra. In the study of representations of a Lie algebra, a particular ring, called the universal enveloping algebra, associated with the Lie algebra plays an important role. The universal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Consequences
Consequence may refer to: * Logical consequence, also known as a ''consequence relation'', or ''entailment'' * In operant conditioning, a result of some behavior * Consequentialism, a theory in philosophy in which the morality of an act is determined by its effects * Unintended consequences * In logic, a consequent is the second half of a hypothetical proposition or consequences * Consequent (music), the second half of a period (music) Games * Consequences (game), a parlour game Fiction * ''Consequences'' (novel), a 1919 novel by E. M. Delafield * "Consequences" (Kipling story), an 1888 short story by Rudyard Kipling * "Consequences" (Cather story), a 1915 short story by Willa Cather Film and TV * ''Die Konsequenz'' (English: ''The Consequence''), a 1977 West German film * ''Anjaam'' (English: ''Consequence''), a 1994 Hindi film * '' Anjaam (1940 film)'', an earlier Hindi film of the same name * Consequence (film), a 2003 action thriller film starring Armand Assante * ''Con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |