"dgla" redirects here. For the fatty acid, see Dihomo-γ-linolenic acid.
In mathematics, in particular abstract algebra and topology, a differential graded Lie algebra (or dg Lie algebra, or dgla) is a graded vector space with added Lie algebra and chain complex structures that are compatible. Such objects have applications in deformation theory[1] and rational homotopy theory.
^Hinich, Vladimir (2001). "DG coalgebras as formal stacks". Journal of Pure and Applied Algebra. 162 (2–3): 209–250. arXiv:math/9812034. doi:10.1016/S0022-4049(00)00121-3. MR 1843805. S2CID 15720862.
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that respects the algebra structure. A differentialgradedalgebra (or DG-algebra for short) A is a gradedalgebra equipped with a map d : A → A {\displaystyle...
mathematics, differentialalgebra is, broadly speaking, the area of mathematics consisting in the study of differential equations and differential operators...
definition of a graded ring; hence, the notion applies to non-associative algebras as well; e.g., one can consider a gradedLiealgebra. Generally, the...
the Lie bracket. For example, a gradedLiealgebra is a Liealgebra (or more generally a Lie superalgebra) with a compatible grading. A differential graded...
differential gradedLiealgebra. To be a little more specific, the Jacobi identity only holds up to homotopy. Therefore, a differentialgradedLiealgebra can...
the algebra of tensor fields of the underlying manifold. The Lie derivative commutes with contraction and the exterior derivative on differential forms...
derivative gives the exterior algebra of differential forms on a manifold the structure of a differentialgradedalgebra. The exterior derivative commutes...
In algebra, a differentialgraded module, or dg-module, is a Z {\displaystyle \mathbb {Z} } -graded module together with a differential; i.e., a square-zero...
graded Lie algebra (say, graded by Z {\displaystyle \mathbb {Z} } or N {\displaystyle \mathbb {N} } ) that is anticommutative and has a graded Jacobi identity...
mathematics and theoretical physics, a superalgebra is a Z2-gradedalgebra. That is, it is an algebra over a commutative ring or field with a decomposition...
algebra. A differentialgradedalgebra is an associative algebra together with a grading and a differential. For example, the de Rham algebra Ω ( M ) =...
representation theory, differential geometry, quantum statistical mechanics, quantum information, and quantum field theory. Operator algebras can be used to study...
of this differential on an exterior algebra makes sense for any Liealgebra, so it is used to define Liealgebra cohomology for all Liealgebras. More generally...
enveloping algebra of a Liealgebra is the unital associative algebra whose representations correspond precisely to the representations of that Liealgebra. Universal...
category of Liealgebras. That is, it is left adjoint to the forgetful functor. The free Liealgebra on a set X is naturally graded. The 1-graded component...
or noncommutative geometry a quantum differential calculus or noncommutative differential structure on an algebra A {\displaystyle A} over a field k {\displaystyle...
pullback. Differential forms are part of the field of differential geometry, influenced by linear algebra. Although the notion of a differential is quite...
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations...
as the algebra of generalized Poisson brackets defined on differential forms. A Gerstenhaber algebra is a graded-commutative algebra with a Lie bracket...
equivalent to the homotopy category of connected differentialgradedLiealgebras. (The associated gradedLiealgebra ker ( d ) / im ( d ) {\displaystyle...
group, and thus is subject to the Dold–Kan correspondence. DifferentialgradedLiealgebra Quillen, Daniel (September 1969). "Rational homotopy theory"...
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