結合代數

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在數學裡，結合代數是指一向量空間(或更一般地，一模)，其允許向量有具分配律和結合律的乘法。因此，它為一特殊的代數。

[编辑] 定義

一於體K上的結合代數A的定義為一於K上的向量空量，其K-雙線性映射A x A → A 具有結合律：

對任何於A內的x、y和z，(x y) z = x (y z)。

此乘法的雙線性性質可表示成

對任何於A內的x、y和z，满足结合率: (x + y) z = x z + y z；
對任何於A內的x、y及於K的a，满足分配率: x (y + z) = x y + x z；
對任何於A內的x、y及於K內的a，满足结合率 a (x y) = (a x) y = x (a y)。

當A含有單位元，即元素1使得對任一於A內的x，1x = x1 = x，則稱A為具一的結合代數或單作結合代數。此一代數為一個環，且包含所以體K內的元素a，由a1相連接。

上述的定義沒有任何改變地廣義化成了於可交換環K上的代數(除了K-線性空間被稱做模而非向量空間之外)。詳述請見代數 (環論)。

於一體K上的結合代數A的維度為其K-向量空間的維度。

[编辑] 例子

其元素為體K的n×n方陣形成了一於K上的單作結合代數。
複數形成了於實數上的二維單作結合代數。
四元數形成了於實數上的四維單作結合代數(但不為一複數上的代數，因為複數和四元數不可交換)。
實係數多項式形成了一於實數上的單作結合代數。
給定一巴拿赫空間X，其連續線性算子 A : n → X形成了一單作結合代數(以算子複合做為乘法)；事實上，這是一個巴拿赫代數。
給定一拓撲空間X，於X上的連續實(複)值函數形成了一單作結合代數；這裡，加法和乘法是對函數的各點相加和相乘。
一非單作的結合代數為所有x趨向無限時的極限為零的函數f: R → R所組成的集合。
克理福代數也是結合代數的一種，在幾何和物理上都很有用。
局部有限偏序集合的相交代數為一組合數學內的單作結合代數。

[编辑] 代數同態

若A和B為體K上的結合代數，代數同態 h: A → B則是一K-線性映射，其對任何於A內的x、y，會有h(xy) = h(x) h(y)的關係。加上態射的概念，於K上的結合代數組成的類便成了一範疇。

舉個例子，設A為所有實值連續函數R → R所組成的代數，及B=R，這兩者都是於R上的代數，且其每一連續函數f指定至數字f(0)的映射會是個由A至B的代數同態。

[编辑] 免指標標記法

前面所述之結合代數的定義，其結合律的定義是對A的所有元素而定的。但有時不涉及A內元素的結合律定義會較方便。這可以由下列方法作到。一定義成在一向量空間A內映射M的代數：

$M: A \times A \rightarrow A$

其為結合代數當M有下面性質：

$M \circ (\mbox {Id} \times M) = M \circ (M \times \mbox {Id})$

其中，符號 $\circ$ 表示函數的複合，而Id則為恆等函數：對所有於A內的x， $I d (x) = x$ 。要了解其定義是等價的，只需要知道上述式子的兩邊都是三個引數的函數。例如，式子左邊為

$( M \circ (\mbox {Id} \times M)) (x,y,z) = M (x, M(y,z))$

類似地，一單作結合代數可以以單位映射 $\eta: K \rightarrow A$ 來定義，其性質如下：

$M \circ (\mbox {Id} \times \eta ) = s = M \circ (\eta \times \mbox {Id})$

其中，單位映射η將K內的元素k映射至A內的元素k1，這裡1是A的單位元。映射s只是個純量乘積： $s:K\times A \rightarrow A$ 。

[编辑] 廣義化

One may consider associative algebras over a commutative ring R: these are modules over R together with a R-bilinear map which yields an associative multiplication. In this case, a unital R-algebra A can equivalently be defined as a ring A with a ring homomorphism R→A.

The n-by-n matrices with integer entries form an associative algebra over the integers and the polynomials with coefficients in the ring Z/nZ (see modular arithmetic) form an associative algebra over Z/nZ.

[编辑] 共代數

An associative unitary algebra over K is based on a morphism A×A→A having 2 inputs (multiplicator and multiplicand) and one output (product), as well as a morphism K→A identifying the scalar multiples of the multiplicative identity. These two morphisms can be dualized using categorial duality by reversing all arrows in the commutative diagrams which describe the algebra axioms; this defines the structure of a coalgebra.

There is also an abstract notion of F-coalgebra.

[编辑] 表示

A representation of an algebra is a linear map $\rho:A\rightarrow gl(V)$ from A to the general linear algebra of some vector space (or module) V that preserves the multiplicative operation: that is, $ρ(x y) = ρ(x)ρ(y)$ .

Note, however, that there is no natural way of defining a tensor product of representations of associative algebras, without somehow imposing additional conditions. Here, by tensor product of representations, the usual meaning is intended: the result should be a linear representation on the product vector space. Imposing such additional structure typically leads to the idea of a Hopf algebra or a Lie algebra, as demonstrated below.

[编辑] Motivation for a Hopf algebra

Consider, for example, two representations $\sigma:A\rightarrow gl(V)$ and $\tau:A\rightarrow gl(W)$ . One might try to form a tensor product representation $\rho: x \mapsto \rho(x) = \sigma(x) \otimes \tau(x)$ according to how it acts on the product vector space, so that

$\rho(x)(v \otimes w) = (\sigma(x)(v)) \otimes (\tau(x)(w))$ .

However, such a map would not be linear, since one would have

$\rho(kx) = \sigma(kx) \otimes \tau(kx) = k\sigma(x) \otimes k\tau(x) = k^2 (\sigma(x) \otimes \tau(x)) = k^2 \rho(x)$

for $k \in K$ . One can rescue this attempt and restore linearity by imposing additional structure, by defining a map $\Delta:A \rightarrow A \times A$ , and defining the tensor product representation as

$\rho = (\sigma\otimes \tau) \circ \Delta$ .

Here, Δ is a comultiplication. The resulting structure is called a bialgebra. To be consistent with the definitions of the associative algebra, the coalgebra must be co-associative, and, if the algebra is unital, then the co-algebra must be unital as well. Note that bialgebras leave multiplication and co-multiplication unrelated; thus it is common to relate the two (by defining an antipode), thus creating a Hopf algebra.

[编辑] Motivation for a Lie algebra

One can try to be more clever in defining a tensor product. Consider, for example,

$x \mapsto \rho (x) = \sigma(x) \otimes \mbox{Id}_W + \mbox{Id}_V \otimes \tau(x)$

so that the action on the tensor product space is given by

$\rho(x) (v \otimes w) = (\sigma(x) v)\otimes w + v \otimes (\tau(x) w)$ .

This map is clearly linear in x, and so it does not have the problem of the earlier definition. However, it fails to preserve multiplication:

$\rho(xy) = \sigma(x) \sigma(y) \otimes \mbox{Id}_W + \mbox{Id}_V \otimes \tau(x) \tau(y)$ .

But, in general, this does not equal

$\rho(x)\rho(y) = \sigma(x) \sigma(y) \otimes \mbox{Id}_W + \sigma(x) \otimes \tau(y) + \sigma(y) \otimes \tau(x) + \mbox{Id}_V \otimes \tau(x) \tau(y)$ .

Equality would hold if the product xy were antisymmetric (if the product were the Lie bracket, that is, $xy \equiv M(x,y) = [x,y]$ ), thus turning the associative algebra into a Lie algebra.