Related papers: DG Algebra structures on the quantum affine $n$-sp…
In this paper, we introduce and study differential graded (DG for short) polynomial algebras. In brief, a DG polynomial algebra $\mathcal{A}$ is a connected cochain DG algebra such that its underlying graded algebra $\mathcal{A}^{\#}$ is a…
A differential graded (DG for short) free algebra $\mathcal{A}$ is a connected cochain DG algebra such that its underlying graded algebra is $$\mathcal{A}^{\#}=\k\langle x_1,x_2,\cdots, x_n\rangle,\,\, \text{with}\,\, |x_i|=1,\,\, \forall…
In this paper, we introduce the notion of DG Sklyanin algebras, which are connected cochain DG algebras whose underlying graded algebras are Sklyanin algebras. Let $\mathcal{A}$ be a $3$-dimensional DG Sklyanin algebra with…
Let $\mathcal{A}$ be a connected cochain DG algebra such that its underlying graded algebra $\mathcal{A}^{\#}$ is the graded skew polynomial algebra $$k\langle x_1,x_2, x_3\rangle/\left(\begin{array}{ccc} x_1x_2+x_2x_1\\ x_2x_3+x_3x_2\\…
We define and investigate deformed n-Calabi-Yau completions of homologically smooth differential graded (=dg) categories. Important examples are: deformed preprojective algebras of connected non Dynkin quivers, Ginzburg dg algebras…
Calabi-Yau algebras are particularly symmetric differential graded algebras. There is a construction called `Calabi-Yau completion' which produces a canonical Calabi-Yau algebra from any homologically smooth dg algebra. Homologically smooth…
We introduce some new algebraic structures arising naturally in the geometry of Calabi-Yau manifolds and mirror symmetry. We give a universal construction of Calabi-Yau algebras in terms of a noncommutative symplectic DG algebra resolution.…
Assume that $\mathscr{A}$ is a connected cochain DG algebra. We show that $\mathscr{A}$ is homologically smooth and Gorenstein if and only if its $\mathrm{Ext}$-algebra $H(R\Hom_{\mathscr{A}}(\mathbbm{k},\mathbbm{k}))$ is a Frobenius graded…
Any finite-dimensional commutative (associative) graded algebra with all nonzero homogeneous subspaces one-dimensional is defined by a symmetric coefficient matrix. This algebraic structure gives a basic kind of $A$-graded algebras…
We study differential graded algebras whose homology is an exterior algebra over a commutative ring R on a generator of degree n, and also certain types of differential modules over these DGAs. We obtain a complete classification when R is…
We describe a Lie Algebra on the moduli space of Calabi-Yau threefolds enhanced with differential forms and its relation to the Bershadsky-Cecotti-Ooguri-Vafa holomorphic anomaly equation. In particular, we describe algebraic topological…
We prove that the stable tame isomorphism, quasi-isomorphism, and derived Morita equivalence problems for semifree noncommutative differential graded algebras (DGAs) are all undecidable. This resolves half of Problem 5.16 from the K3…
In this article we study dimer models, as introduced in string theory, which give a way of writing down a class of non-commutative `superpotential' algebras. Some examples are 3-dimensional Calabi-Yau algebras, as defined by Ginzburg, and…
This paper concerns an associative graded algebra A that is the enveloping algebra of a Lie algebra with exponential growth. The algebra A is 3-Calabi-Yau. There is a Z-form of A so for every field k there is an algebra A_k. An algebraic…
In this paper, we prove that the Chekanov-Eliashberg algebra of an horizontally displaceable n-dimensional Legendrian sphere in the contactisation of a Liouville manifold is a (n+1)-Calabi-Yau differential graded algebra. In particular it…
A differential calculus on an associative algebra A is an algebraic analogue of the calculus of differential forms on a smooth manifold. It supplies A with a structure on which dynamics and field theory can be formulated to some extent in…
Let a finite group G act on a differential graded algebra A. This article presents necessary conditions and sufficient conditions for the skew group algebra A*G to be Calabi-Yau. In particular, when A is the Ginzburg dg algebra of a quiver…
A differential graded algebra can be viewed as an A-infinity algebra. By a theorem of Kadeishvili, a dga over a field admits a quasi-isomorphism from a minimal A-infinity algebra. We introduce the notion of a derived A-infinity algebra and…
A well-known and old result of Hazewinkel and Koszul states that the cohomology of a finite-dimensional Lie algebra is isomorphic, up to a suitable shift, to its twisted homology, a Lie-theoretical version of Poincare duality. This paper…
In this paper we relate the deformation theory of Ginzburg Calabi-Yau algebras to negative cyclic homology. We do this by exhibiting a DG-Lie algebra that controls this deformation theory and whose homology is negative cyclic homology. We…