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Poincaré-Birkhoff-Witt Algebras

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

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. (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.

     

Keywords

Quantum Relation Reduction System Weyl Algebra Group Order Irreducible Element 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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