Poincaré-Birkhoff-Witt Deformations of a Class ofN-Homogeneous Algebras

Abstract

In this paper, we mainly focus on the Poincaré–Birkhoff–Witt (PBW) deformation theory for a class of N-homogeneous algebras; here N ≥ 2 is an integer, which generalizes the results in [2 Berger , R. , Ginzburg , V. ( 2006 ). Higher symplectic reflection algebras and non-homogeneous N-Koszul property . J. Algebra 304 : 577 – 601 .[Crossref], [Web of Science ®] , [Google Scholar]] and [7 Fløtystad , G. , Vatne , J. E. ( 2006 ). PBW-deformations of N-Koszul algebras . J. Algebra 302 : 116 – 155 .[Crossref], [Web of Science ®] , [Google Scholar]]. More precisely, let k be a field of characteristic zero, V a finite dimensional vector space over k, and A = T(V)/(R) an N-homogeneous algebra (i.e., R ⊆ V ⊗N ) with being supported in a single degree d such that d > N. Set . Assume that P is a subspace of F N and (P) is a two sided ideal of T(V). Let U = T(V)/(P) be the deformation algebra of A. It is proved that U is the PBW-deformation of A if and only if J n ∩ F n−1 = J n−1 for any N ≤ n ≤ d. And if in particular d ≤ 2N, then U is the PBW-deformation of A if and only if P ∩ F N−1 = 0 and J d ∩ F d−1 = J d−1, where for n ≥ 0 and J n = 0 for n < N.