Central simple algebra
A simple associative algebra with a unit element that is a central algebra. Every finite-dimensional central simple algebra over a field
is isomorphic to a matrix algebra
over a finite-dimensional central division algebra
over
. In particular, if
is algebraically closed, then every finite-dimensional central simple algebra
over
is isomorphic to
, and if
, then
is isomorphic to the algebra of real or quaternion matrices. The tensor product of a central simple algebra
and an arbitrary simple algebra
is a simple algebra, which is central if
is central. Two finite-dimensional central simple algebras
and
over
are called equivalent if
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for certain and
, or, which is equivalent, if
and
are isomorphic matrix algebras over one and the same central division algebra. The equivalence classes of central simple algebras over
form the Brauer group of
relative to the operation induced by tensor multiplication.
References
[1] | B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German) |
[2] | Yu.A. Drozd, V.V. Kirichenko, "Finite-dimensional algebras" , Kiev (1980) (In Russian) |
Comments
References
[a1] | R.S. Peirce, "Associative algebras" , Springer (1980) |
[a2] | A.A. Albert, "Structure of algebras" , Amer. Math. Soc. (1939) |
[a3] | M. Deuring, "Algebren" , Springer (1935) |
[a4] | I.N. Herstein, "Noncommutative rings" , Math. Assoc. Amer. (1968) |
[a5] | N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956) |
Central simple algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Central_simple_algebra&oldid=21373