Algebras over a field:
Basic definitions and constructions

Fix a (commutative) field k, which will be our ``base field''. An algebra over k, or more simply a k-algebra, is an associative ring A with unit together with a copy of k in the center of A (whose unit element coincides with that of A). Thus A is a k-vector space and the multiplication map from AxA to A is k-bilinear. Indeed, we could equivalently define a k-algebra as an associative ring A with nonzero unit that has the structure of a k-vector space with bilinear multiplication; the embedding of k into A would then send each field element c to c1=1c where 1 is the unit element of A.

k-algebras are ubiquitous in mathematics, and occur (when k=R or C) even in the mathematical sciences -- you may have encountered the algebra of operators generated by x and d/dx in quantum mechanics. We'll generally be concerned only with algebras that are finite-dimensional as k-vector spaces, and will have to assume some further structures or conditions on the algebras to get reasonable descriptions.

Here are some examples of algebras that will be relevant to our investigation:

• k itself.
• More generally, any (commutative) field K containing k.
• Generalizing in a differrent direction, the ring Mn(k) of square matrices of order n with entries in k. Abstractly, this is End(V) where V is an n-dimensional vector space over k. Note that k itself is just M1(k).
• Generalizing in both directions at once, Mn(K) for any field extension K/k, or even Mn(A) for any k-algebra A (yes, this works!).
• The ``group ring'' (here a group algebra) k[G], for any group G.
• If A is any algebra, so is its ``opposite algebra'' Ao, which is the same as a vector space, but with multiplication defined so that the product xy in Ao is yx.
• The direct sum of any two (or more) k-algebras is again a k-algebra, with unit (1,1) (or (1,1,...,1)).
• If A is any k-algebra, and B is a vector subspace of A that contains 1 and is closed under multiplication, then B is itself a k-algebra (a ``subalgebra'' of A).
• In particular, the subalgebra of A generated by any subset S of A. Note that if S is a singleton (that is, |S|=1), then this subalgebra is commutative.
• If I is any two-sided ideal of a k-algebra A, and I does not contain 1, then A/I inherits the structure of a k-algebra. [Note that an ideal (even a one-sided one) in a k-algebra is automatically a k-vector subspace of A.]
• The Hamilton quaternions constitute a 4-dimensional algebra H over R.

The center of a k-algebra A is a commutative k-subalgebra of A. We say that a k-algebra is central if its center is k. Examples in our above list are k, Mn(k), and H; The k-algebras K and Mn(K) are not central unless K=k. A k-algebra A is said to be simple if its only two-sided ideals are A itself and {0}; equivalently, if its only quotient algebra is A/{0}=A itself. Examples are k, K, or indeed any division algebra (also known as a ``skew field'', ``corps gauche'' in French), such as H; a division algebra A doesn't even have any one-sided ideals other than A and {0}. The matrix algebra Mn(K) (which does have one-sided ideals) is also simple. The direct sum of two or more algebras is not simple (why?). Clearly any algebra A and its opposite Ao have the same center, and A is central/simple/division if and only if Ao is.

A representation of a k-algebra A is a k-vector space E together with an action of A on E by endomorphisms, i.e., a homomorphism from A to Endk(E). [Note that the identity element of A must act as the identity on E.] A ``subrepresentation'' is a subspace E' stable under the action of A. A representation E is irreducible if its only subrepresentations are {0} and E itself. The representation E is said to be faithful if the homomorphism from A to Endk(E) is injective. Since the kernel of a homomorphism is a two-sided ideal in A, every positive-dimensional representation of a simple algebra is automatically faithful.