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Authors
Pedro Tamaroff
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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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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.
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.
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.
Let $W$ be a Coxeter group. The goal of the paper is to construct new Hopf algebras that contain Hecke algebras $H_{bf q}(W)$ as (left) coideal subalgebras. Our Hecke-Hopf algebras ${bf H}(W)$ have a number of applications. In particular they provide new solutions of quantum Yang-Baxter equation and lead to a construction of a new family of endo-functors of the category of $H_{bf q}(W)$-modules. Hecke-Hopf algebras for the symmetric group are related to Fomin-Kirillov algebras, for an arbitrary Coxeter group $W$ the Demazure part of ${bf H}(W)$ is being acted upon by generalized braided derivatives which generate the corresponding (generalized) Nichols algebra.
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