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

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

2008

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

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.

Lie-Massey brackets and n-homotopically multiplicative maps of differential graded Lie algebras

Journal of Pure and Applied Algebra, 1993

For Alex Heller on his 65th birthday Retakh, V.S.. Lie-Massey brackets and n-homotopically multiplicative maps of differential graded Lie algebras, Journal of Pure and Applied Algebra 89 (1993) 217-229. The definition of rr-homotopically multiplicative maps of differential graded Lie algebras is given. It is shown that such maps conserve n-Lie-Massey brackets.

Graded Lie-Cartan pairs I

Reports on mathematical physics, 1988

The Lie-Cartan pairs proposed in [l] as an algebraic frame for the classical operators of differential geometry are generalized to the Z/Z-graded case (graded Lie-Cartan pairs of a graded Lie algebra and a graded commutative algebra). The generalized case is reduced to the Abelian case by tensoring with arbitrary graded commutative algebras.

Graded Poisson structures on the algebra of differential forms

Commentarii Mathematici Helvetici, 1995

One-to-one correspondences are established between the set of all nondegenerate graded Jacobi operators of degree 1 defined on the graded algebra (M) of differential forms on a smooth, oriented, Riemannian manifold M, the space of bundle isomorphisms L: T M ! T M , and the space of nondegenerate derivations of degree 1 having null square. Derivations with this property, and Jacobi structures of odd Z2-degree are also studied through the action of the automorphism group of (M).

Koszul differential graded algebras and BGG correspondence II

Chinese Annals of Mathematics, Series B, 2010

The concept of Koszul differential graded algebra (Koszul DG algebra) is introduced. Koszul DG algebras exist extensively, and have nice properties similar to the classic Koszul algebras. A DG version of the Koszul duality is proved. When the Koszul DG algebra A is AS-regular, the Ext-algebra E of A is Frobenius. In this case, similar to the classical BGG correspondence, there is an equivalence between the stable category of finitely generated left E-modules, and the quotient triangulated category of the full triangulated subcategory of the derived category of right DG A-modules consisting of all compact DG modules modulo the full triangulated subcategory consisting of all the right DG modules with finite dimensional cohomology. The classical BGG correspondence can be derived from the DG version. n i=0 A i (M ), d) be the de Rham complex of M , then (A * (M ), d) is a commutative DG algebra and by de Rham theorem ([M]) the 0-th cohomology group H 0 (A * (M )) ∼ = R. Hence the DG algebra A * (M ) has 2000 Mathematics Subject Classification. Primary 16E45, 16E10.

Universal enveloping algebras of differential graded Poisson algebras

2014

In this paper, we introduce the notion of differential graded Poisson algebra and study its universal enveloping algebra. From any differential graded Poisson algebra AAA, we construct two isomorphic differential graded algebras: AeA^eAe and AEA^EAE. It is proved that the category of differential graded Poisson modules over AAA is isomorphic to the category of differential graded modules over AeA^eAe, and AeA^eAe is the unique universal enveloping algebra of AAA up to isomorphisms. As applications of the universal property of AeA^eAe, we prove that (Ae)opcong(Aop)e(A^e)^{op}\cong (A^{op})^e(Ae)opcong(Aop)e and (AotimesBbbkB)econgAeotimesBbbkBe(A\otimes_{\Bbbk}B)^e\cong A^e\otimes_{\Bbbk}B^e(AotimesBbbkB)econgAeotimesBbbkBe as differential graded algebras. As consequences, we obtain that ``$e$'' is a monoidal functor and establish links among the universal enveloping algebras of differential graded Poisson algebras, differential graded Lie algebras and associative algebras.

Deformations of Modules of Differential Forms

Journal of Nonlinear Mathematical Physics, 2003

We study non-trivial deformations of the natural action of the Lie algebra Vect(R n ) on the space of differential forms on R n . We calculate abstractions for integrability of infinitesimal multi-parameter deformations and determine the commutative associative algebra corresponding to the miniversal deformation in the sense of .

A ] 1 1 Ja n 20 07 Higher Derived Brackets and Deformation Theory I

2008

The existing constructions of derived Lie and sh-Lie brackets involve multilinear maps that are used to define higher order differential operators. In this paper, we prove the equivalence of three different definitions of higher order operators. We then introduce a unifying theme for building derived brackets and show that two prevalent derived Lie bracket constructions are equivalent. Two basic methods of constructing derived strict sh-Lie brackets are also shown to be essentially the same. So far, each of these derived brackets is defined on an abelian subalgebra of a Lie algebra. We describe, as an alternative, a cohomological construction of derived sh-Lie brackets. Namely, we prove that a differential algebra with a graded homotopy commutative and associative product and an odd, square-zero operator (that commutes with the differential) gives rise to an sh-Lie structure on the cohomology via derived brackets. The method is in particular applicable to differential vertex operato...

Generalization of connection based on the concept of graded q-differential algebra

2010

We propose a generalization of the concept of connection form by means of a graded q-differential algebra Ω q , where q is a primitive Nth root of unity, and develop the concept of curvature N-form for this generalization of the connection form. The Bianchi identity for a curvature N-form is proved. We study an Ω q -connection on module and prove that every projective module admits an Ω q -connection. If the module is equipped with a Hermitian structure, we introduce a notion of an Ω q -connection consistent with the Hermitian structure.