Finite-dimensional associative algebra

An associative algebra (cf. Associative rings and algebras) that is also a finite-dimensional vector space over a field such that

for all , . The dimension of the space over is called the dimension of the algebra over . It is also customary to say that the algebra is -dimensional. Every -dimensional associative algebra over a field has a faithful representation by matrices of order over , that is, there is an isomorphism of the algebra onto a subalgebra of the algebra of all square -matrices over . If has an identity, then it has a faithful representation by matrices of order over .

Let be a basis of the vector space over (it is also called a basis of the algebra ), and suppose that

The elements of are called the structure constants of the algebra in the given basis. They form a tensor of rank three in the space .

Main theorems concerning finite-dimensional associative algebras.

The Jacobson radical of a finite-dimensional associative algebra is nilpotent and, if the ground field is separable, it splits off as a semi-direct summand (see Wedderburn–Mal'tsev theorem). A semi-simple finite-dimensional associative algebra over a field splits into a direct sum of matrix algebras over skew-fields. If the ground field is algebraically closed, then a semi-simple finite-dimensional associative algebra splits into a direct sum of full matrix algebras over . The simple finite-dimensional algebras are just the full matrix algebras over skew-fields (Wedderburn's theorem). In particular, a finite-dimensional associative algebra without zero divisors is a skew-field. The following are the only finite-dimensional associative algebras with division (that is, skew-fields) over the real field: the real field, the complex field and the skew-field of quaternions (Frobenius' theorem).

Many of the structural properties of finite-dimensional associative algebras mentioned here also hold in the larger classes of Noetherian and Artinian rings (see, e.g., Wedderburn–Artin theorem).

References

[1]	B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German)
[2]	A.A. Albert, "Structure of algebras" , Amer. Math. Soc. (1939)

Comments

Skew-fields are also known as division algebras, cf. Division algebra.

The representation theory of finite-dimensional (associative) algebras is a very active branch of mathematics nowadays (1988). Cf., e.g., [a1]–[a2] and Quiver and Representation of an associative algebra.

References

[a1]	R. Pierce, "Associative algebras" , Springer (1980)
[a2]	C.M. Ringel, "Tame algebras and integral quadratic forms" , Lect. notes in math. , 1099 , Springer (1984)