mathematics, in particular abstract algebra and topology, a homotopy Lie algebra (or L ∞ {\displaystyle L_{\infty }} -algebra) is a generalisation of the concept...

the two algebraic descriptions of the rational homotopy category. In short, a Lie algebra determines a graded-commutative algebra by Lie algebra cohomology...

mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was first introduced in 1929 by Élie Cartan to study the topology of Lie groups...

a homotopy equivalence, then the ∞-category of algebras over O in C is equivalent to the ∞-category of algebras over O' in C. En-ring Homotopy Lie algebra...

mathematics, a Lie algebra (pronounced /liː/ LEE) is a vector space g {\displaystyle {\mathfrak {g}}} together with an operation called the Lie bracket, an...

have applications in deformation theory and rational homotopy theory. A differential graded Lie algebra is a graded vector space L = ⨁ L i {\displaystyle...

whose inverse equals its transpose). The orthogonal group is an algebraic group and a Lie group. It is compact. The orthogonal group in dimension n has...

mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained...

holds, but only holds up to a homotopy, which is a way to say after an operation "compressing" the information in the algebra, the multiplication is associative...

infinite-dimensional Lie algebras Free Lie algebra Graded Lie algebra Differential graded Lie algebra Homotopy Lie algebra Malcev Lie algebra Modular Lie algebra Monster...

up to homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems...

The foundation of Lie theory is the exponential map relating Lie algebras to Lie groups which is called the Lie group–Lie algebra correspondence. The...

This (real) Lie algebra has dimension n2 − 1. More information about the structure of this Lie algebra can be found below in § Lie algebra structure. In...

being the use of the corresponding 'infinitesimal' representations of Lie algebras. A complex representation of a group is an action by a group on a finite-dimensional...

the rational homotopy type of X {\displaystyle X} . Differential graded Lie algebra Rational homotopy theory Homotopy associative algebra Sullivan 1977...

In mathematics, the Whitehead product is a graded quasi-Lie algebra structure on the homotopy groups of a space. It was defined by J. H. C. Whitehead...

using Clifford algebras. Hurwitz's theorem has been applied in algebraic topology to problems on vector fields on spheres and the homotopy groups of the...

the case of a Lie 2-algebra, the Jacobi identity is replaced by an isomorphism called a Jacobiator. 2-ring homotopy Lie algebra Baez & Crans 2004, 1...

Hamiltonian formulation has constraints not related to a Lie algebra (i.e., the role of Lie algebra structure constants are played by more general structure...

from a homotopy point of view, Lecture Notes in Mathematics, vol. 161, Berlin-New York: Springer-Verlag. Switzer, Robert M. (1975). Algebraic topology—homotopy...

homological algebra in mathematics, the homotopy category K(A) of chain complexes in an additive category A is a framework for working with chain homotopies and...

The Lie algebra of the diffeomorphism group of M {\displaystyle M} consists of all vector fields on M {\displaystyle M} equipped with the Lie bracket...

Algebraic Topology: Higher homotopy groupoids of filtered spaces[usurped] Brown, Ronald; Higgins, Philip; Sivera, Rafael (2011). Nonabelian Algebraic...

widely-known fundamental group; as such, it captures information about the homotopy type of a topological space. In terms of category theory, the fundamental...

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...

on the context, "algebra" can also refer to other algebraic structures, like a Lie algebra or an associative algebra. The word algebra comes from the Arabic...

if SO = PSO) of the compact Lie algebra s o ( n , R ) . {\displaystyle {\mathfrak {so}}(n,\mathbf {R} ).} The homotopy groups of the cover and the quotient...

Galois theory Algebraic stack Gerbe Étale cohomology Motive (algebraic geometry) Motivic cohomology A¹ homotopy theory Homotopical algebra Niels Henrik...

a Lie algebra. There exists some structure theory for such algebras that generalizes the analogous results for Lie algebras and associative algebras.[citation...

spin group and its Lie algebra are embedded inside the Clifford algebra in a natural way, and in applications the Clifford algebra is often the easiest...