Algebra over a field

(Formerly referred to as linear algebra) is an algebra over a field, or algebra over - algebra is a vector space over a field, which has been extended to a compatible with the vector space structure of multiplication.

Definition

An algebra over a field or short - algebra is a vector space with a bilinear - linking

Called multiplication, which is symbolized by or. ( This link is independent of the multiplication in the body and that of body elements with vectors, but the use of the same symbol does not lead to confusion, since it is apparent from the context, which linking is meant. )

Explicitly means the bilinearity that applies to all elements and all scalars:

If the underlying field of the real numbers, it is called the algebra also real algebra.

Generalization

General can be a commutative ring, then " vector space " to replace "module" and an algebra is obtained over a commutative ring.

Subalgebras and ideals

A sub algebra an algebra over a field is a sub-space of which that is adjacent to the addition and multiplication by a scalar, that is a member of, and is completed in under the defined multiplication, . Then a separate algebra. For example, it sums the complex numbers as real algebra, to form the real and not the algebra of the imaginary number of a sub- complex numbers.

If, in addition

With any element of so called a left -sided ideal of. Accordingly is, if

Right sided ideal of is. If both are the case, or even commutative, so is simply called an ideal of. If the algebra has no ideals, it is called simple.

Other attributes and examples

Associative algebras

An associative algebra is an algebra in the associative law holds for multiplication and thus is a ring. Examples:

• The algebra of matrices over a field; In this case, the multiplication is matrix multiplication.
• The incidence algebra of a partially ordered set.
• Algebras of linear operators from a vector space into itself; the multiplication is the sequential execution here.
• The group algebra of a group; Here, the group elements form a base of the vector space, and multiplying the algebra is the continuation of the bilinear group multiplication.
• The algebra of polynomials with coefficients in a stranger.
• The algebra of polynomials with coefficients in several unknowns.
• A function algebra is obtained by providing a functional space of functions from a set into a body with the following pointwise multiplication:

Funktionenalgebren are associative, because the body multiplication underlying is associative.

• A body extension is an associative algebra over. Thus, for example, a - algebra and algebra can be used as or considered as algebra.

Commutative algebras

A commutative algebra is an algebra in which the commutative law for multiplication. Examples:

• In the mathematical branch Commutative Algebra algebras are considered, which are associative and commutative. These include the above-mentioned polynomial algebras, the Funktionenalgebren and field extensions.
• Genetic algebras are commutative algebras with some additional properties in which the associative law is generally not fulfilled.

Unitary algebras

A unitary algebra is an algebra with a neutral element of the multiplication, the identity element ( cf. unitary ring). Examples:

• Matrizenalgebren with the identity matrix as the identity element.
• Each group algebra is unitary: the identity element of the group is the unit element of the algebra.
• The constant polynomial 1 is one element of a polynomial algebra.
• The field K with his body as a multiplication algebra multiplication is as algebra associative, commutative and unitary.

If that is clear from the respective context, the properties " associative ", " commutative " and " unitary " is not explicitly mentioned in the rule. Has an algebra no identity, then you can adjoin one; each algebra is therefore in a unitary included.

Non - associative algebras

Some authors refer to an algebra as a non- associative, if the associative law is not required. ( However, this leads to the conceptualization somewhat confusing result that any particular algebra associative and non- associative is. ) Some examples of algebras that are not necessarily associative:

• A division algebra is an algebra in which to " divide " can, ie in all equations and are always uniquely solvable. A division algebra must be neither commutative nor associative nor unitary.
• A Lie algebra is an algebra in which both of the following conditions apply ( in Lie algebras, the product is usually written as ):
• ( Jacobi identity)