Graded Calabi Yau algebras of dimension 3

By Raf Bocklandt. 2006

224 citations recorded by the source corpus.

Abstract In this paper, we prove that Graded Calabi Yau algebras of dimension 3 are isomorphic to path algebras of quivers with relations derived from a superpotential. We show that for a given quiver Q and a degree d , the set of good superpotentials of degree d , i.e. those that give rise to Calabi Yau algebras, is either empty or almost everything (in the measure theoretic sense). We also give some constraints on the structure of quivers that allow good superpotentials, and for the simplest quivers we give a complete list of the degrees for which good superpotentials exist.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4b91e30b4b0747c251d99713b09bf104f3e69c5b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

On cluster algebras with coefficients and 2-Calabi-Yau categories (2007, background): “Following [42], cf. also [32], we deﬁne the category C to be d -Calabi-Yau if there exists a family of linear forms such that the bilinear forms Let us assume that C is 2-Calabi-Yau.”
Calabi–Yau triangulated categories (2008, background): “Let k be a field of characteristic 0. Let Q be a finite connected quiver and I an ideal of the path algebra kQ which is homogeneous (for path length) and generated in degrees 2. Let A = kQ/I be the quotient of the path algebra by the ideal I. Theorem 4.2 ([ 8 ]).”
Quivers with potentials and their representations I: Mutations (2007, background): “ith the completed path algebra, so our potentials are possibly inﬁnite linear combinations of cyclic paths. The Jacobian algebras also play an important role in the recent work on Calabi-Yau algebras [4, 25, 26, 27]. In this paper we introduce and study mutations for QPs and their (decorated) repres”
CONSISTENCY CONDITIONS FOR DIMER MODELS (2011, methodology): “p[n−1] oo p[n−2] oo p[1] oo p[0] oo andp = p[n− 1]p[n− 2] .”
On the noncommutative Donaldson-Thomas invariants arising from brane tilings (2008, unknown): “1], who proved that a graded quiver potential algebra can be a graded 3-Calabi-Yau algebra only under some rather restrictive conditions (for example, the algebra C[x, y, z] does not satisfy those conditions) because the grading group in [2] can only be Z and all arrows have there degree one.”
Classifying tilting complexes over preprojective algebras of Dynkin type (2015, background): “We refer to [5,10,15] and see [20] for the detailed proofs.”
Mutation of cluster-tilting objects and potentials (2008, background): “We call � 3-Calabi-Yau (3-CY for short) if the bounded derived category Db(f.l.�) is a 3-CY category [ Boc , CR, G, IR].”
A theory of generalized Donaldson–Thomas invariants (2008, background): “For more details, see Ginzburg [30], Bocklandt [8], and Segal [95].”
Calabi-Yau algebras and superpotentials (2010, background): “Note that the complete case encompasses the graded case which has been treated in [2].”
Non-commutative Donaldson-Thomas theory and the conifold (2007, background): “t to an edge e∈ Eis deﬁned to be zero if edoes not appear in the monomial, and otherwise deﬁned by cyclically permuting the monomial until the edge eis in the ﬁrst position, and deleting it; see e.g. [4, 12]. Let A= CQ/IW be the quotient of the path algebra CQby the ideal IW of relations deﬁned by t”
Dimer Models and Calabi-Yau Algebras (2009, background): “In [4], Bocklandt proved that every graded Calabi-Yau algebra of global dimension 3 is isomorphic to a superpotential algebra.”