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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, an associative algebra ''A'' over a
commutative ring In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
(often a field) ''K'' is a ring ''A'' together with a
ring homomorphism In mathematics, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function that preserves addition, multiplication and multiplicative identity ...
from ''K'' into the center of ''A''. This is thus an
algebraic structure In mathematics, an algebraic structure or algebraic system consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplicatio ...
with an addition, a multiplication, and a
scalar multiplication In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector ...
(the multiplication by the image of the ring homomorphism of an element of ''K''). The addition and multiplication operations together give ''A'' the structure of a ring; the addition and scalar multiplication operations together give ''A'' the structure of a module or
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
over ''K''. In this article we will also use the term ''K''-algebra to mean an associative algebra over ''K''. A standard first example of a ''K''-algebra is a ring of square matrices over a commutative ring ''K'', with the usual
matrix multiplication In mathematics, specifically in linear algebra, matrix multiplication is a binary operation that produces a matrix (mathematics), matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the n ...
. A commutative algebra is an associative algebra for which the multiplication is
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...
, or, equivalently, an associative algebra that is also a
commutative ring In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
. In this article associative algebras are assumed to have a multiplicative identity, denoted 1; they are sometimes called unital associative algebras for clarification. In some areas of mathematics this assumption is not made, and we will call such structures non-unital associative algebras. We will also assume that all rings are unital, and all ring homomorphisms are unital. Every ring is an associative algebra over its center and over the integers.


Definition

Let ''R'' be a
commutative ring In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
(so ''R'' could be a field). An associative ''R''-algebra ''A'' (or more simply, an ''R''-algebra ''A'') is a ring ''A'' that is also an ''R''-module in such a way that the two additions (the ring addition and the module addition) are the same operation, and
scalar multiplication In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector ...
satisfies : r\cdot(xy) = (r\cdot x)y = x(r\cdot y) for all ''r'' in ''R'' and ''x'', ''y'' in the algebra. (This definition implies that the algebra, being a ring, is unital, since rings are supposed to have a
multiplicative identity In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...
.) Equivalently, an associative algebra ''A'' is a ring together with a
ring homomorphism In mathematics, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function that preserves addition, multiplication and multiplicative identity ...
from ''R'' to the center of ''A''. If ''f'' is such a homomorphism, the scalar multiplication is (here the multiplication is the ring multiplication); if the scalar multiplication is given, the ring homomorphism is given by . (See also ' below). Every ring is an associative Z-algebra, where Z denotes the ring of the
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s. A is an associative algebra that is also a
commutative ring In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
.


As a monoid object in the category of modules

The definition is equivalent to saying that a unital associative ''R''-algebra is a monoid object in ''R''-Mod (the
monoidal category In mathematics, a monoidal category (or tensor category) is a category (mathematics), category \mathbf C equipped with a bifunctor :\otimes : \mathbf \times \mathbf \to \mathbf that is associative up to a natural isomorphism, and an Object (cate ...
of ''R''-modules). By definition, a ring is a monoid object in the
category of abelian groups In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab. Properties The zero object o ...
; thus, the notion of an associative algebra is obtained by replacing the category of abelian groups with the
category of modules In algebra, given a ring ''R'', the category of left modules over ''R'' is the category whose objects are all left modules over ''R'' and whose morphisms are all module homomorphisms between left ''R''-modules. For example, when ''R'' is the ...
. Pushing this idea further, some authors have introduced a "generalized ring" as a monoid object in some other category that behaves like the category of modules. Indeed, this reinterpretation allows one to avoid making an explicit reference to elements of an algebra ''A''. For example, the associativity can be expressed as follows. By the universal property of a
tensor product of modules In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction is analogous to the construction of the tensor produ ...
, the multiplication (the ''R''-bilinear map) corresponds to a unique ''R''-linear map : m : A \otimes_R A \to A. The associativity then refers to the identity: : m \circ ( \otimes m) = m \circ (m \otimes \operatorname).


