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A simple associative algebra with a unit element that is a [[Central algebra|central algebra]]. Every finite-dimensional central simple algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021210/c0212101.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021210/c0212102.png" /> is isomorphic to a matrix algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021210/c0212103.png" /> over a finite-dimensional central division algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021210/c0212104.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021210/c0212105.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021210/c0212106.png" /> is algebraically closed, then every finite-dimensional central simple algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021210/c0212107.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021210/c0212108.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021210/c0212109.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021210/c02121010.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021210/c02121011.png" /> is isomorphic to the algebra of real or quaternion matrices. The tensor product of a central simple algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021210/c02121012.png" /> and an arbitrary simple algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021210/c02121013.png" /> is a simple algebra, which is central if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021210/c02121014.png" /> is central. Two finite-dimensional central simple algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021210/c02121015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021210/c02121016.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021210/c02121017.png" /> are called equivalent if
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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/c/c021/c021210/c02121018.png" /></td> </tr></table>
 
 
 
for certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021210/c02121019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021210/c02121020.png" />, or, which is equivalent, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021210/c02121021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021210/c02121022.png" /> are isomorphic matrix algebras over one and the same central division algebra. The equivalence classes of central simple algebras over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021210/c02121023.png" /> form the [[Brauer group|Brauer group]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021210/c02121024.png" /> relative to the operation induced by tensor multiplication.
 
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.L. van der Waerden,  "Algebra" , '''1–2''' , Springer  (1967–1971)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  Yu.A. Drozd,  V.V. Kirichenko,  "Finite-dimensional algebras" , Kiev  (1980)  (In Russian)</TD></TR></table>
 
 
 
 
 
 
 
====Comments====
 
  
 +
A central simple algebra is a simple associative algebra with a unit element that is a
 +
[[Central algebra|central algebra]]. Every finite-dimensional central
 +
simple algebra $A$ over a field $K$ is isomorphic to a matrix algebra
 +
$M_n(C)$ over a finite-dimensional central division algebra $C$ over
 +
$K$. In particular, if $K$ is algebraically closed, then every
 +
finite-dimensional central simple algebra $A$ over $K$ is isomorphic
 +
to $M_n(K)$, and if $K=\R$, then $A$ is isomorphic to the algebra of real or
 +
quaternion matrices. The tensor product of a central simple algebra
 +
$A$ and an arbitrary simple algebra $B$ is a simple algebra, which is
 +
central if $B$ is central. Two finite-dimensional central simple
 +
algebras $A$ and $B$ over $K$ are called equivalent if
 +
$$A\otimes_K M_m(K) \cong B\otimes M_n(K)$$
 +
for
 +
certain $m$ and $n$, or, which is equivalent, if $A$ and $B$ are
 +
isomorphic matrix algebras over one and the same central division
 +
algebra. The equivalence classes of central simple algebras over $K$
 +
form the
 +
[[Brauer group|Brauer group]] of $K$ relative to the operation induced
 +
by tensor multiplication.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R.S. Peirce,   "Associative algebras" , Springer (1980)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"A.A. Albert,   "Structure of algebras" , Amer. Math. Soc.  (1939)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Deuring,   "Algebren" , Springer (1935)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"I.N. Herstein,   "Noncommutative rings" , Math. Assoc. Amer.  (1968)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"N. Jacobson,   "Structure of rings" , Amer. Math. Soc.  (1956)</TD></TR></table>
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{|
 +
|-
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|valign="top"|{{Ref|Al}}||valign="top"| A.A. Albert, "Structure of algebras", Amer. Math. Soc. (1939) {{MR|0000595}}  {{ZBL|0023.19901}}
 +
|-
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|valign="top"|{{Ref|De}}||valign="top"| M. Deuring, "Algebren", Springer (1935)   {{ZBL|0011.19801}}
 +
|-
 +
|valign="top"|{{Ref|DrKi}}||valign="top"| Yu.A. Drozd, V.V. Kirichenko, "Finite-dimensional algebras", Kiev (1980) (In Russian) {{MR|0591671}}  {{ZBL|0469.16001}}
 +
|-
 +
|valign="top"|{{Ref|Pe}}||valign="top"| R.S. Peirce, "Associative algebras", Springer (1980)  
 +
|-
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|valign="top"|{{Ref|He}}||valign="top"| I.N. Herstein, "Noncommutative rings", Math. Assoc. Amer.  (1968) {{MR|0227205}}  {{ZBL|0177.05801}}
 +
|-
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|valign="top"|{{Ref|Ja}}||valign="top"| N. Jacobson, "Structure of rings", Amer. Math. Soc.  (1956) {{MR|0081264}}  {{ZBL|0073.02002}}
 +
|-
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|}

Revision as of 23:48, 29 February 2012

2010 Mathematics Subject Classification: Primary: 16-XX [MSN][ZBL]

A central simple algebra is a simple associative algebra with a unit element that is a central algebra. Every finite-dimensional central simple algebra $A$ over a field $K$ is isomorphic to a matrix algebra $M_n(C)$ over a finite-dimensional central division algebra $C$ over $K$. In particular, if $K$ is algebraically closed, then every finite-dimensional central simple algebra $A$ over $K$ is isomorphic to $M_n(K)$, and if $K=\R$, then $A$ is isomorphic to the algebra of real or quaternion matrices. The tensor product of a central simple algebra $A$ and an arbitrary simple algebra $B$ is a simple algebra, which is central if $B$ is central. Two finite-dimensional central simple algebras $A$ and $B$ over $K$ are called equivalent if $$A\otimes_K M_m(K) \cong B\otimes M_n(K)$$ for certain $m$ and $n$, or, which is equivalent, if $A$ and $B$ are isomorphic matrix algebras over one and the same central division algebra. The equivalence classes of central simple algebras over $K$ form the Brauer group of $K$ relative to the operation induced by tensor multiplication.

References

[Al] A.A. Albert, "Structure of algebras", Amer. Math. Soc. (1939) MR0000595 Zbl 0023.19901
[De] M. Deuring, "Algebren", Springer (1935) Zbl 0011.19801
[DrKi] Yu.A. Drozd, V.V. Kirichenko, "Finite-dimensional algebras", Kiev (1980) (In Russian) MR0591671 Zbl 0469.16001
[Pe] R.S. Peirce, "Associative algebras", Springer (1980)
[He] I.N. Herstein, "Noncommutative rings", Math. Assoc. Amer. (1968) MR0227205 Zbl 0177.05801
[Ja] N. Jacobson, "Structure of rings", Amer. Math. Soc. (1956) MR0081264 Zbl 0073.02002
How to Cite This Entry:
Central simple algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Central_simple_algebra&oldid=13839
This article was adapted from an original article by r group','../u/u095350.htm','Unitary transformation','../u/u095590.htm','Vector bundle, analytic','../v/v096400.htm')" style="background-color:yellow;">A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article