Derived Representation Schemes and Noncommutative Geometry

@article{Berest2013DerivedRS,
  title={Derived Representation Schemes and Noncommutative Geometry},
  author={Yuri Yu. Berest and Giovanni Felder and Ajay C. Ramadoss},
  journal={arXiv: K-Theory and Homology},
  year={2013}
}
Some 15 years ago M. Kontsevich and A. Rosenberg [KR] proposed a heuristic principle according to which the family of schemes ${Rep_n(A)}$ parametrizing the finite-dimensional represen- tations of a noncommutative algebra A should be thought of as a substitute or "approximation" for Spec(A). The idea is that every property or noncommutative geometric structure on A should induce a corresponding geometric property or structure on $Rep_n(A)$ for all n. In recent years, many interesting structures… 
Stable representation homology and Koszul duality
This paper is a sequel to [BKR], where we studied the derived affine scheme DRep_n(A) of the classical representation scheme Rep_n(A) for an associative k-algebra A. In [BKR], we have constructed
Pre-Calabi-Yau structures and moduli of representations
We give a new characterization of weak Calabi-Yau structures in the sense of [KV13] and establish some relations with other concepts. We start by developing some noncommutative calculus for DG
Derived representation schemes and Nakajima quiver varieties
We introduce a derived representation scheme associated with a quiver, which may be thought of as a derived version of a Nakajima variety. We exhibit an explicit model for the derived representation
On the noncommutative Poisson geometry of certain wild character varieties
To show that certain wild character varieties are multiplicative analogues of quiver varieties, Boalch introduced colored multiplicative quiver varieties. They form a class of (nondegenerate) Poisson
Multiplicative preprojective algebras are 2-Calabi-Yau.
We prove that multiplicative preprojective algebras, defined by Crawley-Boevey and Shaw, are 2-Calabi-Yau algebras, in the case of quivers containing unoriented cycles. If the quiver is not itself a
Pre-Calabi-Yau algebras and noncommutative calculus on higher cyclic Hochschild cohomology
We prove $L_{\infty}$-formality for the higher cyclic Hochschild complex $\chH$ over free associative algebra or path algebra of a quiver. The $\chH$ complex is introduced as an appropriate tool for
J ul 2 02 1 Multiplicative preprojective algebras are 2-Calabi – Yau
We prove that multiplicative preprojective algebras, defined by Crawley-Boevey and Shaw, are 2-Calabi–Yau algebras, in the case of quivers containing unoriented cycles. If the quiver is not itself a
Chern-Simons forms and higher character maps of Lie representations
This paper is a sequel to our earlier work [BFPRW], where we study the derived representation scheme DRep_{g}(A) parametrizing the representations of a Lie algebra A in a finite-dimensional reductive
Large $N$ phenomena and quantization of the Loday-Quillen-Tsygan theorem
We offer a new approach to large N limits using the Batalin-Vilkovisky formalism, both commutative and noncommutative, and we exhibit how the Loday-Quillen-Tsygan Theorem admits BV quantizations in
...
...

References

SHOWING 1-10 OF 101 REFERENCES
Noncommutative Poisson structures, derived representation schemes and Calabi-Yau algebras
Recantly, William Crawley-Boevey proposed the definition of a Poisson structure on a noncommutative algebra $A$ based on the Kontsevich principle. His idea was to find the {\it weakest} possible
Qurves and quivers
Noncommutative geometry based on commutator expansions
We develop an approach to noncommutative algebraic geometry ``in the perturbative regime" around ordinary commutative geometry. Let R be a noncommutative algebra and A=R/[R,R] its commutativization.
Stable representation homology and Koszul duality
This paper is a sequel to [BKR], where we studied the derived affine scheme DRep_n(A) of the classical representation scheme Rep_n(A) for an associative k-algebra A. In [BKR], we have constructed
Non-commutative Symplectic Geometry, Quiver varieties,$\,$ and$\,$ Operads
Quiver varieties have recently appeared in various different areas of Mathematics such as representation theory of Kac-Moody algebras and quantum groups, instantons on 4-manifolds, and resolutions
Cyclic homology and nonsingularity
From the pioneering work of Connes [Col] one knows that periodic cyclic homology can be regarded as a natural extension of de Rham cohomology to the realm of noncommutative geometry. Our aim in this
Algebra extensions and nonsingularity
This paper is concerned with a notion of nonsingularity for noncommutative algebras, which arises naturally in connection with cyclic homology. Let us consider associative unital algebras over the
Noncommutative complete intersections and matrix integrals
We introduce a class of noncommutatative algebras called representation complete intersections (RCI). A graded associative algebra A is said to be RCI provided there exist arbitrarily large positive
Noncommutative smooth spaces
We will work in the category Al gk of associative unital algebras over a fixed base field k. If A € Ob(Algk), we denote by 1A € A the unit in A and by m A : A ⊗ A—→A the product. For an algebra A, we
Graded algebras and their differential graded extensions
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
...
...