- Associative algebra
In

mathematics , an**associative algebra**is avector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner. They are thus special algebras.**Definition**An associative algebra "A" over a field "K" is defined to be a vector space over "K" together with a "K"-bilinear multiplication "A" x "A" → "A" (where the image of ("x","y") is written as "xy") such that the associative law holds:

* ("x y") "z" = "x" ("y z") for all "x", "y" and "z" in "A". The bilinearity of the multiplication can be expressed as

* ("x" + "y") "z" = "x z" + "y z" for all "x", "y", "z" in "A",

* "x" ("y" + "z") = "x y" + "x z" for all "x", "y", "z" in "A",

* "a" ("x y") = ("a" "x") "y" = "x" ("a" "y") for all "x", "y" in "A" and "a" in "K".If "A" contains an identity element, i.e. an element 1 such that 1"x" = "x"1 = "x" for all "x" in "A", then we call "A" an "associative algebra with one" or a(orunital **unitary**)**associative algebra**.Such an algebra is a ring, and contains all elements "a" of the field "K" by identification with "a"1.The "dimension" of the associative algebra "A" over the field "K" is its dimension as a "K"-vector space.

**Modules**The preceding definition generalizes without any change to an algebra over a

commutative ring "K". Such a space is then a "module", rather than a vector space, over "K" with a bilinear form. A unital "R"-algebra "A" can equivalently be defined as a ring "A" with a ring homomorphism "R"→"A". For instance:* The "n"-by-"n" matrices with

integer entries form a unital associative algebra over the integers.

* The polynomials with coefficients in the ring**Z**/"n**"Z**, the integers modulo "n", form a unital associative algebra over**Z**/"n**"Z**.See

algebra (ring theory) for more.**Examples*** The square "n"-by-"n" matrices with entries from the field "K" form a unitary associative algebra over "K".

* Thecomplex number s form a 2-dimensional unitary associative algebra over thereal number s.

* Thequaternions form a 4-dimensional unitary associative algebra over the reals (but not an algebra over the complex numbers, since if complex numbers are treated as a subset of the quaternions, complex numbers and quaternions don't commute).

* Thereal matrices (2 x 2) form an associative algebra useful in plane mapping.

* Thepolynomial s with real coefficients form a unitary associative algebra over the reals.

* Given anyBanach space "X", the continuouslinear operator s "A" : "X" → "X" form a unitary associative algebra (using composition of operators as multiplication); this is in fact aBanach algebra .

* Given any topological space "X", the continuous real- (or complex-) valued functions on "X" form a real (or complex) unitary associative algebra; here we add and multiply functions pointwise.

* An example of a non-unitary associative algebra is given by the set of all functions "f":**R**→**R**whose limit as "x" nears infinity is zero.

* TheClifford algebra s are useful ingeometry andphysics .

*Incidence algebra s of locally finitepartially ordered set s are unitary associative algebras considered incombinatorics .**Algebra homomorphisms**If "A" and "B" are associative algebras over the same field "K", an "algebra homomorphism" "h": "A" → "B" is a "K"-linear map which is also multiplicative in the sense that "h"("xy") = "h"("x") "h"("y") for all "x", "y" in "A". With this notion of morphism, the class of all associative algebras over "K" becomes a category.

Take for example the algebra "A" of all real-valued continuous functions

**R**→**R**, and "B" =**R**. Both are algebras over**R**, and the map which assigns to every continuous function "f" the number "f"(0) is an algebra homomorphism from "A" to "B".**Associativity and the multiplication mapping**Associativity was defined above quantifying over all "elements" of "A". It is possible to define associativity in a way that does not explicitly refer to elements. An algebra is defined as a map "M" (multiplication) on a vector space "A"::$M:\; A\; imes\; A\; ightarrow\; A$An associative algebra is an algebra where the map "M" has the property:$M\; circ\; (mbox\; \{Id\}\; imes\; M)\; =\; M\; circ\; (M\; imes\; mbox\; \{Id\})$Here, the symbol $circ$ refers to

function composition , and Id : "A" → "A" is theidentity map on "A".To see the equivalence of the definitions, we need only understand that each side of the above equation is a function that takes three arguments. For example, the left-hand side acts as:$(\; M\; circ\; (mbox\; \{Id\}\; imes\; M))\; (x,y,z)\; =\; M\; (x,\; M(y,z))$

