Namespaces
Variants
Actions

Difference between revisions of "Division algebra"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (MR/ZBL numbers added)
m (fixing spaces)
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
An algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d0336801.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d0336802.png" /> such that for any elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d0336803.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d0336804.png" /> the equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d0336805.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d0336806.png" /> are solvable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d0336807.png" />. An associative division algebra, considered as a ring, is a skew-field, its centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d0336808.png" /> is a field, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d0336809.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d03368010.png" />, the division algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d03368011.png" /> is called a central division algebra. Finite-dimensional central associative division algebras over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d03368012.png" /> may be identified, up to an isomorphism, with the elements of the [[Brauer group|Brauer group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d03368013.png" /> of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d03368014.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d03368015.png" /> denote the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d03368016.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d03368017.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d03368018.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d03368019.png" /> is the maximal subfield in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d03368020.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d03368021.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d03368022.png" />. According to the [[Frobenius theorem|Frobenius theorem]], all associative finite-dimensional division algebras over the field of real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d03368023.png" /> are exhausted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d03368024.png" /> itself, the field of complex numbers, and the [[Quaternion|quaternion]] algebra. For this reason the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d03368025.png" /> is cyclic of order two. If the associativity requirement is dropped, there is yet another example of a division algebra over the field of real numbers: the [[Cayley–Dickson algebra|Cayley–Dickson algebra]]. This algebra is alternative, and its dimension over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d03368026.png" /> is 8. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d03368027.png" /> is a finite-dimensional (not necessarily associative) division algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d03368028.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d03368029.png" /> has one of the values 1, 2, 4, or 8.
+
<!--
 +
d0336801.png
 +
$#A+1 = 44 n = 0
 +
$#C+1 = 44 : ~/encyclopedia/old_files/data/D033/D.0303680 Division algebra
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
 +
An algebra  $  A $
 +
over a field $  F $
 +
such that for any elements $  a \neq 0 $
 +
and $  b $
 +
the equations $  ax = b $,  
 +
$  ya = b $
 +
are solvable in $  A $.  
 +
An associative division algebra, considered as a ring, is a skew-field, its centre $  C $
 +
is a field, and $  C \supseteq F $.  
 +
If $  C = F $,  
 +
the division algebra $  A $
 +
is called a central division algebra. Finite-dimensional central associative division algebras over $  F $
 +
may be identified, up to an isomorphism, with the elements of the [[Brauer group|Brauer group]] $  B( F  ) $
 +
of the field $  F $.  
 +
Let $  [ A: F ] $
 +
denote the dimension of $  A $
 +
over $  F $.  
 +
If $  A \in B( F  ) $
 +
and if $  L $
 +
is the maximal subfield in $  A $ ($  L \supseteq F $),  
 +
then $  [ A:  F ] = {[ L: F ] }  ^ {2} $.  
 +
According to the [[Frobenius theorem|Frobenius theorem]], all associative finite-dimensional division algebras over the field of real numbers $  \mathbf R $
 +
are exhausted by $  \mathbf R $
 +
itself, the field of complex numbers, and the [[Quaternion|quaternion]] algebra. For this reason the group $  B( \mathbf R ) $
 +
is cyclic of order two. If the associativity requirement is dropped, there is yet another example of a division algebra over the field of real numbers: the [[Cayley–Dickson algebra|Cayley–Dickson algebra]]. This algebra is alternative, and its dimension over $  \mathbf R $
 +
is 8. If $  A $
 +
is a finite-dimensional (not necessarily associative) division algebra over $  \mathbf R $,  
 +
then $  [ A: \mathbf R ] $
 +
has one of the values 1, 2, 4, or 8.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian) {{MR|0158000}} {{ZBL|0121.25901}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.A. Albert, "Structure of algebras" , Amer. Math. Soc. (1939) {{MR|0000595}} {{ZBL|0023.19901}} {{ZBL|65.0094.02}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> I.N. Herstein, "Noncommutative rings" , Math. Assoc. Amer. (1968) {{MR|1535024}} {{MR|0227205}} {{ZBL|0177.05801}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> J.F. Adams, "On the non-existence of elements of Hopf invariant one" ''Ann. of Math.'' , '''72''' : 1 (1960) pp. 20–104 {{MR|0141119}} {{ZBL|0096.17404}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian) {{MR|0158000}} {{ZBL|0121.25901}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.A. Albert, "Structure of algebras" , Amer. Math. Soc. (1939) {{MR|0000595}} {{ZBL|0023.19901}} {{ZBL|65.0094.02}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> I.N. Herstein, "Noncommutative rings" , Math. Assoc. Amer. (1968) {{MR|1535024}} {{MR|0227205}} {{ZBL|0177.05801}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> J.F. Adams, "On the non-existence of elements of Hopf invariant one" ''Ann. of Math.'' , '''72''' : 1 (1960) pp. 20–104 {{MR|0141119}} {{ZBL|0096.17404}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
Over a finite field every finite-dimensional central division algebra is automatically commutative. For infinite-dimensional division algebras the situation is quite different, because a result of Mokar–Limonov states that such an algebra contains a free algebra in two variables.
 
