Poincaré-Birkhoff-Witt Algebras

Abstract

A PBW algebra R over a field k may be viewed as an associative algebra generated by finitely many elements x ₁,…, x _n subject to the relations

$$Q = \{ {x_j}{x_i} = {q_{ji}}{x_i}{x_j} + {p_{ji}}\} \;(1 \leqslant i < j \leqslant n)$$

, where each p _ji is a finite k-linear combination of standard terms ${x^\alpha } = x_1^{{\alpha _1}} \cdots x_n^{{\alpha _n}}$, with $\alpha = ({\alpha _1}, \ldots ,{\alpha _n}) \in {\mathbb{N}^n}$, and where each qji is a non-zero scalar in the field k. The algebra is required to satisfy the following two conditions:

• (1) there is an admissible order ≤ on ℕⁿ such that $\exp ({p_{ji}}) \prec {\varepsilon _i} + {\varepsilon _j}$ for every l ≤ i < j ≤ n;

• (2) the standard terms x ^α, with $\alpha \; \in \;{\mathbb{N}^n}$>, form a basis of R as a k-vectorspace.