HOME

TheInfoList



OR:

In ring theory and related areas of
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
a central simple algebra (CSA) over a field ''K'' is a finite-dimensional associative ''K''-algebra ''A'' that is
simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by John ...
, and for which the center is exactly ''K''. (Note that ''not'' every simple algebra is a central simple algebra over its center: for instance, if ''K'' is a field of characteristic 0, then the Weyl algebra K ,\partial_X/math> is a simple algebra with center ''K'', but is ''not'' a central simple algebra over ''K'' as it has infinite dimension as a ''K''-module.) For example, the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s C form a CSA over themselves, but not over the
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s R (the center of C is all of C, not just R). The
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...
s H form a 4-dimensional CSA over R, and in fact represent the only non-trivial element of the
Brauer group In mathematics, the Brauer group of a field ''K'' is an abelian group whose elements are Morita equivalence classes of central simple algebras over ''K'', with addition given by the tensor product of algebras. It was defined by the algebraist ...
of the reals (see below). 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 class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
es 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 In mathematics, the Brauer group of a field ''K'' is an abelian group whose elements are Morita equivalence classes of central simple algebras over ''K'', with addition given by the tensor product of algebras. It was defined by the algebraist ...
Br(''F'') of the field ''F''.Lorenz (2008) p.159 It is always a torsion group.Lorenz (2008) p.194


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 In algebra, a division ring, also called a skew field (or, occasionally, a sfield), is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicativ ...
''S''. Hence, there is a unique division algebra in each Brauer equivalence class.Lorenz (2008) p.160 * Every
automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphism ...
of a central simple algebra is an inner automorphism (this follows from the Skolem–Noether theorem). * The
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
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.Gille & Szamuely (2006) p.21 The Schur index of a central simple algebra is the degree of the equivalent division algebra:Lorenz (2008) p.163 it depends only on the Brauer class of the algebra.Gille & Szamuely (2006) p.100 * 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,Jacobson (1996) p.60 and the two numbers are composed of the same prime factors.Jacobson (1996) p.61Gille & Szamuely (2006) p.104 * If ''S'' is a simple subalgebra of a central simple algebra ''A'' then dim''F'' ''S'' divides dim''F'' ''A''. * Every 4-dimensional central simple algebra over a field ''F'' is isomorphic to a
quaternion algebra In mathematics, a quaternion algebra over a field (mathematics), field ''F'' is a central simple algebra ''A'' over ''F''See Milies & Sehgal, An introduction to group rings, exercise 17, chapter 2. that has dimension (vector space), dimension 4 ove ...
; 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(D) = \prod_^r p_i^ \ :then ''D'' has a tensor product decomposition ::D = \bigotimes_^r D_i \ :where each component ''D''''i'' is a central division algebra of index p_i^, and the components are uniquely determined up to isomorphism.Gille & Szamuely (2006) p.105


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''.Jacobson (1996) pp.27-28Gille & Szamuely (2006) p.101 As an example, the field C splits the quaternion algebra H over R with : t + x \mathbf + y \mathbf + z \mathbf \leftrightarrow \left(\right) . We can use the existence of the splitting field to define reduced norm and reduced trace for a CSA ''A''.Gille & Szamuely (2006) pp.37-38 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 in non-zero: hence a CSA is a division algebra if and only if the reduced norm is non-zero on the non-zero elements.Gille & Szamuely (2006) p.38


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.


See also

* Azumaya algebra, generalization of CSAs where the base field is replaced by a commutative local ring * Severi–Brauer variety * Posner's theorem


References

* * * *


Further reading

* * {{cite book , last1=Gille , first1=Philippe , last2=Szamuely , first2=Tamás , title=Central simple algebras and Galois cohomology , series=Cambridge Studies in Advanced Mathematics , volume=101 , location=Cambridge , publisher=
Cambridge University Press Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...
, year=2006 , isbn=0-521-86103-9 , zbl=1137.12001 Algebras Ring theory