Similarly, a unital associative algebra can be defined in terms of a unit

$eta:\; K\; ightarrow\; A$which has the property:$M\; circ\; (mbox\; \{Id\}\; imes\; eta\; )\; =\; s\; =\; M\; circ\; (eta\; imes\; mbox\; \{Id\})$Here, the unit map η takes an element "k" in "K" to the element "k1" in "A", where "1" is the unit element of "A". The map "s" is just plain-old scalar multiplication: $s:K\; imes\; A\; ightarrow\; A$; thus, the above identity is sometimes written with Id standing in the place of "s", with scalar multiplication being implicitly understood.**Coalgebras**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 usingcategorial duality by reversing all arrows in thecommutative diagram s which describe the algebraaxiom s; this defines the structure of acoalgebra .There is also an abstract notion of

F-coalgebra .**Representations**A representation of an algebra is a linear map ρ: "A" → gl("V") from "A" to the general linear algebra of some vector space (or module) "V" that preserves the multiplicative operation: that is, ρ("xy")=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 aHopf algebra or aLie algebra , as demonstrated below.**Motivation for a Hopf algebra**Consider, for example, two representations $sigma:A\; ightarrow\; gl(V)$ and $au:A\; ightarrow\; gl(W)$. One might try to form a tensor product representation $ho:\; x\; mapsto\; ho(x)\; =\; sigma(x)\; otimes\; au(x)$ according to how it acts on the product vector space, so that

:$ho(x)(v\; otimes\; w)\; =\; (sigma(x)(v))\; otimes\; (\; au(x)(w)).$

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

:$ho(kx)\; =\; sigma(kx)\; otimes\; au(kx)\; =\; ksigma(x)\; otimes\; k\; au(x)\; =\; k^2\; (sigma(x)\; otimes\; au(x))\; =\; k^2\; ho(x)$

for "k" ∈ "K". One can rescue this attempt and restore linearity by imposing additional structure, by defining a map Δ: "A" → "A" × "A", and defining the tensor product representation as

:$ho\; =\; (sigmaotimes\; au)\; circ\; Delta.$

Here, Δ is a

comultiplication . The resulting structure is called abialgebra . 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 aHopf algebra .**Motivation for a Lie algebra**One can try to be more clever in defining a tensor product. Consider, for example,

:$x\; mapsto\; ho\; (x)\; =\; sigma(x)\; otimes\; mbox\{Id\}\_W\; +\; mbox\{Id\}\_V\; otimes\; au(x)$

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

:$ho(x)\; (v\; otimes\; w)\; =\; (sigma(x)\; v)otimes\; w\; +\; v\; otimes\; (\; au(x)\; w)$.

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:

:$ho(xy)\; =\; sigma(x)\; sigma(y)\; otimes\; mbox\{Id\}\_W\; +\; mbox\{Id\}\_V\; otimes\; au(x)\; au(y)$.

But, in general, this does not equal

:$ho(x)\; ho(y)\; =\; sigma(x)\; sigma(y)\; otimes\; mbox\{Id\}\_W\; +\; sigma(x)\; otimes\; au(y)\; +\; sigma(y)\; otimes\; au(x)\; +\; mbox\{Id\}\_V\; otimes\; au(x)\; au(y)$.

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 aLie algebra .**References***

* Ross Street, " [*http://www-texdev.ics.mq.edu.au/Quantum/Quantum.ps Quantum Groups: an entrée to modern algebra*] " (1998). "(Provides a good overview of index-free notation)"

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