Algebra over a body

An algebra over a field ${\ displaystyle K}$ , algebra over ${\ displaystyle K}$ or algebra (formerly also known as linear algebra ) is a vector space over a field , which has been extended by a multiplication compatible with the vector space structure . Depending on the context, it is sometimes also required that the multiplication fulfills the associative law or the commutative law or that the algebra has a unity with regard to the multiplication . ${\ displaystyle K}$ ${\ displaystyle K}$

definition

An algebra over a field or short -algebra is a - vector space with a - bilinear connection ${\ displaystyle A}$ ${\ displaystyle K}$ ${\ displaystyle K}$ ${\ displaystyle K}$ ${\ displaystyle K}$

{\ displaystyle A \ times A \ to A,}

Called multiplication, which is symbolized by or . (This link is independent of the multiplication in the body and that of body elements with vectors; however, the use of the same symbol does not lead to confusion, since the context makes it clear which link is meant.) ${\ displaystyle x \ cdot y}$ ${\ displaystyle xy}$

Explicitly, bilinearity means that for all elements and all scalars : ${\ displaystyle x, y, z \ in A}$ ${\ displaystyle \ lambda \ in K}$

${\ displaystyle (x + y) \ cdot z = x \ cdot z + y \ cdot z}$
${\ displaystyle x \ cdot (y + z) = x \ cdot y + x \ cdot z}$
${\ displaystyle \ lambda (x \ cdot y) = (\ lambda x) \ cdot y = x \ cdot (\ lambda y)}$

If the underlying field is the field of real numbers , the algebra is also called real algebra. ${\ displaystyle \ mathbb {R}}$

generalization

A commutative ring can be more general , then “vector space” has to be replaced by “ module ”, and one obtains an algebra over a commutative ring . ${\ displaystyle K}$

Sub-algebras and ideals

A sub- ${\ displaystyle U}$ algebra of an algebra over a field is a subspace of which , in addition to addition and multiplication with a scalar , i.e. an element of , is also closed under the multiplication defined in, i.e. H. . Then there is an algebra of its own. If the complex numbers are understood as real algebra, then, for example, the real, but not the imaginary numbers, form a sub-algebra of the complex numbers. ${\ displaystyle A}$ ${\ displaystyle K}$ ${\ displaystyle A}$ ${\ displaystyle K}$ ${\ displaystyle A}$ ${\ displaystyle u, v \ in U \ Rightarrow uv \ in U}$ ${\ displaystyle U}$

Is beyond that

{\ displaystyle v \ in U \ Rightarrow av \ in U}

with any element of , then a left-sided ideal of is called . Correspondingly means if ${\ displaystyle a}$ ${\ displaystyle A}$ ${\ displaystyle U}$ ${\ displaystyle A}$ ${\ displaystyle U}$

{\ displaystyle v \ in U \ Rightarrow va \ in U}

right-sided ideal of is. If both are the case or even commutative, then simply is called an ideal of . If algebra has no non-trivial ideals, it is called simple . ${\ displaystyle A}$ ${\ displaystyle A}$ ${\ displaystyle U}$ ${\ displaystyle A}$ ${\ displaystyle A}$

More attributes and examples

Associative algebras

An associative algebra is an algebra in which the associative law applies to multiplication and which is therefore a ring . Examples: ${\ displaystyle K}$

The algebra of - matrices over a field; the multiplication is here the matrix multiplication . ${\ displaystyle n \ times n}$

The incidence algebra of a partially ordered set .

Algebras of linear operators of a vector space in itself; the multiplication is here the sequential execution . ${\ displaystyle K}$

The group algebra to a group ; Here, the group elements form a - base of -Vektorraums , and the algebra multiplication is the continuation of the bilinear group multiplication. ${\ displaystyle K [G]}$ ${\ displaystyle G}$ ${\ displaystyle K}$ ${\ displaystyle K}$ ${\ displaystyle K [G]}$

The algebra of polynomials with coefficients in in an unknown . ${\ displaystyle K [x]}$ ${\ displaystyle K}$ ${\ displaystyle x}$

The algebra of polynomials with coefficients in in several unknowns . ${\ displaystyle K [x_ {1}, \ dotsc, x_ {n}]}$ ${\ displaystyle K}$ ${\ displaystyle x_ {1}, \ dotsc, x_ {n}}$

A function algebra is obtained by providing a function space of functions from a set in a field with the following pointwise multiplication : ${\ displaystyle M}$ ${\ displaystyle K}$

{\ displaystyle (f \ cdot g) (x): = f (x) \ cdot g (x), \ qquad f, g \ colon M \ to K, x \ in M}

.

