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Register a new userComparison morphisms between two projective resolutions of monomial algebras
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We construct comparison morphisms between two well-known projective resolutions of a monomial algebra $A$: the bar resolution and Bardzells resolution; the first one is used to define the cup product and the Lie bracket on the Hochschild cohomology $HH^*(A)$ and the second one has been shown to be an efficient tool for computations of these cohomology groups. The constructed comparison morphisms allow us to show that the cup product restricted to even degrees of the Hochschild cohomology has a very simple description. Moreover, for $A=k Q/I$ a monomial algebra such that $dim_k e_i A e_j = 1$ whenever there exists an arrow $alpha: i to j in Q_1$, we describe the Lie action of the Lie algebra $HH^1(A)$ on $HH^{ast}(A)$.
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An explicit combinatorial minimal free resolution of an arbitrary monomial ideal $I$ in a polynomial ring in $n$ variables over a field of characteristic $0$ is defined canonically, without any choices, using higher-dimensional generalizations of combined spanning trees for cycles and cocycles (hedges) in the upper Koszul simplicial complexes of $I$ at lattice points in $mathbb{Z}^n$. The differentials in these sylvan resolutions are expressed as matrices whose entries are sums over lattice paths of weights determined combinatorially by sequences of hedges (hedgerows) along each lattice path. This combinatorics enters via an explicit matroidal expression for the Moore-Penrose pseudoinverses of the differentials in any CW complex as weighted averages of splittings defined by hedges. This Hedge Formula also yields a projection formula from CW chains to boundaries. The translation from Moore-Penrose combinatorics to free resolutions relies on Wall complexes, which construct minimal free resolutions of graded ideals from vertical splittings of Koszul bicomplexes. The algebra of Wall complexes applied to individual hedgerows yields explicit but noncanonical combinatorial minimal free resolutions of arbitrary monomial ideals in any characteristic.
Let $mathbf{k}$ be a field of arbitrary characteristic, let $Lambda$ be a finite dimensional $mathbf{k}$-algebra, and let $V$ be an indecomposable Gorenstein-projective $Lambda$-module with finite dimension over $mathbf{k}$. It follows that $V$ has a well-defined versal deformation ring $R(Lambda, V)$, which is complete local commutative Noetherian $mathbf{k}$-algebra with residue field $mathbf{k}$, and which is universal provided that the stable endomorphism ring of $V$ is isomorphic to $mathbf{k}$. We prove that if $Lambda$ is a monomial algebra without overlaps, then $R(Lambda,V)$ is universal and isomorphic either to $mathbf{k}$ or to $mathbf{k}[[t]]/(t^2)$
This paper is devoted to the description of complex finite-dimensional algebras of level two. We obtain the classification of algebras of level two in the varieties of Jordan, Lie and associative algebras.
We compare the restriction to the context of weak Hopf algebras of the notion of crossed product with a Hopf algebroid introduced in cite{BB} with the notion of crossed product with a weak Hopf algebra introduced in~cite{AG}
Recently the first two authors constructed an L-infinity morphism using the S^1-equivariant version of the Poisson Sigma Model (PSM). Its role in deformation quantization was not entirely clear. We give here a good interpretation and show that the resulting formality statement is equivalent to formality on cyclic chains as conjectured by Tsygan and proved recently by several authors.
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