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A.1 G-algebras

Definition

Let $K$ be a field, and let a $K$-algebra $A$ be given in terms of generators and relations: $A= K \langle x_1, \ldots ,x_n \mid$ $\{ x_j x_i=c_{ij} \cdot x_i x_j + d_{ij}\}, 1 \leq i <j \leq n \rangle$, where $c_{ij} \in K^{*}, d_{ij} \in K[x_1, \ldots, x_n]$. $A$ is called a $G$-algebra, if the following conditions hold:

  • there is a monomial well-ordering $<_A$ such that $\forall i<j \; \hbox{lm}(d_{ij})<_A x_i x_j$,

  • non-degeneracy conditions: $\forall \; 1 \leq i<j<k \leq n \; \; : \; {\cal NDC}_{ijk} =0$, where

    \begin{displaymath}
{\cal NDC}_{ijk} = c_{ik}c_{jk} \cdot d_{ij}x_k - x_k d_{ij}...
... \cdot d_{ik}x_j + d_{jk}x_i
- c_{ij}c_{ik} \cdot x_i d_{jk}.
\end{displaymath}

Theorem

Let $A$ be a $G$-algebra. Then

  • $A$ has a PBW (Poincaré-Birkhoff-Witt) basis,

  • $A$ is left and right noetherian,

  • $A$ is an integral domain,

  • $A$ has left and right quotient rings.


In order to set up a $G$-algebra in PLURAL, one has to do the following steps:

  • define a commutative ring $R= K[x_1, \ldots, x_n]$, equipped with a global monomial ordering,

  • define strictly $n\times n$ upper triangular matrices

    1. $C=\{c_{ij}, i<j\}$, with nonzero entries of type number,

    2. $D=\{d_{ij}, i<j\}$, with polynomial entries from $R$.

  • call the initialization function ncalgebra ( ncalgebra) with the data $C$ and $D$.

At present, we do not check automatically whether non-degeneracy conditions hold but provide a procedure ndc from the library nctools.lib instead.


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