使用者:Råy kuø/沙盒2

體上的代數（algebra over a field）或體代數，一般可簡稱為代數，是在向量空間的基礎上定義了一個雙線性的乘法運算而構成的代數結構。根據此乘法是否具有結合律，可以進一步地分成結合代數以及非結合代數兩類。如果乘法單位元包含在此代數裡，則稱為單位代數。

若沒有特別指明，通常假設此代數為結合代數。而在一些代數幾何的討論框架下，會假設此代數是、單位結合且交換。在更一般的情況下，會討論將向量空間換成環所形成的代數，稱為環上的代數（algebra over a ring）。

需要注意的是這裡的雙線性乘法運算跟向量空間上的雙線性形式是不一樣的。具體而言，雙線性乘法運算是一個在向量空間裡的向量，而雙線性形式所給出的是在體K上的純量。

定義

例子

代數	向量空間	雙線性乘法	結合律	交換律
複數	$\mathbb {R} ^{2}$	複數裡的乘法 $\left(a+ib\right)\cdot \left(c+id\right)$	有	有
三圍向量的外積	$\mathbb {R} ^{3}$	外積 ${\vec {a}}\times {\vec {b}}$	無	無
四元數s	$\mathbb {R} ^{4}$	Hamilton product $(a+{\vec {v}})(b+{\vec {w}})$	有	無
多項式	$\mathbb {R} [X]$	多項式乘法	有	有
方塊矩陣	$\mathbb {R} ^{n\times n}$	矩陣乘法	有	無

定義

令K為一個體, A為 K上的向量空間，且有二元乘法，記為 $\cdot :A\times A\to A\ ;\ (x,y)\mapsto A$ 則A 是一個K-代數（K-algebra）如果滿足以下幾點：

對於所有 A中的向量 x, y, z ，所有K中的純量 a,b，有

• 右分配律：(x + y) · z = x · z + y · z
• 左分配律：z · (x + y) = z · x + z · y
• 與純量相容：(ax) · (by) = (ab) (x · y)

此時的K稱為A的基體（base field）。此外，當A有交換律時，左分配律等價於右分配律。

基本概念

代數上的同態

給定 K-代數 A 以及 B,兩個 K-algebras 上的同態（ K-algebra homomorphism）是一個K-線性映射 f: A → B 使得對於所有 A 中的 x, y ，都有 f(xy) = f(x) f(y)。若 A 、B 都是單位代數，則滿足f(1_A) = 1_B 的同態稱為單位同態（unital homomorphism）。所有K-algebras 上的同態所構成的空間通常寫成：

$\mathbf {Hom} _{K{\text{-alg}}}(A,B).$

K-algebras 上的同構（ K-algebra isomorphism）是雙射的K-algebras 上的同態。

子代數

一個K-代數的子代數是一個線性子空間，且具有乘法封閉性（任何兩個元素的乘積仍然在這個子空間內）。換句話說，代數的一個子代數是一個在加法、乘法和純量乘法下閉合的非空子集。

形式上，給定 K-代數 A ，$L\subseteq A$ 為一子集，且滿足以下條件

$\forall x,y\in L\ ,\ \forall c\in K\quad x\cdot y,\ y\cdot x,\ cx\in L$，則稱L 是一個子代數。一個例子是考慮 $\mathbb {C} =\mathbb {R} ^{2}$，則$\mathbb {R}$形成一個子代數。

理想

一個 K 代數的左理想是一個線性子空間，具有這樣的性質：子空間中的任何元素與代數中的任何元素在左側相乘後仍在這個子空間內。用符號來表示，我們說 K-代數 A 的一個子集 L 是一個左理想，如果對於 L 中的所有元素 x 和 y，代數 A 中的元素 z 和 K 中的純量 c，滿足以下三點：

1. x+y 在 L 中（L 在加法下閉合），
2. cx 在 L 中（L 在純量乘法下閉合），
3. z⋅x 在 L 中（L 在左乘任意元素下閉合）。

若將（3）替換為 x⋅z 在 L 中，則這將定義右理想。雙邊理想是同時是左理想和右理想的子集。理想這個術語通常指雙邊理想，而當代數是交換代數時，左理想等價於右理想等價於雙邊理想。條件（1）和（2）一起等價於 L 是 A 的線性子空間。根據條件（3），每個左理想或右理想都是子代數。這個定義與環的理想的定義不同，因為在這裡我們要求條件（2），包含了額外的純量。當然，如果代數是帶單位元的，則條件（3）蘊含條件（2）。

Extension of scalars

If we have a field extension F/K, which is to say a bigger field F that contains K, then there is a natural way to construct an algebra over F from any algebra over K. It is the same construction one uses to make a vector space over a bigger field, namely the tensor product $V_{F}:=V\otimes _{K}F$. So if A is an algebra over K, then $A_{F}$ is an algebra over F.

