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Complex Lie Algebra
In mathematics, a complex Lie algebra is a Lie algebra over the complex numbers. Given a complex Lie algebra \mathfrak, its conjugate \overline is a complex Lie algebra with the same underlying real vector space but with i = \sqrt acting as -i instead. As a real Lie algebra, a complex Lie algebra \mathfrak is trivially isomorphic to its conjugate. A complex Lie algebra is isomorphic to its conjugate if and only if it admits a real form (and is said to be defined over the real numbers). Real form Given a complex Lie algebra \mathfrak, a real Lie algebra \mathfrak_0 is said to be a real form of \mathfrak if the complexification \mathfrak_0 \otimes_\mathbb is isomorphic to \mathfrak. A real form \mathfrak_0 is abelian (resp. nilpotent, solvable, semisimple) if and only if \mathfrak is abelian (resp. nilpotent, solvable, semisimple). On the other hand, a real form \mathfrak_0 is simple if and only if either \mathfrak is simple or \mathfrak is of the form \mathfrak \times \overli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Without Loss Of Generality
''Without loss of generality'' (often abbreviated to WOLOG, WLOG or w.l.o.g.; less commonly stated as ''without any loss of generality'' or ''with no loss of generality'') is a frequently used expression in mathematics. The term is used to indicate the assumption that what follows is chosen arbitrarily, narrowing the premise to a particular case, but does not affect the validity of the proof in general. The other cases are sufficiently similar to the one presented that proving them follows by essentially the same logic. As a result, once a proof is given for the particular case, it is trivial to adapt it to prove the conclusion in all other cases. In many scenarios, the use of "without loss of generality" is made possible by the presence of symmetry. For example, if some property ''P''(''x'',''y'') of real numbers is known to be symmetric in ''x'' and ''y'', namely that ''P''(''x'',''y'') is equivalent to ''P''(''y'',''x''), then in proving that ''P''(''x'',''y'') holds for every ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Semidirect Product
In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. It is usually denoted with the symbol . There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in which a group can be made up of two subgroups, one of which is a normal subgroup. * an ''outer'' semidirect product is a way to construct a new group from two given groups by using the Cartesian product as a set and a particular multiplication operation. As with direct products, there is a natural equivalence between inner and outer semidirect products, and both are commonly referred to simply as ''semidirect products''. For finite groups, the Schur–Zassenhaus theorem provides a sufficient condition for the existence of a decomposition as a semidirect product (also known as splitting extension). Inner semidirect product definitions Given a group with identity element , a subgroup , and a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Exponential Map (Lie Theory)
In the theory of Lie groups, the exponential map is a map from the Lie algebra \mathfrak g of a Lie group G to the group, which allows one to recapture the local group structure from the Lie algebra. The existence of the exponential map is one of the primary reasons that Lie algebras are a useful tool for studying Lie groups. The ordinary exponential function of mathematical analysis is a special case of the exponential map when G is the multiplicative group of positive real numbers (whose Lie algebra is the additive group of all real numbers). The exponential map of a Lie group satisfies many properties analogous to those of the ordinary exponential function, however, it also differs in many important respects. Definitions Let G be a Lie group and \mathfrak g be its Lie algebra (thought of as the tangent space to the identity element of G). The exponential map is a map :\exp\colon \mathfrak g \to G which can be defined in several different ways. The typical modern definition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Cartan Subgroup
In the theory of algebraic groups, a Cartan subgroup of a connected linear algebraic group G over a (not necessarily algebraically closed) field k is the centralizer of a maximal torus. Cartan subgroups are smooth (equivalently reduced), connected and nilpotent. If k is algebraically closed, they are all conjugate to each other. Notice that in the context of algebraic groups a ''torus'' is an algebraic group T such that the base extension T_ (where \bar is the algebraic closure of k) is isomorphic to the product of a finite number of copies of the \mathbf_m=\mathbf_1. Maximal such subgroups have in the theory of algebraic groups a role that is similar to that of maximal tori in the theory of Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...s. If G is reductive (in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Cartan Subalgebra
