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Difference between revisions of "Structure constant"

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''of an algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090680/s0906801.png" /> over a field or over a commutative associative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090680/s0906802.png" />''
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''of an algebra $A$ over a field or over a commutative associative ring $P$''
 
 
An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090680/s0906803.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090680/s0906804.png" />, defined by the equality
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090680/s0906805.png" /></td> </tr></table>
 
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090680/s0906806.png" /> is a fixed base of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090680/s0906807.png" />. The structure constants determine the algebra uniquely. If the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090680/s0906808.png" /> are the structure constants of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090680/s0906809.png" /> in another base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090680/s09068010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090680/s09068011.png" />, then
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090680/s09068012.png" /></td> </tr></table>
 
 
 
Every identity that is true in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090680/s09068013.png" /> can be expressed by relations between structure constants. For example,
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090680/s09068014.png" /></td> </tr></table>
 
  
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An element $c_{\alpha\beta}^\gamma \in P$, $\alpha, \beta, \gamma \in I$, defined by the equality
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$$
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e_\alpha e_\beta = \sum_\gamma c_{\alpha\beta}^\gamma e_\gamma
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$$
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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
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$$
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\sum_\xi d_{\xi\eta}^\zeta t_\xi^\gamma = \sum_{\alpha,\beta} t_\xi^\alpha t_\eta^\beta c_{\alpha\beta}^\gamma \ .
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$$
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Every identity that is true in $A$ can be expressed by relations between structure constants. For example,
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$$
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c_{\alpha\beta}^\gamma = c_{\beta\alpha}^\gamma
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$$
 
(commutativity);
 
(commutativity);
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090680/s09068015.png" /></td> </tr></table>
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\sum_\xi c_{\alpha\beta}^\xi c_{\xi\lambda}^\gamma = \sum_\sigma c_{\alpha\sigma}^\lambda c_{\beta\gamma}^\sigma
 
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$$
 
(associativity);
 
(associativity);
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090680/s09068016.png" /></td> </tr></table>
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\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)
 
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$$
 
(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>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Cohn,  "Algebra" , '''2''' , Wiley  (1989)  pp. 167ff</TD></TR>
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</table>
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 +
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Revision as of 19:15, 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
How to Cite This Entry:
Structure constant. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Structure_constant&oldid=38895
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article