Murray Gerstenhaber

ON THE DEFORMATION OF RINGS AND ALGEBRAS

CHAPTER I. The deformation theory for algebras 1. Infinitesimal deformations of an algebra 2. Obstructions 3. Trivial deformations 4. Obstructions to derivations and the squaring operation 5. Obstructions are cocycles 6. Additivity and integrability of the square 7. Restricted deformation theories and their cohomology theories 8. Rigidity of fields in the commutative theory CHAPTER II. The parameter space 1. The set of structure constants as parameter space for the deformation theory 2. Central algebras and an example justifying the choice of parameter space 3. The automorphism group as a parameter space, and examples of obstructions to derivations 4. A fiber space over the parameter space, and the upper semicontinuity theorem 5. An example of a restricted theory and the corresponding modular group CHAPTER III. The deformation theory for graded and filtered rings 1. Graded, filtered, and developable rings 2. The Hochschild theory for developable rings 3. Developable rings as deformations of their associated graded rings 4. Trivial deformations and a criterion for rigidity 5. Restriction to the commutative theory 6. Deformations of power series rings

Algebraic Cohomology and Deformation Theory

We should state at the outset that the present article, intended primarily as a survey, contains many results which are new and have not appeared elsewhere. Foremost among these is the reduction, in §28, of the formal deformation theory of a smooth compact complex algebraic variety χ to that of a single ring built from χ. Others include the relationship between the classical Hodge decomposition of the cohomology of an analytic manifold and the more recent Hodge decomposition of the cohomology of a commutative algebra, the invariance of the Euler characteristic of an algebra under deformation, the correspondence between the deformation theories for Morita equivalent algebras, much of the work on the deformation of presheaves (diagrams) of algebras, and the explicit description of the (algebraic) Hodge decomposition for regular affine algebras. However, in line with the goals of a survey article, we have tried to maximize the exposition, including details only in so far as they aid in this purpose. Many proofs are sketched; many others, including the most difficult, are omitted altogether.

Homotopy G -algebras and moduli space operad

This paper emphasizes the ubiquitous role of mod- uli spaces of algebraic curves in associative algebra and algebraic topology. The main results are: (1) the space of an operad with multiplication is a homotopy G- (i.e., homotopy graded Poisson) algebra; (2) the singular cochain complex is naturally an operad; (3) the operad of decorated moduli spaces acts naturally on the de Rham complexX of a Kahler manifold X, thereby yielding the most general type of homotopy G-algebra structure onX. This latter statement is based on a typical construction of supersymmet- ric sigma-model, the construction of Gromov-Witten invariants in Kontsevichs version.

A hodge-type decomposition for commutative algebra cohomology

Abstract The Hochschild cohomology of a commutative algebra A of characteristic zero, with coefficients in a symmetric module M, decomposes into a direct sum Hn (A, M) = H1, n−1 + ∣ + Hn,0, where Hi, n−i is the eigenspace for the eigenvalue 2i−2 of the ‘shuffle operator‘ sn. Harrisons cohomology is H1. (=Σi H1, i). Replacing every module in a long exact sequence 0→M→M n ∣→M 1 a A→0 of A-bimodules (with Mi not necessarily symmetric) by its opposite induces an involution, op, on H.. This is an automorphism of the cup product when M = A. The set of fixed elements is the direct sum of Hi. when i even.

Deformation theory of algebras and structures and applications

The philosophy of deformations: introductory remarks and a guide to this volume.- A. Deformations of algebras.- Algebraic cohomology and deformation theory.- Perturbations of Lie algebra structures.- Cohomology of current Lie algebras.- An example of formal deformations of Lie algebras.- On the rigidity of solvable Lie algebras.- Triangular algebras.- B. Perturbations of algebras in functional analysis and operator theory.- Deformation theory for algebras of analytic functions.- Close operator algebras.- Perturbations of function algebras.- Perturbations of multiplication and homomorphisms.- C. Deformations and moduli in geometry and differential equations, and algebras.- Local isoformal deformation theory for meromorphic differential equations near an irregular singularity.- Geometric and Lie-theoretic principles in pure and applied deformation theory.- Complexes of differential operators and symmetric spaces.- Deformation theory of geometric and algebraic structures.- Some rigidity results in the deformation theory of symmetric spaces.- D. Deformations of algebras and mathematical and quantum physics.- Applications of the deformations of the algebraic structures to geometry and mathematical physics.- Formal deformations of the Poisson Lie algebra of a symplectic manifold and star-products. Existence, equivalence, derivations.- Invariant deformations of the Poisson Lie algebra of a symplectic manifold and star-products.- E. Deformations elsewhere.- A remarkable matrix.- Deformation stability of periodic and quasi periodic motion in dissipative systems.- List of participants.

The hidden group structure of quantum groups: strong duality, rigidity and preferred deformations

A notion of well-behaved Hopf algebra is introduced; reflexivity (for strong duality) between Hopf algebras of Drinfeld-type and their duals, algebras of coefficients of compact semi-simple groups, is proved. A hidden classical group structure is clearly indicated for all generic models of quantum groups. Moyal-product-like deformations are naturally found for all FRT-models on coefficients andC∞-functions. Strong rigidity (Hbi2={0}) under deformations in the category of bialgebras is proved and consequences are deduced.

Quantum groups and deformation quantization: Explicit approaches and implicit aspects

Deformation quantization, which gives a development of quantum mechanics independent of the operator algebra formulation, and quantum groups, which arose from the inverse scattering method and a study of Yang–Baxter equations, share a common idea abstracted earlier in algebraic deformation theory: that algebraic objects have infinitesimal deformations which may point in the direction of certain continuous global deformations, i.e., “quantizations.” In deformation quantization the algebraic object is the algebra of “observables” (functions) on symplectic phase space, whose infinitesimal deformation is the Poisson bracket and global deformation a “star product,” in quantum groups it is a Hopf algebra, generally either of functions on a Lie group or (often its dual in the topological vector space sense, as we briefly explain) a completed universal enveloping algebra of a Lie algebra with, for infinitesimal, a matrix satisfying the modified classical Yang–Baxter equation (MCYBE). Frequently existence proofs a...

Cohomology and deformation of Leibniz pairs

Cohomology and deformation theories are developed for Poisson algebras starting with the more general concept of a Leibniz pair, namely of an associative algebraA together with a Lie algebraL mapped into the derivations ofA. A bicomplex (with both Hochschild and Chevalley-Eilenberg cohomologies) is essential.

On the cohomology of an algebra morphism

1. The techniques suggest a spectral sequence argument for a valuable generalization of the theorems (discussed below). 2. In [3] we introduced a cochain map of some importance but were unable to work with it directly. Here that is precisely what we do. 3. Of necessity [3] is rather densely packed. We hope that a detailed discussion of a special case-namely, algebra morphisms-will make it more accessible.

Boundary Solutions of the Quantum Yang–Baxter Equation and Solutions in Three Dimensions

Boundary solutions to the quantum Yang–Baxter (qYB) equation are defined to be those in the boundary of (but not in) the variety of solutions to the ‘modified’ qYB equation, the latter being analogous to the modified classical Yang–Baxter (cYB) equation. We construct, for a large class of solutions r to the modified cYB equation, explicit ‘boundary quantizations’, i.e., boundary solutions to the qYB equation of the form I + tr + t2r2 +⋯. In the last section we list and give quantizations for all classical r-matrices in sl(3) ∧ sl(3).