MasterPVA and WAlg: Mathematica packages for Poisson vertex algebras and classical affine $${\mathcal {W}}$$ W -algebras

Abstract

We give an introduction to the Mathematica packages MasterPVA and MasterPVAmulti used to compute λ-brackets in Poisson vertex algebras, which play an important role in the theory of infinite-dimensional Hamiltonian systems. As an application, we give an introduction to the Mathematica package WAlg aimed to compute the λ-brackets among the generators of classical affine W-algebras. The use of these packages is shown by providing some explicit examples. arXiv:1603.05028v2 [math-ph]

Computing with Hamiltonian operators

“…6. In the framework of the algebraic approach to the Hamiltonian formalism for PDEs [6] the Mathematica packages MasterPVA and WAlg [17] provide procedures for computing the Schouten bracket between local differential operators. The recent paper [16] shows that the algebraic approach to Schouten brackets leads to the same computations and results as more traditional approaches, also for nonlocal operators.…”

Higher-Order Dispersive Deformations of Multidimensional Poisson Brackets of Hydrodynamic Type

“…The resulting set of equation has then been analysed using the Maple package Janet [25]. We used Mathematica computer algebra system with the package MasterPVAmulti [8], developed to perform computations with the λ bracket formalism, for the other two cases.…”

Weakly nonlocal Poisson brackets: tools, examples, computations

“…In general computation of the Schouten bracket is a highly nontrivial computational task and most of the examples can be handled only with the help of computer algebra systems. Recently in [5,27] two computer algebra packages for calculations of Schouten bracket for local differential operators have been introduced and described. The packages have been developed following two different mathematical approaches to differential operators: the algebraic approach through Poisson Vertex Algebras [5] and the supermanifold approach [27].…”

“…Recently in [5,27] two computer algebra packages for calculations of Schouten bracket for local differential operators have been introduced and described. The packages have been developed following two different mathematical approaches to differential operators: the algebraic approach through Poisson Vertex Algebras [5] and the supermanifold approach [27]. A fairly complete list of packages for local operators can be found in [27].…”

“…• Mathematica implementation. This implementation extends the package MasterPVA.m 3 [5], devoted to the computation of λ-brackets in local PVAs, to the case of weakly non-local PVAs. The new added package nlPVA.m (developed for Mathematica 11) provides the syntax and the command for computing the three terms appearing in the Jacobi identity (40) for weakly non-local λ-brackets.…”