Central simple algebra
Encyclopedia
In ring theory
and related areas of mathematics
a central simple algebra (CSA) over a field
K is a finite-dimensional associative algebra
A, which is simple, and for which the center is exactly K. In other words, any simple algebra is a central simple algebra over its center.
For example, the complex number
s C form a CSA over themselves, but not over the real number
s R (the center of C is all of C, not just R). The quaternion
s 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).
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.
According to the Artin–Wedderburn theorem
a 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.
Ring theory
In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...
and related areas of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
a central simple algebra (CSA) over a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
K is a finite-dimensional associative algebra
Associative algebra
In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...
A, which is simple, and for which the center is exactly K. In other words, any simple algebra is a central simple algebra over its center.
For example, the complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s C form a CSA over themselves, but not over the real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s R (the center of C is all of C, not just R). The quaternion
Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...
s H form a 4 dimensional CSA over R, and in fact represent the only non-trivial element of the Brauer group
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 of finite rank over K and addition is induced by the tensor product of algebras. It arose out of attempts to classify division algebras over a field and is...
of the reals (see below).
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
Division algebra
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field, in which division is possible.- Definitions :...
). This is of particular interest in noncommutative number theory as generalizations of number fields (extensions of the rationals Q); see noncommutative number field.
According to the Artin–Wedderburn theorem
Artin–Wedderburn theorem
In abstract algebra, the Artin–Wedderburn theorem is a classification theorem for semisimple rings. The theorem states that an Artinian semisimple ring R is isomorphic to a product of finitely many ni-by-ni matrix rings over division rings Di, for some integers ni, both of which are uniquely...
a simple algebra A is isomorphic to the matrix algebra M(n,S)
Matrix ring
In abstract algebra, a matrix ring is any collection of matrices forming a ring under matrix addition and matrix multiplication. The set of n×n matrices with entries from another ring is a matrix ring, as well as some subsets of infinite matrices which form infinite matrix rings...
for some division ring
Division ring
In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a non-trivial ring in which every non-zero element a has a multiplicative inverse, i.e., an element x with...
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
Tensor product of algebras
In mathematics, the tensor product of two R-algebras is also an R-algebra. This gives us a tensor product of algebras. The special case R = Z gives us a tensor product of rings, since rings may be regarded as Z-algebras....
. The resulting group is called the Brauer group
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 of finite rank over K and addition is induced by the tensor product of algebras. It arose out of attempts to classify division algebras over a field and is...
Br(F) of the field F.
Properties
- Every automorphismAutomorphismIn 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 automorphisms of an object forms a group, called the automorphism...
of a central simple algebra is an inner automorphismInner automorphismIn abstract algebra an inner automorphism is a functionwhich, informally, involves a certain operation being applied, then another one performed, and then the initial operation being reversed...
(follows from Skolem-Noether theorem) - The dimension of a central simple algebra as a vector space over its centre is always a square
- If S is a simple subalgebraSubalgebraIn mathematics, the word "algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operation. Algebras in universal algebra are far more general: they are a common generalisation of all algebraic structures...
of a central simple algebra A then dimFS divides dimFA - Every 4 dimensional central simple algebra over a field F is isomorphic to a quaternion algebraQuaternion algebraIn mathematics, a quaternion algebra over a field F is a central simple algebra A over F that has dimension 4 over F. Every quaternion algebra becomes the matrix algebra by extending scalars , i.e...
; in fact, it is either a two-by-two matrix algebraMatrix algebraMatrix algebra may refer to:*Matrix theory, is the branch of mathematics that studies matrices*Matrix ring, thought of as an algebra over a field or a commutative ring...
, or a division algebraDivision algebraIn the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field, in which division is possible.- Definitions :...
See also
- Azumaya algebraAzumaya algebraIn mathematics, an Azumaya algebra is a generalization of central simple algebras to R-algebras where R need not be a field. Such a notion was introduced in a 1951 paper of Goro Azumaya, for the case where R is a commutative local ring...
, generalization of CSAs where the base field is replaced by a commutative local ring - Brauer groupBrauer groupIn mathematics, the Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras of finite rank over K and addition is induced by the tensor product of algebras. It arose out of attempts to classify division algebras over a field and is...
- Severi–Brauer variety