Finite-dimensional associative algebra

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

$$ \alpha ( ab) = \ ( \alpha a) b = \ a ( \alpha b) $$

for all $ a, b \in A $, $ \alpha \in F $. The dimension $ n = \mathop{\rm dim} _ {F} A $ of the space $ A $ over $ F $ is called the dimension of the algebra $ A $ over $ F $. It is also customary to say that the algebra $ A $ is $ n $- dimensional. Every $ n $- dimensional associative algebra $ A $ over a field $ F $ has a faithful representation by matrices of order $ n + 1 $ over $ F $, that is, there is an isomorphism of the algebra $ A $ onto a subalgebra of the algebra of all square $ ( n + 1) $- matrices over $ F $. If $ A $ has an identity, then it has a faithful representation by matrices of order $ n $ over $ F $.

Let $ e _ {1} \dots e _ {n} $ be a basis of the vector space $ A $ over $ F $( it is also called a basis of the algebra $ A $), and suppose that

$$ e _ {i} e _ {j} = \ \sum _ {k = 1 } ^ { n } \gamma _ {ij} ^ {k} e _ {k} ,\ \ \gamma _ {ij} ^ {k} \in F. $$

The elements $ \gamma _ {ij} ^ {k} $ of $ F $ are called the structure constants of the algebra $ A $ in the given basis. They form a tensor of rank three in the space $ A $.

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 $ F $ is algebraically closed, then a semi-simple finite-dimensional associative algebra splits into a direct sum of full matrix algebras over $ F $. 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) MR1541390 Zbl 1032.00002 Zbl 1032.00001 Zbl 0903.01009 Zbl 0781.12003 Zbl 0781.12002 Zbl 0724.12002 Zbl 0724.12001 Zbl 0569.01001 Zbl 0534.01001 Zbl 0997.00502 Zbl 0997.00501 Zbl 0316.22001 Zbl 0297.01014 Zbl 0221.12001 Zbl 0192.33002 Zbl 0137.25403 Zbl 0136.24505 Zbl 0087.25903 Zbl 0192.33001 Zbl 0067.00502
[2]	A.A. Albert, "Structure of algebras" , Amer. Math. Soc. (1939) MR0000595 Zbl 0023.19901 Zbl 65.0094.02

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) MR0890331 MR0674652 Zbl 0671.16001 Zbl 0497.16001
[a2]	C.M. Ringel, "Tame algebras and integral quadratic forms" , Lect. notes in math. , 1099 , Springer (1984) MR0774589 Zbl 0546.16013