|
Malcev Lie Algebra
In mathematics, a Malcev Lie algebra, or Mal'tsev Lie algebra, is a generalization of a rational nilpotent Lie algebra, and Malcev groups are similar. Both were introduced by , based on the work of . Definition According to a Malcev Lie algebra is a rational Lie algebra L together with a complete, descending -vector space filtration \_ , such that: * F_1 L = L * _rL, F_sLsubset F_L * the associated graded Lie algebra \oplus_ F_rL/F_L is generated by elements of degree one. Applications Relation to Hopf algebras showed that Malcev Lie algebras and Malcev groups are both equivalent to complete Hopf algebras, i.e., Hopf algebras ''H'' endowed with a filtration so that ''H'' is isomorphic to \varprojlim H / F_n H. The functors involved in these equivalences are as follows: a Malcev group ''G'' is mapped to the completion (with respect to the augmentation ideal) of its group ring Q''G'', with inverse given by the group of ''grouplike elements'' of a Hopf algebra ''H'', ess ... [...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 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] |
Hopf Algebra
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously a ( unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antihomomorphism satisfying a certain property. The representation theory of a Hopf algebra is particularly nice, since the existence of compatible comultiplication, counit, and antipode allows for the construction of tensor products of representations, trivial representations, and dual representations. Hopf algebras occur naturally in algebraic topology, where they originated and are related to the H-space concept, in group scheme theory, in group theory (via the concept of a group ring), and in numerous other places, making them probably the most familiar type of bialgebra. Hopf algebras are also studied in their own right, with much work on specific classes of examples on the one hand and classification problems o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Filtration (mathematics)
In mathematics, a filtration \mathcal is, informally, like a set of ever larger Russian dolls, each one containing the previous ones, where a "doll" is a subobject of an algebraic structure. Formally, a filtration is an indexed family (S_i)_ of subobjects of a given algebraic structure S, with the index i running over some totally ordered index set I, subject to the condition that ::if i\leq j in I, then S_i\subseteq S_j. If the index i is the time parameter of some stochastic process, then the filtration can be interpreted as representing all historical but not future information available about the stochastic process, with the algebraic structure S_i gaining in complexity with time. Hence, a process that is adapted to a filtration \mathcal is also called non-anticipating, because it cannot "see into the future". Sometimes, as in a filtered algebra, there is instead the requirement that the S_i be subalgebras with respect to some operations (say, vector addition), but n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Augmentation Ideal
In algebra, an augmentation ideal is an ideal that can be defined in any group ring. If ''G'' is a group and ''R'' a commutative ring, there is a ring homomorphism \varepsilon, called the augmentation map, from the group ring R /math> to R, defined by taking a (finiteWhen constructing , we restrict to only finite (formal) sums) sum \sum r_i g_i to \sum r_i. (Here r_i\in R and g_i\in G.) In less formal terms, \varepsilon(g)=1_R for any element g\in G, \varepsilon(rg)=r for any elements r\in R and g\in G, and \varepsilon is then extended to a homomorphism of ''R''- modules in the obvious way. The augmentation ideal is the kernel of \varepsilon and is therefore a two-sided ideal in ''R'' 'G'' is generated by the differences g - g' of group elements. Equivalently, it is also generated by \, which is a basis as a free ''R''-module. For ''R'' and ''G'' as above, the group ring ''R'' 'G''is an example of an ''augmented'' ''R''-algebra. Such an algebra comes equipped with a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the given group. As a ring, its addition law is that of the free module and its multiplication extends "by linearity" the given group law on the basis. Less formally, a group ring is a generalization of a given group, by attaching to each element of the group a "weighting factor" from a given ring. If the ring is commutative then the group ring is also referred to as a group algebra, for it is indeed an algebra over the given ring. A group algebra over a field has a further structure of a Hopf algebra; in this case, it is thus called a group Hopf algebra. The apparatus of group rings is especially useful in the theory of group representations. Definition Let G be a group, written multiplicatively, and let R be a ring. The group ring of G ove ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primitive Element (coalgebra)
Primitive may refer to: Mathematics * Primitive element (field theory) * Primitive element (finite field) * Primitive cell (crystallography) * Primitive notion, axiomatic systems * Primitive polynomial (other), one of two concepts * Primitive function or antiderivative, ' = ''f'' * Primitive permutation group * Primitive root of unity; See Root of unity * Primitive triangle, an integer triangle whose sides have no common prime factor Sciences * Primitive (phylogenetics), characteristic of an early stage of development or evolution * Primitive equations, a set of nonlinear differential equations that are used to approximate atmospheric flow * Primitive change, a general term encompassing a number of basic molecular alterations in the course of a chemical reaction Computing * Cryptographic primitives, low-level cryptographic algorithms frequently used to build computer security systems * Geometric primitive, the simplest kinds of figures in computer graphics * Language pri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