From ring homomorphisms

An associative algebra amounts to a
ring homomorphism In mathematics, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function that preserves addition, multiplication and multiplicative identity ...
whose image lies in the center. Indeed, starting with a ring ''A'' and a ring homomorphism whose image lies in the center of ''A'', we can make ''A'' an ''R''-algebra by defining : r\cdot x = \eta(r)x for all and . If ''A'' is an ''R''-algebra, taking , the same formula in turn defines a ring homomorphism whose image lies in the center. If a ring is commutative then it equals its center, so that a commutative ''R''-algebra can be defined simply as a commutative ring ''A'' together with a commutative ring homomorphism . The ring homomorphism ''η'' appearing in the above is often called a structure map. In the commutative case, one can consider the category whose objects are ring homomorphisms for a fixed ''R'', i.e., commutative ''R''-algebras, and whose morphisms are ring homomorphisms that are under ''R''; i.e., is (i.e., the coslice category of the category of commutative rings under ''R''.) The prime spectrum functor Spec then determines an anti-equivalence of this category to the category of
affine scheme In commutative algebra, the prime spectrum (or simply the spectrum) of a commutative ring R is the set of all prime ideals of R, and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with ...
s over Spec ''R''. How to weaken the commutativity assumption is a subject matter of noncommutative algebraic geometry and, more recently, of derived algebraic geometry. See also: '' Generic matrix ring''.


Algebra homomorphisms

A
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
between two ''R''-algebras is an ''R''-linear
ring homomorphism In mathematics, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function that preserves addition, multiplication and multiplicative identity ...
. Explicitly, is an associative algebra homomorphism if : \begin \varphi(r \cdot x) &= r \cdot \varphi(x) \\ \varphi(x + y) &= \varphi(x) + \varphi(y) \\ \varphi(xy) &= \varphi(x)\varphi(y) \\ \varphi(1) &= 1 \end The class of all ''R''-algebras together with algebra homomorphisms between them form a
category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...
, sometimes denoted ''R''-Alg. The subcategory of commutative ''R''-algebras can be characterized as the coslice category ''R''/CRing where CRing is the category of commutative rings.


Examples

The most basic example is a ring itself; it is an algebra over its center or any subring lying in the center. In particular, any commutative ring is an algebra over any of its subrings. Other examples abound both from algebra and other fields of mathematics.


Algebra

* Any ring ''A'' can be considered as a Z-algebra. The unique ring homomorphism from Z to ''A'' is determined by the fact that it must send 1 to the identity in ''A''. Therefore, rings and Z-algebras are equivalent concepts, in the same way that
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...
s and Z-modules are equivalent. * Any ring of characteristic ''n'' is a (Z/''n''Z)-algebra in the same way. * Given an ''R''-module ''M'', the
endomorphism ring In mathematics, the endomorphisms of an abelian group ''X'' form a ring. This ring is called the endomorphism ring of ''X'', denoted by End(''X''); the set of all homomorphisms of ''X'' into itself. Addition of endomorphisms arises naturally in ...
of ''M'', denoted End''R''(''M'') is an ''R''-algebra by defining . * Any ring of matrices with coefficients in a commutative ring ''R'' forms an ''R''-algebra under matrix addition and multiplication. This coincides with the previous example when ''M'' is a finitely-generated, free ''R''-module. ** In particular, the square ''n''-by-''n'' matrices with entries from the field ''K'' form an associative algebra over ''K''. * The
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s form a 2-dimensional commutative algebra over the
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s. * The
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...
s form a 4-dimensional associative algebra over the reals (but not an algebra over the complex numbers, since the complex numbers are not in the center of the quaternions). * Every
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, ...
is a commutative ''R''-algebra. In fact, this is the free commutative ''R''-algebra on the set . * The free ''R''-algebra on a set ''E'' is an algebra of "polynomials" with coefficients in ''R'' and noncommuting indeterminates taken from the set ''E''. * The
tensor algebra In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra over a field, algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', ...
of an ''R''-module is naturally an associative ''R''-algebra. The same is true for quotients such as the exterior and symmetric algebras. Categorically speaking, the
functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
that maps an ''R''-module to its tensor algebra is
left adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are k ...
to the functor that sends an ''R''-algebra to its underlying ''R''-module (forgetting the multiplicative structure). * Given a module ''M'' over a commutative ring ''R'', the direct sum of modules has a structure of an ''R''-algebra by thinking ''M'' consists of infinitesimal elements; i.e., the multiplication is given as . The notion is sometimes called the algebra of dual numbers. * A quasi-free algebra, introduced by Cuntz and Quillen, is a sort of generalization of a free algebra and a semisimple algebra over an algebraically closed field.


Representation theory

* The universal enveloping algebra of a Lie algebra is an associative algebra that can be used to study the given Lie algebra. * If ''G'' is a group and ''R'' is a commutative ring, the set of all functions from ''G'' to ''R'' with finite support form an ''R''-algebra with the convolution as multiplication. It is called the group algebra of ''G''. The construction is the starting point for the application to the study of (discrete) groups. * If ''G'' is an algebraic group (e.g., semisimple complex Lie group), then the
coordinate ring In algebraic geometry, an affine variety or affine algebraic variety is a certain kind of algebraic variety that can be described as a subset of an affine space. More formally, an affine algebraic set is the set of the common zeros over an algeb ...
of ''G'' is the Hopf algebra ''A'' corresponding to ''G''. Many structures of ''G'' translate to those of ''A''. * A quiver algebra (or a path algebra) of a directed graph is the free associative algebra over a field generated by the paths in the graph.


