A pr 2 00 5 N-differential graded algebras (original) (raw)

D G ] 6 M ay 2 00 5 N-differential graded algebras

2008

We introduce the concept of N-differential graded algebras (N-dga), and study the moduli space of deformations of the differential of an N-dga. We prove that it is controlled by what we call the N-Maurer-Cartan equation.

Ja n 20 06 N-differential graded algebras

2008

We introduce the concept of N-differential graded algebras (N-dga), and study the moduli space of deformations of the differential of a N-dga. We prove that it is controlled by what we call the N-Maurer-Cartan equation.

On N-differential graded algebras

2008

We introduce the concept of N-differential graded algebras (N-dga), and study the moduli space of deformations of the differential of an N-dga. We prove that it is controlled by what we call the (M,N)-Maurer-Cartan equation.

O ct 2 00 6 On N-differential graded algebras

2008

We introduce the concept of N -differential graded algebras (N-dga), and study the moduli space of deformations of the differential of an N-dga. We prove that it is controlled by what we call the (M,N)-Maurer-Cartan equation. Introduction The goal of this paper is to take the first step towards finding a generalization of Homological Mirror Symmetry (HMS) [11] to the context of N -homological algebra [5]. In [7] Fukaya introduced HMS as the equivalence of the deformation functor of the differential of a differential graded algebra associated with the holomorphic structure, with the deformation functor of an A∞-algebra associated with the symplectic structure of a Calabi-Yau variety. This idea motivated us to define deformation functors of the differential of an N -differential graded algebra. An N -dga is a graded associative algebra A, provided with an operator d : A → A of degree 1 such that d(ab) = d(a)b + (−1)ad(b) and d = 0. A nilpotent differential graded algebra (Nil-dga) wil...

Graded differential geometry of graded matrix algebras

Journal of Mathematical Physics, 1999

We study the graded derivation-based noncommutative differential geometry of the Z 2-graded algebra M(n|m) of complex (n + m) × (n + m)-matrices with the "usual block matrix grading" (for n = m). Beside the (infinite-dimensional) algebra of graded forms the graded Cartan calculus, graded symplectic structure, graded vector bundles, graded connections and curvature are introduced and investigated. In particular we prove the universality of the graded derivation-based first-order differential calculus and show, that M(n|m) is a "noncommutative graded manifold" in a stricter sense: There is a natural body map and the cohomologies of M(n|m) and its body coincide (as in the case of ordinary graded manifolds).

Graded algebras and their differential graded extensions

Journal of Mathematical Sciences, 2007

In the survey, we deal with the following situation. Let A be a graded algebra or a differential graded algebra. Adjoining a set x of free (in any sense) indeterminates, we make a new differential graded algebra A x by setting the differential values d : x → A on x. In the general case, such a construction is called the Shafarevich complex. Beginning with classical examples like the bar-complex, Koszul complex, and Tate resolution, we discuss noncommutative (and sometimes even nonassociative) versions of these notions. The comparison with the Koszul complex leads to noncommutative regular sequences and complete intersections; Tate's process of killing cycles gives noncommutative DG resolutions and minimal models. The applications include the Golod-Shafarevich theorem, growth measures for graded algebras, characterizations of algebras of low homological dimension, and a homological description of Gröbner bases. The same constructions for categories of algebras with identities (like Lie or Jordan algebras) allow one to give a homological description of extensions and deformations of PI-algebras.

A pr 2 00 7 On the ( 3 , N ) Maurer-Cartan equation

We introduce geometric examples of N-differential graded algebras. Deformations, of the differential, of 3-differential graded algebras are controlled by the (3, N) Maurer-Cartan equation. We find explicit formulae for the coefficients appearing in that equation. We use these results to introduce and study N Lie algebroids.

Q A ] 1 9 O ct 2 00 7 On the ( 3 , N ) Maurer-Cartan equation

Deformations of the 3-differential of 3-differential graded algebras are controlled by the (3, N) Maurer-Cartan equation. We find explicit formulae for the coefficients appearing in that equation, introduce new geometric examples of N-differential graded algebras, and use these results to study N Lie algebroids.

Resonance varieties of differential graded algebras

2022

Under suitable finiteness assumptions, one may define the Koszul modules and the resonance varieties of any differential graded algebra. When such a dga models a space or a group, the geometry of these varieties mirror topological and group-theoretical properties of those objects. In characteristic 0, rational homotopy theory methods provide a useful bridge between resonance and the various Lie algebras associated to a group. In characteristic 2, the action of the Bockstein homomorphism on the mod 2 cohomology ring yields new resonance varieties, which bear witness to orientability and Poincaré duality for closed manifolds.