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Publication# Gerstenhaber structure on Hochschild cohomology of the Fomin–Kirillov algebra on 3 generators

2022

Journal paper

Journal paper

Abstract

The goal of this article is to compute the Gerstenhaber bracket of the Hochschild cohomology of the Fomin–Kirillov algebra on three generators over a field of characteristic different from 2 and 3. This is in part based on a general method we introduce to easily compute the Gerstenhaber bracket between elements of HH0(A) and elements of HHn(A) for n∈N0, the method by M. Suárez-Álvarez [J. Pure Appl. Algebra 221, No. 8, 1981–1998 (2017; Zbl 1392.16009)] to calculate the Gerstenhaber bracket between elements of HH1(A) and elements of HHn(A) for any n∈N0, as well as an elementary result that allows to compute the remaining brackets from the previous ones. We also show that the Gerstenhaber bracket of HH∙(A) is not induced by any Batalin–Vilkovisky generator.

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Gerstenhaber algebra

In mathematics and theoretical physics, a Gerstenhaber algebra (sometimes called an antibracket algebra or braid algebra) is an algebraic structure discovered by Murray Gerstenhaber (1963) that combines the structures of a supercommutative ring and a graded Lie superalgebra. It is used in the Batalin–Vilkovisky formalism. It appears also in the generalization of Hamiltonian formalism known as the De Donder–Weyl theory as the algebra of generalized Poisson brackets defined on differential forms.

Poisson algebra

In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz's law; that is, the bracket is also a derivation. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central in the study of quantum groups. Manifolds with a Poisson algebra structure are known as Poisson manifolds, of which the symplectic manifolds and the Poisson–Lie groups are a special case. The algebra is named in honour of Siméon Denis Poisson.

Euler method

In mathematics and computational science, the Euler method (also called the forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit method for numerical integration of ordinary differential equations and is the simplest Runge–Kutta method. The Euler method is named after Leonhard Euler, who first proposed it in his book Institutionum calculi integralis (published 1768–1870).

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