Analysis

* Given any
Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
''X'', the continuous
linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
s form an associative algebra (using composition of operators as multiplication); this is a Banach algebra. * Given any
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
''X'', the continuous real- or complex-valued functions on ''X'' form a real or complex associative algebra; here the functions are added and multiplied pointwise. * The set of semimartingales defined on the filtered probability space forms a ring under stochastic integration. * The Weyl algebra * An Azumaya algebra


Geometry and combinatorics

* The
Clifford algebra In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure of a distinguished subspace. As -algebras, they generalize the real number ...
s, which are useful in
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
and
physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
. *
Incidence algebra In order theory, a field of mathematics, an incidence algebra is an associative algebra, defined for every locally finite partially ordered set and commutative ring with unity. Subalgebra#Subalgebras_for_algebras_over_a_ring_or_field, Subalgebras c ...
s of locally finite
partially ordered set In mathematics, especially order theory, a partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements need ...
s are associative algebras considered in
combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...
. * The partition algebra and its subalgebras, including the Brauer algebra and the Temperley-Lieb algebra. * A
differential graded algebra In mathematics – particularly in homological algebra, algebraic topology, and algebraic geometry – a differential graded algebra (or DGA, or DG algebra) is an algebraic structure often used to capture information about a topological or geo ...
is an associative algebra together with a grading and a differential. For example, the de Rham algebra \Omega(M) = \bigoplus_^n \Omega^p(M), where \Omega^p(M) consists of differential ''p''-forms on a manifold ''M'', is a differential graded algebra.


Mathematical physics

* A Poisson algebra is a commutative associative algebra over a field together with a structure of a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
so that the Lie bracket satisfies the Leibniz rule; i.e., . * Given a Poisson algebra \mathfrak a, consider the vector space \mathfrak ![u!">.html" ;"title="![u">![u!/math> of formal power series">">![u<_a>!.html" ;"title=".html" ;"title="![u">![u!">.html" ;"title="![u">![u!/math> of formal power series over \mathfrak. If \mathfrak ![u!">.html" ;"title="![u">![u!/math> has a structure of an associative algebra with multiplication * such that, for f, g \in \mathfrak, *: f * g = f g - \frac \ u + \cdots, : then \mathfrak ![u!">.html" ;"title="![u">![u!/math> is called a deformation quantization of \mathfrak a. * A quantized enveloping algebra. The dual of such an algebra turns out to be an associative algebra (see ) and is, philosophically speaking, the (quantized) coordinate ring of a quantum group. * Gerstenhaber algebra


Constructions

; Subalgebras : A subalgebra of an ''R''-algebra ''A'' is a subset of ''A'' which is both a
subring In mathematics, a subring of a ring is a subset of that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and that shares the same multiplicative identity as .In general, not all s ...
and a submodule of ''A''. That is, it must be closed under addition, ring multiplication, scalar multiplication, and it must contain the identity element of ''A''. ; Quotient algebras : Let ''A'' be an ''R''-algebra. Any ring-theoretic ideal ''I'' in ''A'' is automatically an ''R''-module since . This gives the
quotient ring In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. ...
the structure of an ''R''-module and, in fact, an ''R''-algebra. It follows that any ring homomorphic image of ''A'' is also an ''R''-algebra. ; Direct products : The direct product of a family of ''R''-algebras is the ring-theoretic direct product. This becomes an ''R''-algebra with the obvious scalar multiplication. ; Free products: One can form a free product of ''R''-algebras in a manner similar to the free product of groups. The free product is the
coproduct In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The cop ...
in the category of ''R''-algebras. ; Tensor products : The tensor product of two ''R''-algebras is also an ''R''-algebra in a natural way. See tensor product of algebras for more details. Given a commutative ring ''R'' and any ring ''A'' the
tensor product In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of ...
''R'' ⊗Z ''A'' can be given the structure of an ''R''-algebra by defining . The functor which sends ''A'' to is
left adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are k ...
to the functor which sends an ''R''-algebra to its underlying ring (forgetting the module structure). See also: Change of rings. ; Free algebra : A free algebra is an algebra generated by symbols. If one imposes commutativity; i.e., take the quotient by commutators, then one gets a polynomial algebra.


