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Register a new userDeformations of algebras in noncommutative geometry
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Authors
Travis Schedler
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These are significantly expanded lecture notes for the authors minicourse at MSRI in June 2012, as published in the MSRI lecture note series, with some minor additional corrections. In these notes, we give an example-motivated review of the deformation theory of associative algebras in terms of the Hochschild cochain complex as well as quantization of Poisson structures, and Kontsevichs formality theorem in the smooth setting. We then discuss quantization and deformation via Calabi-Yau algebras and potentials. Examples discussed include Weyl algebras, enveloping algebras of Lie algebras, symplectic reflection algebras, quasihomogeneous isolated hypersurface singularities (including du Val singularities), and Calabi-Yau algebras.
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Let $E$ be a Koszul Frobenius algebra. A Clifford deformation of $E$ is a finite dimensional $mathbb Z_2$-graded algebra $E(theta)$, which corresponds to a noncommutative quadric hypersurface $E^!/(z)$, for some central regular element $zin E^!_2$. It turns out that the bounded derived category $D^b(text{gr}_{mathbb Z_2}E(theta))$ is equivalent to the stable category of the maximal Cohen-Macaulay modules over $E^!/(z)$ provided that $E^!$ is noetherian. As a consequence, $E^!/(z)$ is a noncommutative isolated singularity if and only if the corresponding Clifford deformation $E(theta)$ is a semisimple $mathbb Z_2$-graded algebra. The preceding equivalence of triangulated categories also indicates that Clifford deformations of trivial extensions of a Koszul Frobenius algebra are related to the Kn{o}rrer Periodicity Theorem for quadric hypersurfaces. As an application, we recover Kn{o}rrer Periodicity Theorem without using of matrix factorizations.
We compute the Nakayama automorphism of a PBW-deformation of a Koszul Artin-Schelter Gorenstein algebra of finite global dimension, and give a criterion for an augmented PBW-deformation of a Koszul Calabi-Yau algebra to be Calabi-Yau. The relations between the Calabi-Yau property of augmented PBW-deformations and that of non-augmented cases are discussed. The Nakayama automorphisms of PBW-deformations of Koszul Artin-Schelter Gorenstein algebras of global dimensions 2 and 3 are given explicitly. We show that if a PBW-deformation of a graded Calabi-Yau algebra is still Calabi-Yau, then it is defined by a potential under some mild conditions. Some classical results are also recovered. Our main method used in this paper is elementary and based on linear algebra. The results obtained in this paper will be applied in a subsequent paper.
In this work we study the deformations of a Hopf algebra $H$ by partial actions of $H$ on its base field $Bbbk$, via partial smash product algebras. We introduce the concept of a $lambda$-Hopf algebra as a Hopf algebra obtained as a partial smash product algebra, and show that every Hopf algebra is a $lambda$-Hopf algebra. Moreover, a method to compute partial actions of a given Hopf algebra on its base field is developed and, as an application, we exhibit all partial actions of such type for some families of Hopf algebras.
In this paper, we define a new cohomology theory for multiplicative Hom-pre-Lie algebras which controls deformations of Hom-pre-Lie algebra structure. This new cohomology is a natural one considering the structure map $alpha$. We show this new cohomology is deformation cohomology of the Hom-pre-Lie algebra. We also develop cohomology and the associated deformation theory for Hom-pre-Lie algebras in the equivariant context.
We introduce a notion of left-symmetric Rinehart algebras, which is a generalization of a left-symmetric algebras. The left multiplication gives rise to a representation of the corresponding sub-adjacent Lie-Rinehart algebra. We construct left-symmetric Rinehart algebra from $mathcal O$-operators on Lie-Rinehart algebra. We extensively investigate representations of a left-symmetric Rinehart algebras. Moreover, we study deformations of left-symmetric Rinehart algebras, which is controlled by the second cohomology class in the deformation cohomology. We also give the relationships between $mathcal O$-operators and Nijenhuis operators on left-symmetric Rinehart algebras.
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