Kinds of algebras and examples

Algebras over fields come in many different types. These types are specified by insisting on some further axioms, such as commutativity or associativity of the multiplication operation, which are not required in the broad definition of an algebra. The theories corresponding to the different types of algebras are often very different.

Unital algebra

An algebra is unital or unitary if it has a unit or identity element I with Ix = x = xI for all x in the algebra.

Zero algebra

An algebra is called a zero algebra if uv = 0 for all u, v in the algebra, not to be confused with the algebra with one element. It is inherently non-unital (except in the case of only one element), associative and commutative.

One may define a unital zero algebra by taking the direct sum of modules of a field (or more generally a ring) K and a K-vector space (or module) V, and defining the product of every pair of elements of V to be zero. That is, if λ, μ ∈ K and u, v ∈ V, then (λ + u) (μ + v) = λμ + (λv + μu). If e₁, ... e_d is a basis of V, the unital zero algebra is the quotient of the polynomial ring K[E₁, ..., E_n] by the ideal generated by the E_iE_j for every pair (i, j).

An example of unital zero algebra is the algebra of dual numbers, the unital zero R-algebra built from a one dimensional real vector space.

These unital zero algebras may be more generally useful, as they allow to translate any general property of the algebras to properties of vector spaces or modules. For example, the theory of Gröbner bases was introduced by Bruno Buchberger for ideals in a polynomial ring R = K[x₁, ..., x_n] over a field. The construction of the unital zero algebra over a free R-module allows extending this theory as a Gröbner basis theory for submodules of a free module. This extension allows, for computing a Gröbner basis of a submodule, to use, without any modification, any algorithm and any software for computing Gröbner bases of ideals.

Associative algebra

主條目：Associative algebra

Examples of associative algebras include

• the algebra of all n-by-n matrices over a field (or commutative ring) K. Here the multiplication is ordinary matrix multiplication.
• group algebras, where a group serves as a basis of the vector space and algebra multiplication extends group multiplication.
• the commutative algebra K[x] of all polynomials over K (see polynomial ring).
• algebras of functions, such as the R-algebra of all real-valued continuous functions defined on the interval [0,1], or the C-algebra of all holomorphic functions defined on some fixed open set in the complex plane. These are also commutative.
• Incidence algebras are built on certain partially ordered sets.
• algebras of linear operators, for example on a Hilbert space. Here the algebra multiplication is given by the composition of operators. These algebras also carry a topology; many of them are defined on an underlying Banach space, which turns them into Banach algebras. If an involution is given as well, we obtain B*-algebras and C*-algebras. These are studied in functional analysis.

Non-associative algebra

主條目：Non-associative algebra

A non-associative algebra (or distributive algebra) over a field K is a K-vector space A equipped with a K-bilinear map $A\times A\rightarrow A$. The usage of "non-associative" here is meant to convey that associativity is not assumed, but it does not mean it is prohibited – that is, it means "not necessarily associative".

Examples detailed in the main article include:

• Euclidean space R³ with multiplication given by the vector cross product
• Octonions
• Lie algebras
• Jordan algebras
• Alternative algebras
• Flexible algebras
• Power-associative algebras

Algebras and rings

The definition of an associative K-algebra with unit is also frequently given in an alternative way. In this case, an algebra over a field K is a ring A together with a ring homomorphism

$\eta \colon K\to Z(A),$

where Z(A) is the center of A. Since η is a ring homomorphism, then one must have either that A is the zero ring, or that η is injective. This definition is equivalent to that above, with scalar multiplication

$K\times A\to A$

given by

$(k,a)\mapsto \eta (k)a.$

Given two such associative unital K-algebras A and B, a unital K-algebra homomorphism f: A → B is a ring homomorphism that commutes with the scalar multiplication defined by η, which one may write as

$f(ka)=kf(a)$

for all $k\in K$ and $a\in A$. In other words, the following diagram commutes:

${\begin{matrix}&&K&&\\&\eta _{A}\swarrow &\,&\eta _{B}\searrow &\\A&&{\begin{matrix}f\\\longrightarrow \end{matrix}}&&B\end{matrix}}$

Structure coefficients

主條目：Structure constants

For algebras over a field, the bilinear multiplication from A × A to A is completely determined by the multiplication of basis elements of A. Conversely, once a basis for A has been chosen, the products of basis elements can be set arbitrarily, and then extended in a unique way to a bilinear operator on A, i.e., so the resulting multiplication satisfies the algebra laws.

