Difference between revisions of "Structure constant"
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− | ''of an algebra | + | ''of an algebra $A$ over a field or over a commutative associative ring $P$'' |
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+ | 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); | (commutativity); | ||
− | + | $$ | |
− | + | \sum_\xi c_{\alpha\beta}^\xi c_{\xi\lambda}^\gamma = \sum_\sigma c_{\alpha\sigma}^\lambda c_{\beta\gamma}^\sigma | |
− | + | $$ | |
(associativity); | (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). | (Jacobi's identity). | ||
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====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P.M. Cohn, "Algebra" , '''2''' , Wiley (1989) pp. 167ff</TD></TR></table> | + | <table> |
+ | <TR><TD valign="top">[a1]</TD> <TD valign="top"> P.M. Cohn, "Algebra" , '''2''' , Wiley (1989) pp. 167ff</TD></TR> | ||
+ | </table> | ||
+ | |||
+ | {{TEX|done}} |
Revision as of 17:15, 31 May 2016
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
[a1] | P.M. Cohn, "Algebra" , 2 , Wiley (1989) pp. 167ff |
Structure constant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Structure_constant&oldid=12496