Dual of an associative algebra

Let ''A'' be an associative algebra over a commutative ring ''R''. Since ''A'' is in particular a module, we can take the dual module ''A''* of ''A''. A priori, the dual ''A''* need not have a structure of an associative algebra. However, ''A'' may come with an extra structure (namely, that of a Hopf algebra) so that the dual is also an associative algebra. For example, take ''A'' to be the ring of continuous functions on a compact group ''G''. Then, not only ''A'' is an associative algebra, but it also comes with the co-multiplication and co-unit . The "co-" refers to the fact that they satisfy the dual of the usual multiplication and unit in the algebra axiom. Hence, the dual ''A''* is an associative algebra. The co-multiplication and co-unit are also important in order to form a tensor product of representations of associative algebras (see ' below).


Enveloping algebra

Given an associative algebra ''A'' over a commutative ring ''R'', the enveloping algebra ''A''e of ''A'' is the algebra or , depending on authors. Note that a
bimodule In abstract algebra, a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible. Besides appearing naturally in many parts of mathematics, bimodules play a clarifying role, i ...
over ''A'' is exactly a left module over ''A''e.


Separable algebra

Let ''A'' be an algebra over a commutative ring ''R''. Then the algebra ''A'' is a right module over with the action . Then, by definition, ''A'' is said to separable if the multiplication map splits as an ''A''e-linear map, where is an ''A''e-module by . Equivalently, ''A'' is separable if it is a projective module over ; thus, the -projective dimension of ''A'', sometimes called the bidimension of ''A'', measures the failure of separability.


Finite-dimensional algebra

Let ''A'' be a finite-dimensional algebra over a field ''k''. Then ''A'' is an
Artinian ring In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are ...
.


Commutative case

As ''A'' is Artinian, if it is commutative, then it is a finite product of Artinian local rings whose residue fields are algebras over the base field ''k''. Now, a reduced Artinian local ring is a field and thus the following are equivalent # A is separable. # A \otimes \overline is reduced, where \overline is some algebraic closure of ''k''. # A \otimes \overline = \overline^n for some ''n''. # \dim_k A is the number of k-algebra homomorphisms A \to \overline. Let \Gamma = \operatorname(k_s/k) = \varprojlim \operatorname(k'/k), the profinite group of finite Galois extensions of ''k''. Then A \mapsto X_A = \ is an anti-equivalence of the category of finite-dimensional separable ''k''-algebras to the category of finite sets with continuous \Gamma-actions.


Noncommutative case

Since a simple Artinian ring is a (full) matrix ring over a
division ring In algebra, a division ring, also called a skew field (or, occasionally, a sfield), is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicativ ...
, if ''A'' is a simple algebra, then ''A'' is a (full) matrix algebra over a division algebra ''D'' over ''k''; i.e., . More generally, if ''A'' is a semisimple algebra, then it is a finite product of matrix algebras (over various division ''k''-algebras), the fact known as the Artin–Wedderburn theorem. The fact that ''A'' is Artinian simplifies the notion of a Jacobson radical; for an Artinian ring, the Jacobson radical of ''A'' is the intersection of all (two-sided) maximal ideals (in contrast, in general, a Jacobson radical is the intersection of all left maximal ideals or the intersection of all right maximal ideals.) The Wedderburn principal theorem states: for a finite-dimensional algebra ''A'' with a nilpotent ideal ''I'', if the projective dimension of as a module over the enveloping algebra is at most one, then the natural surjection splits; i.e., ''A'' contains a subalgebra ''B'' such that is an isomorphism. Taking ''I'' to be the Jacobson radical, the theorem says in particular that the Jacobson radical is complemented by a semisimple algebra. The theorem is an analog of Levi's theorem for
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
s.


Lattices and orders

Let ''R'' be a
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite leng ...
integral domain In mathematics, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibilit ...
with field of fractions ''K'' (for example, they can be Z, Q). A '' lattice'' ''L'' in a finite-dimensional ''K''-vector space ''V'' is a finitely generated ''R''-submodule of ''V'' that spans ''V''; in other words, . Let ''A''''K'' be a finite-dimensional ''K''-algebra. An '' order'' in ''A''''K'' is an ''R''-subalgebra that is a lattice. In general, there are a lot fewer orders than lattices; e.g., Z is a lattice in Q but not an order (since it is not an algebra). A ''maximal order'' is an order that is maximal among all the orders.


