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Lie Operad
In mathematics, the Lie operad is an operad theory, operad Operad algebra, whose algebras are Lie algebras. The notion (at least one version) was introduced by in their formulation of Koszul duality. Definition à la Ginzburg–Kapranov Fix a base field ''k'' and let \mathcal(x_1, \dots, x_n) denote the free Lie algebra over ''k'' with generators x_1, \dots, x_n and \mathcal(n) \subset \mathcal(x_1, \dots, x_n) the subspace spanned by all the bracket monomials containing each x_i exactly once. The symmetric group S_n acts on \mathcal(x_1, \dots, x_n) by permutations of the generators and, under that action, \mathcal(n) is invariant. The operadic composition is given by substituting expressions (with renumbered variables) for variables. Then, \mathcal = \ is an operad. Koszul-Dual The Koszul-dual of \mathcal is the commutative-ring operad, an operad whose algebras are the commutative rings over ''k.'' Notes References * External links *Todd TrimbleNotes on operads and th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operad Theory
In mathematics, an operad is a structure that consists of abstract Operation (mathematics), operations, each one having a fixed finite number of inputs (arguments) and one output, as well as a specification of how to compose these operations. Given an operad O, one defines an ''algebra over O'' to be a set together with concrete operations on this set which behave just like the abstract operations of O. For instance, there is a Lie operad L such that the algebras over L are precisely the Lie algebras; in a sense L abstractly encodes the operations that are common to all Lie algebras. An operad is to its algebras as a Group (mathematics), group is to its group representations. History Operads originate in algebraic topology; they were introduced to characterize iterated loop spaces by Michael Boardman, J. Michael Boardman and Rainer M. Vogt in 1969 and by J. Peter May in 1970. The word "operad" was created by May as a portmanteau of "operations" and "monad (category theory), monad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operad Algebra
In algebra, an operad algebra is an "algebra" over an operad. It is a generalization of an associative algebra over a commutative ring ''R'', with an operad replacing ''R''. Definitions Given an operad ''O'' (say, a symmetric sequence in a symmetric monoidal ∞-category ''C''), an algebra over an operad, or ''O''-algebra for short, is, roughly, a left module over ''O'' with multiplications parametrized by ''O''. If ''O'' is a topological operad, then one can say an algebra over an operad is an ''O''-monoid object in ''C''. If ''C'' is symmetric monoidal, this recovers the usual definition. Let ''C'' be symmetric monoidal ∞-category with monoidal structure distributive over colimits. If f: O \to O' is a map of operads and, moreover, if ''f'' is a homotopy equivalence, then the ∞-category of algebras over ''O'' in ''C'' is equivalent to the ∞-category of algebras over ''O in ''C''. See also * En-ring *Homotopy Lie algebra In mathematics, in particular abstract algebra ... [...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] |
Koszul Duality
In mathematics, Koszul duality, named after the French mathematician Jean-Louis Koszul, is any of various kinds of dualities found in representation theory of Lie algebras, abstract algebras (semisimple algebra) and topology (e.g., equivariant cohomology). The prototype example, due to Joseph Bernstein, Israel Gelfand, and Sergei Gelfand, is the rough duality between the derived category of a symmetric algebra and that of an exterior algebra. The importance of the notion rests on the suspicion that Koszul duality seems quite ubiquitous in nature. Koszul duality for modules over Koszul algebras The simplest, and in a sense prototypical case of Koszul duality arises as follows: for a 1-dimensional vector space ''V'' over a field ''k'', with dual vector space V^*, the exterior algebra of ''V'' has two non-trivial components, namely :\bigwedge^1 V=V, \quad \bigwedge^0 V = k. This exterior algebra and the symmetric algebra of V^*, \operatorname(V^*), serve to build a two-step chain comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Lie Algebra
In mathematics, a free Lie algebra over a field ''K'' is a Lie algebra generated by a set ''X'', without any imposed relations other than the defining relations of alternating ''K''-bilinearity and the Jacobi identity. Definition The definition of the free Lie algebra generated by a set ''X'' is as follows: : Let ''X'' be a set and i\colon X \to L a morphism of sets (function) from ''X'' into a Lie algebra ''L''. The Lie algebra ''L'' is called free on ''X'' if i is the universal morphism; that is, if for any Lie algebra ''A'' with a morphism of sets f\colon X \to A, there is a unique Lie algebra morphism g\colon L\to A such that f = g \circ i. Given a set ''X'', one can show that there exists a unique free Lie algebra L(X) generated by ''X''. In the language of category theory, the functor sending a set ''X'' to the Lie algebra generated by ''X'' is the free functor from the category of sets to the category of Lie algebras. That is, it is left adjoint to the forgetful functo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \mathrm_n defined over a finite set of n symbols consists of the permutations that can be performed on the n symbols. Since there are n! (n factorial) such permutation operations, the order (number of elements) of the symmetric group \mathrm_n is n!. Although symmetric groups can be defined on infinite sets, this article focuses on the finite symmetric groups: their applications, their elements, their conjugacy classes, a finite presentation, their subgroups, their automorphism groups, and their representation theory. For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set. The symmetric group is important to diverse areas of mathematics such as Galois theory, invariant theory, the representatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Koszul-dual
In mathematics, Koszul duality, named after the French mathematician Jean-Louis Koszul, is any of various kinds of dualities found in representation theory of Lie algebras, abstract algebras (semisimple algebra) and topology (e.g., equivariant cohomology). The prototype example, due to Joseph Bernstein, Israel Gelfand, and Sergei Gelfand, is the rough duality between the derived category of a symmetric algebra and that of an exterior algebra. The importance of the notion rests on the suspicion that Koszul duality seems quite ubiquitous in nature. Koszul duality for modules over Koszul algebras The simplest, and in a sense prototypical case of Koszul duality arises as follows: for a 1-dimensional vector space ''V'' over a field ''k'', with dual vector space V^*, the exterior algebra of ''V'' has two non-trivial components, namely :\bigwedge^1 V=V, \quad \bigwedge^0 V = k. This exterior algebra and the symmetric algebra of V^*, \operatorname(V^*), serve to build a two-step chain comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative-ring Operad
In mathematics, an operad is a structure that consists of abstract operations, each one having a fixed finite number of inputs (arguments) and one output, as well as a specification of how to compose these operations. Given an operad O, one defines an ''algebra over O'' to be a set together with concrete operations on this set which behave just like the abstract operations of O. For instance, there is a Lie operad L such that the algebras over L are precisely the Lie algebras; in a sense L abstractly encodes the operations that are common to all Lie algebras. An operad is to its algebras as a group is to its group representations. History Operads originate in algebraic topology; they were introduced to characterize iterated loop spaces by J. Michael Boardman and Rainer M. Vogt in 1969 and by J. Peter May in 1970. The word "operad" was created by May as a portmanteau of "operations" and "monad" (and also because his mother was an opera singer). Interest in operads was consider ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