In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra \mathfrak of a Lie algebra \mathfrak that is self-normalising (if ,Y\in \mathfrak for all X \in \mathfrak, then Y \in \mathfrak). They were introduced by Élie Cartan in his doctoral thesis. It controls the representation theory of a semi-simple Lie algebra \mathfrak over a field of characteristic 0 . In a finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic zero (e.g., a Cartan subalgebra is the same thing as a maximal abelian subalgebra consisting of elements ''x'' such that the adjoint endomorphism \operatorname(x) : \mathfrak \to \mathfrak is semisimple (i.e., diagonalizable). Sometimes this characterization is simply taken as the definition of a Cartan subalgebra.pg 231 In general, a subalgebra is called toral if it consists of semisimple elements. Over an algebraically closed field, a toral subalgebra is automatically abelian. Thus, ove ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Complex Lie Group
In geometry, a complex Lie group is a Lie group over the complex numbers; i.e., it is a complex-analytic manifold that is also a group in such a way G \times G \to G, (x, y) \mapsto x y^ is holomorphic. Basic examples are \operatorname_n(\mathbb), the general linear groups over the complex numbers. A connected compact complex Lie group is precisely a complex torus (not to be confused with the complex Lie group \mathbb C^*). Any finite group may be given the structure of a complex Lie group. A complex semisimple Lie group is a linear algebraic group. The Lie algebra of a complex Lie group is a complex Lie algebra. Examples *A finite-dimensional vector space over the complex numbers (in particular, complex Lie algebra) is a complex Lie group in an obvious way. *A connected compact complex Lie group ''A'' of dimension ''g'' is of the form \mathbb^g/L, a complex torus, where ''L'' is a discrete subgroup of rank 2g. Indeed, its Lie algebra \mathfrak can be shown to be abelian a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Identity Function
Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unchanged. That is, when is the identity function, the equality is true for all values of to which can be applied. Definition Formally, if is a set, the identity function on is defined to be a function with as its domain and codomain, satisfying In other words, the function value in the codomain is always the same as the input element in the domain . The identity function on is clearly an injective function as well as a surjective function (its codomain is also its range), so it is bijective. The identity function on is often denoted by . In set theory, where a function is defined as a particular kind of binary relation, the identity function is given by the identity relation, or ''diagonal'' of . Algebraic propert ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Simple Lie Algebra
In algebra, a simple Lie algebra is a Lie algebra that is non-abelian and contains no nonzero proper ideals. The classification of real simple Lie algebras is one of the major achievements of Wilhelm Killing and Élie Cartan. A direct sum of simple Lie algebras is called a semisimple Lie algebra. A simple Lie group is a connected Lie group whose Lie algebra is simple. Complex simple Lie algebras A finite-dimensional simple complex Lie algebra is isomorphic to either of the following: \mathfrak_n \mathbb, \mathfrak_n \mathbb, \mathfrak_ \mathbb ( classical Lie algebras) or one of the five exceptional Lie algebras. To each finite-dimensional complex semisimple Lie algebra In mathematics, a Lie algebra is semisimple if it is a direct sum of modules, direct sum of Simple Lie algebra, simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper Lie algebra#Subalgebras.2C ideals ... \mathfrak, there exists a corresponding diagram ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Lie Algebra
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. In other words, a Lie algebra is an algebra over a field for which the multiplication operation (called the Lie bracket) is alternating and satisfies the Jacobi identity. The Lie bracket of two vectors x and y is denoted ,y/math>. A Lie algebra is typically a non-associative algebra. However, every associative algebra gives rise to a Lie algebra, consisting of the same vector space with the commutator Lie bracket, ,y= xy - yx . Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: every Lie group gives rise to a Lie algebra, which is the tangent space at the identity. (In this case, the Lie bracket measures the failure of commutativity for the Lie group.) Conversely, to any finite-di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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: :\alp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |