central simple algebra

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central simple algebra

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central simple algebra (plural central simple algebras)

1. (algebra, ring theory) A finite-dimensional associative algebra over some field K that is a simple algebra and whose centre is exactly K.

   The complex numbers ${\displaystyle \mathbb {C} }$ form a central simple algebra over themselves, but not over the real numbers ${\displaystyle \mathbb {R} }$ (the centre of ${\displaystyle \mathbb {C} }$ is all of ${\displaystyle \mathbb {C} }$, not just ${\displaystyle \mathbb {R} }$). The quaternions ${\displaystyle \mathbb {H} }$ form a 4-dimensional central simple algebra over ${\displaystyle \mathbb {R} }$.

   The concept of central simple algebra over a field K represents a noncommutative analogue to that of extension field over K. In both cases, the object has no nontrivial two-sided ideals and has a distinguished field in its centre, although a central simple algebra need not be commutative and need not have inverses (does not have be a division algebra).

   • 1987, Gregory Karpilovsky, The Algebraic Structure of Crossed Products‎, Elsevier (North-Holland), page 151:

     This crossed product ${\displaystyle E^{\alpha }G}$ was introduced by Noether and played a significant role in the classical theory of central simple algebras.

   • 2007, Falko Lorenz, Algebra: Volume II: Fields with Structure, Algebras and Advanced Topics, Springer, page 151:

     Because of Wedderburn's theorem it is natural to call two central-simple algebras similar if they are isomorphic to matrix algebras over the same division algebra ${\displaystyle D}$.

   • 2014, Jörg Jahnel, Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties, American Mathematical Society, page 84:

     Let ${\displaystyle A_{1},A_{2}}$ be central simple algebras over a field ${\displaystyle K}$. Then ${\displaystyle A_{1}\otimes _{K}A_{2}}$ can be shown to be a central simple algebra over ${\displaystyle K}$. Further, if ${\displaystyle A}$ is a central simple algebra over a field ${\displaystyle K}$, then ${\displaystyle A\otimes _{K}A^{\operatorname {op} }\cong \operatorname {Aut} _{K\operatorname {-Vect} }(A)}$. I.e., it is isomorphic to a matrix algebra.

• CSA (initialism)

• simple algebra

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