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7.2.7.2 ring operations (plural)

+
construct a tensor product $C = A\otimes_{\bf {K} } B$of two $G$-algebras $A$ and $B$ over the ground field. Let, e.g.,

$A= k_1 \langle x_1, \ldots ,x_n \mid$ $\{ x_j x_i=c_{ij} \cdot x_i x_j + d_{ij}\}, 1 \leq i <j \leq n \rangle$, and $B= k_2 \langle y_1, \ldots ,y_m \mid$ $\{ y_j y_i=q_{ij} \cdot y_i y_j + r_{ij}\}, 1 \leq i <j \leq m \rangle$

be two $G$-algebras, then $C$ is defined to be the algebra

$C = K \langle x_1, \ldots ,x_n, y_1, \ldots ,y_m \mid$ $\{ x_j x_i=c_{ij} \cdot x_i x_j + d_{ij}, 1 \leq i <j \leq n\}$, $\{ y_j y_i=q_{ij} \cdot y_i y_j + r_{ij}, 1 \leq i <j \leq m\}$, $\{ y_j x_i = x_i y_j, 1 \leq j \leq m, 1 \leq i \leq n\} \rangle$.

Concerning the ground fields $k_1$ resp. $k_2$ of $A$ resp. $B$, take the following guidelines for $A\otimes_{\bf {K} } B$ into consideration:

  • Neither $k_1$ nor $k_2$ may be $R$ or $C$.
  • If the characteristic of $k_1$ and $k_2$ differs, then one of them must be $Q$.
  • At most one of $k_1$ and $k_2$ may have parameters.
  • If one of $k_1$ and $k_2$ is an algebraic extension of $Z/p$ it may not be defined by a charstr of type (p^n,a).
One can create a ring using ring(list), see also ringlist.

Example:
 
LIB "ncalg.lib";
def a = makeUsl2();       // U(sl_2) in e,f,h presentation
ring W0 = 0,(x,d),dp;
def W = Weyl();              // 1st Weyl algebra in x,d
def S = a+W;
setring S;
S;
==> // coefficients: QQ
==> // number of vars : 5
==> //        block   1 : ordering dp
==> //                  : names    e f h
==> //        block   2 : ordering dp
==> //                  : names    x d
==> //        block   3 : ordering C
==> // noncommutative relations:
==> //    fe=ef-h
==> //    he=eh+2e
==> //    hf=fh-2f
==> //    dx=xd+1

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