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7.4.1 G-algebras

Definition (PBW basis)

Let $K$ be a field, and let a $K$-algebra $A$ be generated by variables $x_1, \ldots ,x_n$ subject to some relations. We call $A$ an algebra with PBW basis (Poincaré-Birkhoff-Witt basis), if a $K$-basis of $A$ is Mon $(x_1,\dots,x_n)=\{x^{a_1}_1 x^{a_2}_2 \dots x^{a_n}_n \mid a_i \in N \cup \{0\} \}$, where a power-product $x^{a_1}_1 x^{a_2}_2 \dots x^{a_n}_n$ (in this particular order) is called a monomial. For example, $x_1 x_2$ is a monomial, while $x_2 x_1$ is, in general, not a monomial.

Definition (G-algebra)

Let $K$ be a field, and let a $K$-algebra $A$ be given in terms of generators subject to the following 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 $<$ on $K[x_1,x_2,\ldots,x_n]$ such that $\forall i<j \;\; \hbox{LM}(d_{ij})< x_i x_j$,

  • 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}

Note: Note that non-degeneracy conditions ensure associativity of multiplication, defined by the relations. It is also proved, that they are necessary and sufficient to guarantee the PBW property of an algebra, defined via C_ij and D_ij as above.

Theorem (properties of G-algebras)

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.

Setting up a G-algebra

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

  • - define a commutative ring $R= K[x_1, \ldots, x_n]$, equipped with a monomial ordering $<$ (see ring declarations (plural)).
    This provides us with the information on a field $K$ (together with its parameters), variables $\{x_i\}$and an ordering <.
    From the sequence of variables we will build a G-algebra with the Poincaré-Birkhoff-Witt (PBW) basis $\{x^{a_1}_1 x^{a_2}_2 \dots x^{a_n}_n\}$.

  • - define strictly $n\times n$ upper triangular matrices (of type matrix)

    1. $C=\{c_{ij}, i<j\}$, with nonzero entries $c_{ij}$ of type number ($c_{ij}$ for $i\geq j$ will be ignored).

    2. $D=\{d_{ij}, i<j\}$, with polynomial entries $d_{ij}$ from $R$ ($d_{ij}$ for $i\geq j$ will be ignored).

  • - Call the initialization function nc_algebra(C,D) (see nc_algebra) with the data $C$ and $D$.

PLURAL does not check automatically whether the non-degeneracy conditions hold but it provides a procedure ndcond from the library nctools_lib to check this.

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