A field extension of is an associative algebra over . So is z. B. an -algebra and can be thought of as -algebra or -algebra. ${\ displaystyle K}$ ${\ displaystyle K}$ ${\ displaystyle \ mathbb {R}}$ ${\ displaystyle \ mathbb {Q}}$ ${\ displaystyle \ mathbb {C}}$ ${\ displaystyle \ mathbb {Q}}$ ${\ displaystyle \ mathbb {R}}$

Commutative algebras

A commutative algebra is an algebra in which the commutative law applies to multiplication . Examples: ${\ displaystyle K}$

In the mathematical sub-area Commutative Algebra , algebras are considered that are associative and commutative. These include the above-mentioned polynomial algebras, the function algebras, and the field extensions.
Genetic algebras are commutative algebras with some additional properties in which the associative law is generally not satisfied.

Unitary algebras

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

Matrix algebras with the identity matrix as one element.
An algebra of vector space endomorphisms with identity as a unit.
One element of incidence algebra is function ${\ displaystyle \ delta (a, b): = {\ begin {cases} 1 \ quad & {\ mbox {if}} a = b, \\ 0 & {\ mbox {otherwise}} \ end {cases}} }$
Every group algebra is unitary: the one element of the group is also one element of the algebra.
The constant polynomial 1 is one element of a polynomial algebra.
The field K with its field multiplication as algebra multiplication is associative, commutative and unitary as -algebra. ${\ displaystyle K}$

If this is clear from the respective context, the properties “associative”, “commutative” and “unitary” are usually not mentioned explicitly. If an algebra has no unity, one can adjoint one ; every algebra is therefore contained in a unitary one.

Non-associative algebras

Some authors refer to an algebra as non-associative if the associative law is not assumed. (This concept formation, however, leads to the somewhat confusing consequence that every associative algebra in particular is also non-associative.) Some examples of algebras that are not necessarily associative: ${\ displaystyle K}$

A division algebra is an algebra in which one can “divide”; H. in all equations and for are always uniquely solvable. A division algebra does not have to be commutative, associative, or unitary.

{\ displaystyle ax = b}

{\ displaystyle ya = b}

{\ displaystyle a \ neq 0}

A Lie algebra is an algebra in which the following two conditions apply (in Lie algebras the product is usually written as):

{\ displaystyle [x, y]}

${\ displaystyle [x, x] = 0}$
${\ displaystyle [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0}$ ( Jacobi identity )

The real vector space with the cross product . In particular, this real algebra is a Lie algebra . ${\ displaystyle \ mathbb {R} ^ {3}}$

A Baric algebra is an algebra for which there is a nontrivial algebra homomorphism .

{\ displaystyle A}

{\ displaystyle w \ colon A \ to K}

Algebra Homomorphisms

The homomorphisms between -algebras, i.e. the structure-preserving maps, are K -linear maps that are also multiplicative. If the algebras have unity elements, one generally demands that these also be mapped onto one another. This means: ${\ displaystyle K}$

A mapping between two algebras is a homomorphism if the following applies: ${\ displaystyle f \ colon A \ rightarrow B}$ ${\ displaystyle K}$

${\ displaystyle f (\ lambda x) = \ lambda f (x)}$ for all ${\ displaystyle \ lambda \ in K, x \ in A}$
${\ displaystyle f (x + y) = f (x) + f (y)}$ for all ${\ displaystyle x, y \ in A}$
${\ displaystyle f (x \ cdot y) = f (x) \ cdot f (y)}$ for all ${\ displaystyle x, y \ in A}$
Possibly , with 1 denoting the unit elements in the algebras. ${\ displaystyle f (1) = 1}$

The usual rates then apply. The kernels of homomorphisms are precisely the two-sided ideals. If there is a homomorphism, then the analogue to the homomorphism theorem applies , i.e. the induced mapping ${\ displaystyle f \ colon A \ rightarrow B}$

{\ displaystyle {\ overline {f}} \ colon A / \ mathrm {ker} (f) \ rightarrow f (A), \, a + \ mathrm {ker} (f) \ mapsto f (a)}

is well defined and an algebra isomorphism , that is, a bijective algebra homomorphism, the inverse mapping is automatically also an algebra homomorphism. With this, the isomorphism theorems can also be transferred to algebras, because the usual proofs lead them back to the homomorphism theorem. ${\ displaystyle A / \ mathrm {ker} (f) \ cong f (A)}$

Individual evidence

↑ see e.g. B. in Dickson (1905), http://www-groups.dcs.st-and.ac.uk/~history/Extras/Dickson_linear_algebras.html
↑ Real algebra . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 978-3-8274-0439-8 .
↑ see e.g. BR Lidl and J. Wiesenbauer, ring theory and its applications , Wiesbaden 1980, ISBN 3-400-00371-9 , page

[1] see e.g. B. in Dickson (1905), http://www-groups.dcs.st-and.ac.uk/~history/Extras/Dickson_linear_algebras.html

[2] Real algebra . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 978-3-8274-0439-8 .

[3] see e.g. BR Lidl and J. Wiesenbauer, ring theory and its applications , Wiesbaden 1980, ISBN 3-400-00371-9 , page