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Register a new userKontsevich formality and PBW algebras
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Authors
Boris Shoikhet
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This paper is based on the authors paper Koszul duality in deformation quantization, I, with some improvements. In particular, an Introduction is added, and the convergence of the spectral sequence in Lemma 2.1 is rigorously proven. Some informal discussion in Section 1.5 is added.
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We develop an elementary method for proving the PBW theorem for associative algebras with an ascending filtration. The idea is roughly the following. At first, we deduce a proof of the PBW property for the {it ascending} filtration (with the filtered degree equal to the total degree in $x_i$s) to a suitable PBW-like property for the {it descending} filtration (with the filtered degree equal to the power of a polynomial parameter $hbar$, introduced to the problem). This PBW property for the descending filtration guarantees the genuine PBW property for the ascending filtration, for almost all specializations of the parameter $hbar$. At second, we develop some very constructive method for proving this PBW-like property for the descending filtration by powers of $hbar$, emphasizing its integrability nature. We show how the method works in three examples. As a first example, we give a proof of the classical Poincar{e}-Birkhoff-Witt theorem for Lie algebras. As a second, much less trivial example, we present a new proof of a result of Etingof and Ginzburg [EG] on PBW property of algebras with a cyclic non-commutative potential in three variables. Finally, as a third example, we found a criterium, for a general quadratic algebra which is the quotient-algebra of $T(V)[hbar]$ by the two-sided ideal, generated by $(x_iotimes x_j-x_jotimes x_i-hbarphi_{ij})_{i,j}$, with $phi_{ij}$ general quadratic non-commutative polynomials, to be a PBW for generic specialization $hbar=a$. This result seems to be new.
Let $n geq 2$ be an integer. An emph{$n$-potent} is an element $e$ of a ring $R$ such that $e^n = e$. In this paper, we study $n$-potents in matrices over $R$ and use them to construct an abelian group $K_0^n(R)$. If $A$ is a complex algebra, there is a group isomorphism $K_0^n(A) cong bigl(K_0(A)bigr)^{n-1}$ for all $n geq 2$. However, for algebras over cyclotomic fields, this is not true in general. We consider $K_0^n$ as a covariant functor, and show that it is also functorial for a generalization of homomorphism called an emph{$n$-homomorphism}.
We discuss a version of the Chevalley--Eilenberg cohomology in characteristic $2$, where the alternating cochains are replaced by symmetric ones.
We consider a class of extensions of both abstract and pseudocompact algebras, which we refer to as strongly proj-bounded extensions. We prove that the finiteness of the left global dimension and the support of the Hochschild homology is preserved by strongly proj-bounded extensions, generalizing results of Cibils, Lanzillota, Marcos and Solotar. Moreover, we show that the finiteness of the big left finitistic dimension is preserved by strongly proj-bounded extensions. In order to construct examples, we describe a new class of extensions of algebras of finite relative global dimension, which may be of independent interest.
Using combinatorics of chains going back to works of Anick, Green, Happel and Zacharia, we give, for any monomial algebra $A$, an explicit description of its minimal model. This also provides us with formulas for a canonical $A_infty$-structure on the Ext-algebra of the trivial $A$-module. We do this by exploiting the combinatorics of chains going back to works of Anick, Green, Happel and Zacharia, and the algebraic discrete Morse theory of Jollenbeck, Welker and Skoldberg. We then show how this result can be used to obtain models for algebras with a chosen Grobner basis, and briefly outline how to compute some classical homological invariants with it.
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