Poincaré-Birkhoff-Witt Algebras

  • José Bueso
  • José Gómez-Torrecillas
  • Alain Verschoren
Part of the Mathematical Modelling book series (MMTA, volume 17)


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. (1)

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

  2. (2)

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



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Copyright information

© Springer Science+Business Media Dordrecht 2003

Authors and Affiliations

  • José Bueso
    • José Gómez-Torrecillas
      • 1
    • Alain Verschoren
      • 2
    1. 1.University of GranadaGranadaSpain
    2. 2.University of AntwerpAntwerpBelgium

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