We discuss two versions of a conjecture attributed to M Barr The Harrison cohomology of a commutative algebra is known to coincide with the Andr e Quillen cohomology over a eld of characteristic zero but not in prime characteristics The conjecture is that a modi ed version of Harrison cohomology taking into account torsion always agrees with Andr e Quillen cohomology We give a counterexample De...

We study the Lie module structure given by the Gerstenhaber bracket on the Hochschild cohomology groups of a monomial algebra with radical square zero. The description of such Lie module structure will be given in terms of the combinatorics of the quiver. The Lie module structure will be related to the classification of finite dimensional modules over simple Lie algebras when the quiver is give...

Methods of Harder and Narasimhan from the theory of moduli of vector bundles are applied to moduli of quiver representations. Using the Hall algebra approach to quantum groups, an analog of the Harder-Narasimhan recursion is constructed inside the quantized enveloping algebra of a KacMoody algebra. This leads to a canonical orthogonal system, the HN system, in this algebra. Using a resolution o...

Let g be a finite-dimensional complex reductive semi simple Lie algebra. We present a new calculation of the continuous cohomology of the Lie algebra zg[[z]]. In particular, we shall give an explicit formula for the Laplacian on the Lie algebra cochains, from which we can deduce that the cohomology in each dimension is a finite-dimensional representation of g which contains any irreducible repr...

Let g be a finite-dimensional complex reductive semi simple Lie algebra. We present a new calculation of the continuous cohomology of the Lie algebra zg[[z]]. In particular, we shall give an explicit formula for the Laplacian on the Lie algebra cochains, from which we can deduce that the cohomology in each dimension is a finite-dimensional representation of g which contains any irreducible repr...

Given a complex projective algebraic variety, write H•(X,C) for its cohomology with complex coefficients and IH •(X,C) for its Intersection cohomology. We first show that under some fairly general conditions the canonical map H•(X,C) → IH •(X,C) is injective. Now let Gr := G((z))/G[[z]] be the loop Grassmannian for a complex semisimple group G, and let X be the closure of a G[[z]]-orbit in Gr. ...

Recently, relative Rota–Baxter (Lie/associative) algebras are extensively studied in the literature from cohomological points of view. In this paper, we consider Leibniz (rRB algebras) as object our study. We construct an L∞-algebra that characterizes rRB its Maurer–Cartan elements. Then define representations algebra and introduce cohomology with coefficients a representation. As applications ...

We consider the following situation: a field k, a commutative k-algebra R and a left R-module M. Since R is commutative, M may also be considered as an R-R bimodule with the same operation on each side (such modules are often termed symmetric). With these assumptions we have the Harrison (co-) homology groups Harr,(R, M) (Harr*(R, M)), the Hochschild (co-) homology groups Hoch, (R, M) (Hoch*(R,...

Given a complex projective algebraic variety, write H•(X,C) for its cohomology with complex coefficients and IH •(X,C) for its Intersection cohomology. We first show that under some fairly general conditions the canonical map H•(X,C) → IH •(X,C) is injective. Now let Gr := G((z))/G[[z]] be the loop Grassmannian for a complex semisimple group G, and let X be the closure of a G[[z]]-orbit in Gr. ...