Thus, given the field K, any finite-dimensional algebra can be specified up to isomorphism by giving its dimension (say n), and specifying n³ structure coefficients c_i,j,k, which are scalars. These structure coefficients determine the multiplication in A via the following rule:

$\mathbf {e} _{i}\mathbf {e} _{j}=\sum _{k=1}^{n}c_{i,j,k}\mathbf {e} _{k}$

where e₁,...,e_n form a basis of A.

Note however that several different sets of structure coefficients can give rise to isomorphic algebras.

In mathematical physics, the structure coefficients are generally written with upper and lower indices, so as to distinguish their transformation properties under coordinate transformations. Specifically, lower indices are covariant indices, and transform via pullbacks, while upper indices are contravariant, transforming under pushforwards. Thus, the structure coefficients are often written c_i,j^k, and their defining rule is written using the Einstein notation as

e_ie_j = c_i,j^ke_k.

If you apply this to vectors written in index notation, then this becomes

(xy)^k = c_i,j^kxⁱy^j.

If K is only a commutative ring and not a field, then the same process works if A is a free module over K. If it isn't, then the multiplication is still completely determined by its action on a set that spans A; however, the structure constants can't be specified arbitrarily in this case, and knowing only the structure constants does not specify the algebra up to isomorphism.

Classification of low-dimensional unital associative algebras over the complex numbers

Two-dimensional, three-dimensional and four-dimensional unital associative algebras over the field of complex numbers were completely classified up to isomorphism by Eduard Study.

There exist two such two-dimensional algebras. Each algebra consists of linear combinations (with complex coefficients) of two basis elements, 1 (the identity element) and a. According to the definition of an identity element,

$\textstyle 1\cdot 1=1\,,\quad 1\cdot a=a\,,\quad a\cdot 1=a\,.$

It remains to specify

$\textstyle aa=1$   for the first algebra,

$\textstyle aa=0$   for the second algebra.

There exist five such three-dimensional algebras. Each algebra consists of linear combinations of three basis elements, 1 (the identity element), a and b. Taking into account the definition of an identity element, it is sufficient to specify

$\textstyle aa=a\,,\quad bb=b\,,\quad ab=ba=0$   for the first algebra,

$\textstyle aa=a\,,\quad bb=0\,,\quad ab=ba=0$   for the second algebra,

$\textstyle aa=b\,,\quad bb=0\,,\quad ab=ba=0$   for the third algebra,

$\textstyle aa=1\,,\quad bb=0\,,\quad ab=-ba=b$   for the fourth algebra,

$\textstyle aa=0\,,\quad bb=0\,,\quad ab=ba=0$   for the fifth algebra.

The fourth of these algebras is non-commutative, and the others are commutative.

Generalization: algebra over a ring

In some areas of mathematics, such as commutative algebra, it is common to consider the more general concept of an algebra over a ring, where a commutative ring R replaces the field K. The only part of the definition that changes is that A is assumed to be an R-module (instead of a K-vector space).

Associative algebras over rings

主條目：Associative algebra

A ring A is always an associative algebra over its center, and over the integers. A classical example of an algebra over its center is the split-biquaternion algebra, which is isomorphic to $\mathbb {H} \times \mathbb {H}$, the direct product of two quaternion algebras. The center of that ring is $\mathbb {R} \times \mathbb {R}$, and hence it has the structure of an algebra over its center, which is not a field. Note that the split-biquaternion algebra is also naturally an 8-dimensional $\mathbb {R}$-algebra.

In commutative algebra, if A is a commutative ring, then any unital ring homomorphism $R\to A$ defines an R-module structure on A, and this is what is known as the R-algebra structure. So a ring comes with a natural $\mathbb {Z}$-module structure, since one can take the unique homomorphism $\mathbb {Z} \to A$. On the other hand, not all rings can be given the structure of an algebra over a field (for example the integers). See Field with one element for a description of an attempt to give to every ring a structure that behaves like an algebra over a field.

See also

• Algebra over an operad
• Alternative algebra
• Clifford algebra
• Differential algebra
• Free algebra
• Geometric algebra
• Max-plus algebra
• Mutation (algebra)
• Operator algebra
• Zariski's lemma

References

• Hazewinkel, Michiel; Gubareni, Nadiya; Kirichenko, Vladimir V. Algebras, rings and modules 1. Springer. 2004. ISBN 1-4020-2690-0.