and
rational homotopy theory (equivariant, stable, parametrized, equivariant & stable, parametrized & stable)
Examples of Sullivan models in rational homotopy theory:
(also nonabelian homological algebra)
Context
Basic definitions
Stable homotopy theory notions
Constructions
Lemmas
Homology theories
Theorems
A graded algebra is an associative algebra which is with a labelling on its elements by elements of some monoid or group, and such that the multiplication in the algebra is reflected in the multiplication in the labelling group.
Let be a group. (Often will be abelian, and, in fact, one usually takes by default the additive group of integers, in which case the actual group being used is omitted from the terminology and notation.)
A graded ring is a ring equipped with a decomposition of the underlying abelian group as a direct sum such that the product takes .
Analogously there is the notion of graded -associative algebra over any commutative ring .
Specifically for a field a graded algebra is a monoid in graded vector spaces over .
An -graded algebra is called connected if in degree-0 it is just the ground ring.
A differential graded algebra is a graded algebra equipped with a derivation of degree +1 (or -1, depending on conventions) and such that . This is the same as a monoid in the category of chain complexes.
A -graded algebra is called strongly -graded (in Ardizzoni & Menini (2007), Def. 3.2) if for every , the multiplication is an epimorphism.
For a commutative ring write for the corresponding object in the opposite category. Write for the multiplicative group underlying the affine line.
There is a natural isomorphism between
-gradings on ;
-actions on .
The proof is spelled out at affine line in the section Properties.
Let be any (discrete) group and , its group algebra. This has a direct sum decomposition as a -module,
where each is a one dimensional free -module, for which it is convenient, here, to give a basis . The graded algebra structure is obtained by extending the multiplication rule,
given on basis elements, by -linearity.
The Lazard ring, carrying the universal (1-dimensional, commutative) formal group law is naturally an -graded ring.
Textbook account:
For Hopf algebras:
The notion of strongly -graded algebra is defined in:
Last revised on August 19, 2024 at 15:23:24. See the history of this page for a list of all contributions to it.