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Central simple algebra

Finite dimensional algebra over a field whose central elements are that field

Finite dimensional algebra over a field whose central elements are that field

In ring theory and related areas of mathematics a central simple algebra (CSA) over a field K is a finite-dimensional associative K-algebra A that is simple, and for which the center is exactly K.

For example, the complex numbers C form a CSA over themselves, but not over the real numbers R (the center of C is all of C, not just R). The quaternions H form a 4-dimensional CSA over R, and in fact represent the only non-trivial element of the Brauer group of the reals (see below). Finite-dimensionality is essential to the definition: for instance, for a field F of characteristic 0, the Weyl algebra F[X,\partial_X] is a simple algebra with center F, but is not a central simple algebra over F as it has infinite dimension as a F-module.

By the Artin–Wedderburn theorem, a finite-dimensional simple algebra A is isomorphic to the matrix algebra M(n,S) for some division ring S. Given two central simple algebras A ~ M(n,S) and B ~ M(m,T) over the same field F, A and B are called similar (or Brauer equivalent) if their division rings S and T are isomorphic. The set of all equivalence classes of central simple algebras over a given field F, under this equivalence relation, can be equipped with a group operation given by the tensor product of algebras. The resulting group is called the Brauer group Br(F) of the field F. It is always a torsion group.

Properties

  • According to the Artin–Wedderburn theorem a finite-dimensional simple algebra A is isomorphic to the matrix algebra M(n,S) for some division ring S. Hence, there is a unique division algebra in each Brauer equivalence class.
  • Every automorphism of a central simple algebra is an inner automorphism (this follows from the Skolem–Noether theorem).
  • The dimension of a central simple algebra as a vector space over its centre is always a square: the degree is the square root of this dimension. The Schur index of a central simple algebra is the degree of the equivalent division algebra: it depends only on the Brauer class of the algebra.
  • The period or exponent of a central simple algebra is the order of its Brauer class as an element of the Brauer group. It is a divisor of the index, and the two numbers are composed of the same prime factors.
  • If S is a simple subalgebra of a central simple algebra A then dimF S divides dimF A.
  • Every 4-dimensional central simple algebra over a field F is isomorphic to a quaternion algebra; in fact, it is either a two-by-two matrix algebra, or a division algebra.
  • If D is a central division algebra over K for which the index has prime factorisation ::\mathrm{ind}(D) = \prod_{i=1}^r p_i^{m_i} \ :then D has a tensor product decomposition
:where each component *D**i* is a central division algebra of index p_i^{m_i}, and the components are uniquely determined up to isomorphism. ## Splitting field We call a field *E* a *splitting field* for *A* over *K* if *A*⊗*E* is isomorphic to a matrix ring over *E*. Every finite dimensional CSA has a splitting field: indeed, in the case when *A* is a division algebra, then a maximal subfield of *A* is a splitting field. In general by theorems of Wedderburn and Koethe there is a splitting field which is a separable extension of *K* of degree equal to the index of *A*, and this splitting field is isomorphic to a subfield of *A*. As an example, the field **C** splits the quaternion algebra **H** over **R** with : t + x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \leftrightarrow \left({\begin{array}{*{20}c} t + x i & y + z i \\ -y + z i & t - x i \end{array}}\right) . We can use the existence of the splitting field to define **reduced norm** and **reduced trace** for a CSA *A*. Map *A* to a matrix ring over a splitting field and define the reduced norm and trace to be the composite of this map with determinant and trace respectively. For example, in the quaternion algebra **H**, the splitting above shows that the element *t* + *x* **i** + *y* **j** + *z* **k** has reduced norm *t*2 + *x*2 + *y*2 + *z*2 and reduced trace 2*t*. The reduced norm is multiplicative and the reduced trace is additive. An element *a* of *A* is invertible if and only if its reduced norm is non-zero: hence a CSA is a division algebra if and only if the reduced norm is non-zero on the non-zero elements. ## Generalization CSAs over a field *K* are a non-commutative analog to extension fields over *K* – in both cases, they have no non-trivial 2-sided ideals, and have a distinguished field in their center, though a CSA can be non-commutative and need not have inverses (need not be a division algebra). This is of particular interest in noncommutative number theory as generalizations of number fields (extensions of the rationals **Q**); see noncommutative number field. ## References - - - - ## References 1. Lorenz (2008) p.159 2. Lorenz (2008) p.194 3. Lorenz (2008) p.160 4. Gille & Szamuely (2006) p.21 5. Lorenz (2008) p.163 6. Gille & Szamuely (2006) p.100 7. Jacobson (1996) p.60 8. Jacobson (1996) p.61 9. Gille & Szamuely (2006) p.104 10. Cohn, Paul M.. (2003). ["Further Algebra and Applications"](https://books.google.com/books?id=2Z_OC6uGzkwC&q=%22central+simple%22). *[[Springer-Verlag]]*. 11. Gille & Szamuely (2006) p.105 12. Jacobson (1996) pp.27-28 13. Gille & Szamuely (2006) p.101 14. Gille & Szamuely (2006) pp.37-38 15. Gille & Szamuely (2006) p.38 ::callout[type=info title="Wikipedia Source"] This article was imported from [Wikipedia](https://en.wikipedia.org/wiki/Central_simple_algebra) and is available under the [Creative Commons Attribution-ShareAlike 4.0 License](https://creativecommons.org/licenses/by-sa/4.0/). Content has been adapted to SurfDoc format. Original contributors can be found on the [article history page](https://en.wikipedia.org/wiki/Central_simple_algebra?action=history). ::
algebras ring-theory
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