Lie N-algebra

	Lie N-algebra In mathematics, a Lie ''n''-algebra is a generalization of a Lie algebra, a vector space with a bracket, to higher order operations. For example, in the case of a Lie 2-algebra, the Jacobi identity is replaced by an isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ... called a ''Jacobiator''. See also * 2-ring * Homotopy Lie algebra References * Jim Stasheff and Urs SchreiberZoo of Lie n-Algebras *about the paper at the n-category café. Further reading * https://ncatlab.org/nlab/show/Lie+2-algebra * https://golem.ph.utexas.edu/category/2007/08/string_and_chernsimons_lie_3al.html Lie algebras {{algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Generalization Of A Lie Algebra In mathematics, a Lie algebra has been generalized in several ways. Graded Lie algebra and Lie superalgebra A graded Lie algebra is a Lie algebra with grading. When the grading is \mathbb/2, it is also known as a Lie superalgebra. Lie-isotopic algebra A Lie-isotopic algebra is a generalization of Lie algebras proposed by physicist R. M. Santilli in 1978. Definition Recall that a finite-dimensional Lie algebra L with generators X_1, X_2, ..., X_n and commutation rules : _i X_j= X_i X_j - X_j X_i = C_^k X_k, can be defined (particularly in physics) as the totally anti-symmetric algebra A(L)^- attached to the universal enveloping associative algebra A(L)=\ equipped with the associative product X_i \times X_j over a numeric field F with multiplicative unit 1. Consider now the axiom-preserving lifting of A(L) into the form A^(L^)=\, called universal enveloping isoassociative algebra, with isoproduct :X_i\times X_j = X_i T^* X_j, verifying the isoassociative law :X_i ... [...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]
	Isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is derived . The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may often be identified. In mathematical jargon, one says that two objects are the same up to an isomorphism. A common example where isomorphic structures cannot be identified is when the structures are substructures of a larger one. For example, all subspaces of dimension one of a vector space are isomorphic and cannot be identified. An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	2-ring In mathematics, a categorical ring is, roughly, a category equipped with addition and multiplication. In other words, a categorical ring is obtained by replacing the underlying set of a ring by a category. For example, given a ring ''R'', let ''C'' be a category whose objects are the elements of the set ''R'' and whose morphisms are only the identity morphisms. Then ''C'' is a categorical ring. But the point is that one can also consider the situation in which an element of ''R'' comes with a "nontrivial automorphism". This line of generalization of a ring eventually leads to the notion of an ''E''''n''-ring. See also Categorification Higher-dimensional algebra Lie n-algebra In mathematics, a Lie ''n''-algebra is a generalization of a Lie algebra, a vector space with a bracket, to higher order operations. For example, in the case of a Lie 2-algebra, the Jacobi identity is replaced by an isomorphism In mathematics, ... Further reading John Baez2-Rigs in Topology an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Homotopy Lie Algebra In mathematics, in particular abstract algebra and topology, a homotopy Lie algebra (or L_\infty-algebra) is a generalisation of the concept of a differential graded Lie algebra. To be a little more specific, the Jacobi identity only holds up to homotopy. Therefore, a differential graded Lie algebra can be seen as a homotopy Lie algebra where the Jacobi identity holds on the nose. These homotopy algebras are useful in classifying deformation problems over characteristic 0 in deformation theory because deformation functors are classified by quasi-isomorphism classes of L_\infty-algebras. This was later extended to all characteristics by Jonathan Pridham. Homotopy Lie algebras have applications within mathematics and mathematical physics; they are linked, for instance, to the Batalin–Vilkovisky formalism much like differential graded Lie algebras are. Definition There exists several different definitions of a homotopy Lie algebra, some particularly suited to certain situations m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]