Over a finite field every finite-dimensional central division algebra is automatically commutative. For infinite-dimensional division algebras the situation is quite different, because a result of Mokar–Limonov states that such an algebra contains a free algebra in two variables.
  
If a finite-dimensional central division algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d03368030.png" /> contains a maximal commutative subfield <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d03368031.png" /> which is a [[Galois extension|Galois extension]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d03368032.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d03368033.png" /> is a [[Cross product|cross product]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d03368034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d03368035.png" /> in the sense that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d03368036.png" /> is the free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d03368037.png" />-module generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d03368038.png" /> with product determined by:
+
If a finite-dimensional central division algebra $  D $
 +
contains a maximal commutative subfield $  L $
 +
which is a [[Galois extension|Galois extension]] of $  F $,  
 +
then $  D $
 +
is a [[Cross product|cross product]] of $  L $
 +
and $  G = \mathop{\rm Gal} ( L/ F  ) $
 +
in the sense that $  D $
 +
is the free $  L $-module generated by $  \{ {u _  \sigma  } : {\sigma \in G } \} $
 +
with product determined by:
  
<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/d/d033/d033680/d03368039.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
$$ \tag{a1 }
 +
\left .
 +
\begin{array}{ll}
 +
u _  \sigma  u _  \tau  = c ( \sigma , \tau ) u _ {\sigma \tau }  &\textrm{ for  some  }  c ( \sigma , \tau ) \in L  ^ {*} ,  \\
 +
u _  \sigma  \lambda  = \lambda  ^  \sigma  u _  \sigma  &\textrm{ for }  \lambda \in L ,\  \tau \in G . \\
 +
\end{array}
 +
\right \}
 +
$$
  
Associativity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d03368040.png" /> entails that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d03368041.png" /> represents an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d03368042.png" /> (the second [[Galois cohomology|Galois cohomology]] group). One of the basic problems in algebra was formulated by A. Albert (1931): Is every finite-dimensional central division algebra necessarily a cross product? In 1972, S. Amitsur provided a counter-example using properties of generic division algebras resulting from the theory of PI-algebras (see [[PI-algebra|PI-algebra]], [[#References|[a2]]]). Other examples of division algebras were obtain by F. van Ostaeyen (1972 Thesis, cf. [[#References|[a3]]]), i.e. generic cross products, a notion generalized by Amitsur and D. Saltman (1978), describing all cross product division algebras for a given group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d03368043.png" /> over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033680/d03368044.png" /> as reductions of a generic division algebra.
+
Associativity of $  D $
 +
entails that $  c : G \times G \rightarrow L  ^ {*} $
 +
represents an element of $  H  ^ {2} ( G , L  ^ {*} ) $ (the second [[Galois cohomology|Galois cohomology]] group). One of the basic problems in algebra was formulated by A. Albert (1931): Is every finite-dimensional central division algebra necessarily a cross product? In 1972, S. Amitsur provided a counter-example using properties of generic division algebras resulting from the theory of PI-algebras (see [[PI-algebra|PI-algebra]], [[#References|[a2]]]). Other examples of division algebras were obtain by F. van Ostaeyen (1972 Thesis, cf. [[#References|[a3]]]), i.e. generic cross products, a notion generalized by Amitsur and D. Saltman (1978), describing all cross product division algebras for a given group $  G $
 +
over the field $  F $
 +
as reductions of a generic division algebra.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A.H. Schofield, "Representations of rings over skew fields" , London Math. Soc. (1986) {{MR|0800853}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> N. Jacobson, "PI algebras. An introduction" , Springer (1975) {{MR|0369421}} {{ZBL|0326.16013}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> F. van Oystaeyen, "Prime spectra in non-commutative algebra" , Springer (1975) {{MR|}} {{ZBL|0302.16001}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A.H. Schofield, "Representations of rings over skew fields" , London Math. Soc. (1986) {{MR|0800853}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> N. Jacobson, "PI algebras. An introduction" , Springer (1975) {{MR|0369421}} {{ZBL|0326.16013}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> F. van Oystaeyen, "Prime spectra in non-commutative algebra" , Springer (1975) {{MR|}} {{ZBL|0302.16001}} </TD></TR></table>