Related concepts


Coalgebras

An associative algebra over ''K'' is given by a ''K''-vector space ''A'' endowed with a bilinear map having two inputs (multiplicator and multiplicand) and one output (product), as well as a morphism identifying the scalar multiples of the multiplicative identity. If the bilinear map is reinterpreted as a linear map (i.e.,
morphism In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
in the category of ''K''-vector spaces) (by the universal property of the tensor product), then we can view an associative algebra over ''K'' as a ''K''-vector space ''A'' endowed with two morphisms (one of the form and one of the form ) satisfying certain conditions that boil down to the algebra axioms. These two morphisms can be dualized using categorial duality by reversing all arrows in the
commutative diagram 350px, The commutative diagram used in the proof of the five lemma In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the s ...
s that describe the algebra
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
s; this defines the structure of a coalgebra. There is also an abstract notion of ''F''-coalgebra, where ''F'' is a
functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
. This is vaguely related to the notion of coalgebra discussed above.


Representations

A representation of an algebra ''A'' is an algebra homomorphism from ''A'' to the endomorphism algebra of some vector space (or module) ''V''. The property of ''ρ'' being an algebra homomorphism means that ''ρ'' preserves the multiplicative operation (that is, for all ''x'' and ''y'' in ''A''), and that ''ρ'' sends the unit of ''A'' to the unit of End(''V'') (that is, to the identity endomorphism of ''V''). If ''A'' and ''B'' are two algebras, and and are two representations, then there is a (canonical) representation of the tensor product algebra on the vector space . However, there is no natural way of defining a
tensor product In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of ...
of two representations of a single associative algebra in such a way that the result is still a representation of that same algebra (not of its tensor product with itself), without somehow imposing additional conditions. Here, by '' tensor product of representations'', the usual meaning is intended: the result should be a linear representation of the same algebra on the product vector space. Imposing such additional structure typically leads to the idea of a Hopf algebra or a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
, as demonstrated below.


Motivation for a Hopf algebra

Consider, for example, two representations and . One might try to form a tensor product representation 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 . One can rescue this attempt and restore linearity by imposing additional structure, by defining an algebra homomorphism , and defining the tensor product representation as : \rho = (\sigma\otimes \tau) \circ \Delta. Such a homomorphism Δ is called a comultiplication if it satisfies certain axioms. The resulting structure is called a
bialgebra In mathematics, a bialgebra over a Field (mathematics), field ''K'' is a vector space over ''K'' which is both a unital algebra, unital associative algebra and a coalgebra, counital coassociative coalgebra. The algebraic and coalgebraic structure ...
. 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 co-unital as well. A Hopf algebra is a bialgebra with an additional piece of structure (the so-called antipode), which allows not only to define the tensor product of two representations, but also the Hom module of two representations (again, similarly to how it is done in the representation theory of groups).


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_W + \mbox_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_W + \mbox_V \otimes \tau(x) \tau(y). But, in general, this does not equal : \rho(x)\rho(y) = \sigma(x) \sigma(y) \otimes \mbox_W + \sigma(x) \otimes \tau(y) + \sigma(y) \otimes \tau(x) + \mbox_V \otimes \tau(x) \tau(y). This shows that this definition of a tensor product is too naive; the obvious fix is to define it such that it is antisymmetric, so that the middle two terms cancel. This leads to the concept of a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
.


Non-unital algebras

Some authors use the term "associative algebra" to refer to structures which do not necessarily have a multiplicative identity, and hence consider homomorphisms which are not necessarily unital. One example of a non-unital associative algebra is given by the set of all functions whose limit as ''x'' nears infinity is zero. Another example is the vector space of continuous periodic functions, together with the convolution product.


See also

*
Abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
*
Algebraic structure In mathematics, an algebraic structure or algebraic system consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplicatio ...
*
Algebra over a field In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear map, bilinear product (mathematics), product. Thus, an algebra is an algebraic structure consisting of a set (mathematics), set to ...
* Sheaf of algebras, a sort of an algebra over a ringed space * Deligne's conjecture on Hochschild cohomology


Notes


Citations


References

* * * * * James Byrnie Shaw (1907
A Synopsis of Linear Associative Algebra
link from
Cornell University Cornell University is a Private university, private Ivy League research university based in Ithaca, New York, United States. The university was co-founded by American philanthropist Ezra Cornell and historian and educator Andrew Dickson W ...
Historical Math Monographs. * Ross Street (1998)
Quantum Groups: an entrée to modern algebra
', an overview of index-free notation. * * * {{DEFAULTSORT:Associative Algebra Algebras Algebraic geometry