View source for Structure constant

''of an algebra $A$ over a field or over a commutative associative ring $P$''

An element $c_{\alpha\beta}^\gamma \in P$, $\alpha, \beta, \gamma \in I$, defined by the equality
$$
e_\alpha e_\beta = \sum_\gamma c_{\alpha\beta}^\gamma e_\gamma
$$
where $\{ e_\alpha : \alpha \in I \}$  is a fixed base of $A$. The structure constants determine the algebra uniquely. If the $d_{\xi\eta}^\zeta$ are the structure constants of the algebra $A$ in another base $\{ f_\xi : \xi \in I \}$, where $f_\xi = \sum t_\xi^\alpha e_\alpha$, then
$$
\sum_\xi d_{\xi\eta}^\zeta t_\xi^\gamma = \sum_{\alpha,\beta} t_\xi^\alpha t_\eta^\beta c_{\alpha\beta}^\gamma \ .
$$
Every identity that is true in $A$ can be expressed by relations between structure constants. For example,
$$
c_{\alpha\beta}^\gamma = c_{\beta\alpha}^\gamma
$$
(commutativity);
$$
\sum_\xi c_{\alpha\beta}^\xi c_{\xi\lambda}^\gamma = \sum_\sigma c_{\alpha\sigma}^\lambda c_{\beta\gamma}^\sigma
$$
(associativity);
$$
\sum_\xi \left({ c_{\alpha\beta}^\xi c_{\xi\gamma}^\lambda + c_{\beta\gamma}^\xi c_{\xi\alpha}^\lambda + c_{\gamma\alpha}^\xi c_{\xi\beta}^\lambda }\right)
$$
(Jacobi's identity).



====Comments====


====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Cohn,   "Algebra" , '''2''' , Wiley  (1989)  pp. 167ff</TD></TR>
</table>

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