Latest revision as of 09:14, 28 June 2022


An algebra $ A $ over a field $ F $ such that for any elements $ a \neq 0 $ and $ b $ the equations $ ax = b $, $ ya = b $ are solvable in $ A $. An associative division algebra, considered as a ring, is a skew-field, its centre $ C $ is a field, and $ C \supseteq F $. If $ C = F $, the division algebra $ A $ is called a central division algebra. Finite-dimensional central associative division algebras over $ F $ may be identified, up to an isomorphism, with the elements of the Brauer group $ B( F ) $ of the field $ F $. Let $ [ A: F ] $ denote the dimension of $ A $ over $ F $. If $ A \in B( F ) $ and if $ L $ is the maximal subfield in $ A $ ($ L \supseteq F $), then $ [ A: F ] = {[ L: F ] } ^ {2} $. According to the Frobenius theorem, all associative finite-dimensional division algebras over the field of real numbers $ \mathbf R $ are exhausted by $ \mathbf R $ itself, the field of complex numbers, and the quaternion algebra. For this reason the group $ B( \mathbf R ) $ is cyclic of order two. If the associativity requirement is dropped, there is yet another example of a division algebra over the field of real numbers: the Cayley–Dickson algebra. This algebra is alternative, and its dimension over $ \mathbf R $ is 8. If $ A $ is a finite-dimensional (not necessarily associative) division algebra over $ \mathbf R $, then $ [ A: \mathbf R ] $ has one of the values 1, 2, 4, or 8.

References

[1] A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian) MR0158000 Zbl 0121.25901
[2] A.A. Albert, "Structure of algebras" , Amer. Math. Soc. (1939) MR0000595 Zbl 0023.19901 Zbl 65.0094.02
[3] I.N. Herstein, "Noncommutative rings" , Math. Assoc. Amer. (1968) MR1535024 MR0227205 Zbl 0177.05801
[4] J.F. Adams, "On the non-existence of elements of Hopf invariant one" Ann. of Math. , 72 : 1 (1960) pp. 20–104 MR0141119 Zbl 0096.17404

Comments

Over a finite field every finite-dimensional central division algebra is automatically commutative. For infinite-dimensional division algebras the situation is quite different, because a result of Mokar–Limonov states that such an algebra contains a free algebra in two variables.

If a finite-dimensional central division algebra $ D $ contains a maximal commutative subfield $ L $ which is a Galois extension of $ F $, then $ D $ is a cross product of $ L $ and $ G = \mathop{\rm Gal} ( L/ F ) $ in the sense that $ D $ is the free $ L $-module generated by $ \{ {u _ \sigma } : {\sigma \in G } \} $ with product determined by:

$$ \tag{a1 } \left . \begin{array}{ll} u _ \sigma u _ \tau = c ( \sigma , \tau ) u _ {\sigma \tau } &\textrm{ for some } c ( \sigma , \tau ) \in L ^ {*} , \\ u _ \sigma \lambda = \lambda ^ \sigma u _ \sigma &\textrm{ for } \lambda \in L ,\ \tau \in G . \\ \end{array} \right \} $$

Associativity of $ D $ entails that $ c : G \times G \rightarrow L ^ {*} $ represents an element of $ H ^ {2} ( G , L ^ {*} ) $ (the second Galois cohomology group). One of the basic problems in algebra was formulated by A. Albert (1931): Is every finite-dimensional central division algebra necessarily a cross product? In 1972, S. Amitsur provided a counter-example using properties of generic division algebras resulting from the theory of PI-algebras (see PI-algebra, [a2]). Other examples of division algebras were obtain by F. van Ostaeyen (1972 Thesis, cf. [a3]), i.e. generic cross products, a notion generalized by Amitsur and D. Saltman (1978), describing all cross product division algebras for a given group $ G $ over the field $ F $ as reductions of a generic division algebra.

References

[a1] A.H. Schofield, "Representations of rings over skew fields" , London Math. Soc. (1986) MR0800853
[a2] N. Jacobson, "PI algebras. An introduction" , Springer (1975) MR0369421 Zbl 0326.16013
[a3] F. van Oystaeyen, "Prime spectra in non-commutative algebra" , Springer (1975) Zbl 0302.16001
How to Cite This Entry:
Division algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Division_algebra&oldid=24066
This article was adapted from an original article by E.N. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article