An Introduction to Coalgebras

I’ve just dumped this here, I wrote it for a project in Luxembourg a few months ago. Its not well edited, and there are errors. I may fix these in the future.
However it may be of benefit to someone.

Introduction to various classical notions of algebras. Coalgebras are defined and introduced, as are various Commutative and Associative Algberas. An introduction to Tensors, Lie Algebra and Quantum Groups is also included. For pedagogical reasons many examples are included from Mechanics and Physics.

1. Introduction to Algebras and Coalgebras

Abstract Algebra is increasingly important in Mathematics, Physics, Computer Science, Linguistics, and other subjects. Its quite suprising when you first learn that Algebra has moved beyond mere group and ring theory. This essay is to look at some of the more exciting structures in this area of Mathematics. The author makes no apology for lack of originality in this piece, and states a lot of the results without proof. The interested reader who wishes to find the proofs can consult the various references which are included in the bibliography. Once again, the aim is to introduce these topics to a good Masters student so that they may gain both an appreciation of Algebra and Coalgebra in itself, or alternatively to understand the literature in Theoretical Physics or Mechanics

1.1. Associative Algebras

An associative algebra over {\mathbb{K}} is a vector space A equipped with a binary operation (linear map) \mu:A \otimes A \rightarrow A which is associative, i.e. {\mu \circ(\mu \otimes id) = \mu \circ (id \otimes \mu).} Here id is the identity map from A to A (sometimes denoted by {id_{A}}), and the operation {\mu} is called the product. Denoting ab:={\mu(a \otimes b)}, associativity reads just like Kindergarden: (ab)c = a(bc). An associative algebra is said to be unital if there is a map {u:\mathbb{K} \rightarrow A} such that {\mu \circ (u \otimes id) = id = \mu \circ (id \otimes u).} We denote by {1_{A}} or simply by 1 the image of {1_{\mathbb{K}}} in A under {u}. With this notation unitality reads: {1a = a = a1.} An algebra morphims (or simply morphism) is a linear map {f:A \rightarrow A'} such that f(ab) = f(a)f(b). If A is also unital, then we further assume f(1) = 1. \phantom{xxx}The category of nonunital associative algebras is denoted by As-alg, and teh category of unital associative algebras by uAs-alg.

1.2. Free Associative Algebras

The free associative algebra over the vector space V is an associative algebra F(v) equipped with a liner map i:{V \rightarrow \mathcal{F}}(V) which satisfied the following universal condition: any map {f:V\rightarrow A}, where A is an associative algebra, extends uniquely into an algebra morphism {\hat{f}:\mathcal{F}(V)\rightarrow A}. We can draw a commutative diagram for this as well. An important observation is that the free algebra over V is well defined up to a unique isomorphism. Categorically {\mathcal{F}} is a functor from the category of vector spaces to the category of associative algebras, which is left adjoint to the forgetful functor assigning to A its underlying vector space: Hom_{As-alg}(F(V),A) \cong Hom_{Vect}(V,a) These are forgetful functors {As-alg \rightarrow Vect \rightarrow Set} and sometimes it is useful to consider the free associative algebra over a set X. This is the same as the associative algebra over the space { \mathbb{K}[X]} spanned by X since the functor is given by {X \rightarrow \mathbb{K}[X]} is left adjoint to the functor Vect {\rightarrow} Set.

2. Tensor Algbera

In this section we introduce some facts from Tensor Algebra, and subsequently Tensor products over noncommutative rings.

2.1. Tensors over Linear Spaces

Before dealing with Tensors over noncommutative rings we need to start with the simplest case. This subject is an extension of linear algebra sometimes called ‘multilinear algebra’. Due to the historical background of Tensors (i.e. the intertwining of the subject with Mathematical Physics and Mechanics) a reference used in this writing was by a Mathematician working in Mechanics ~\cite{Marsden}. We will neglect Manifolds in this work, however we shall note that – ultimately our constructions will be done on each fiber of the tangent bundle, producing a new vector bundle~\cite{GeometricPhasesinClassicalandQuantum, Marsden,FoundationsDiffMan}. Let {\mathbf{E,F,...}} denote Banach Spaces , and {L^{k}(\mathbf{E_{1},...,E_{k};F}}) denotes the vector space of continuous k-multilinear maps of {\mathbf{E_{1}}\times...\times\mathbf{E_{k}}} to {\mathbf{F}} The special case {L(\mathbf{E},\mathbb{R})} is denoted {\mathbf{E^{*}}}, the dual space of E. If E is finite dimensional and {{e_{1},...,e_{n}}} there is a unique ordered basis {{e^{1},...,e^{n}}}, such that {\braket{e^{j},e_{i}} = \delta^{j}_{i}} where the Kronecker delta follows the standard rules. Furthermore , for each {v \in \mathbf{E}}, {v=\Sigma_{i=1}^{n}\braket{e^{i},v}e_{i}} and {\alpha=\Sigma_{i=1}^{n}\braket{\alpha,e_{i}}e^{i}} for each {\alpha = \mathbf{E}^{*}}, whereas denotes the pairing between E and {\mathbf{E^{*}}}. Employing the summation convention we can drop the summation symbols

2.2. Tensor Products

When learning Abstract Algebra, it becomes evident that Tensor Products are essential knowledge.

2.3. Tensor module, tensor algebra

By definition the tensor algebra over the vector space V is the tensor module

\displaystyle  T(V):=\mathbb{K}1\oplus V \oplus ... \oplus V^{\otimes n} \oplus ... \ \ \ \ \ (1)

equipped with the concatenation product {T(V)\otimes T(V) \rightarrow T(V)} given by {v_{1}...v_{p}\otimes v_{p+1} ... v_{p+q} \Rightarrow v_{1}...v_{p}v_{p+1}...v_{p+q}.} This operation is clearly associative and 1 is taken as a unit. Observer that T(V) is augmented by {\epsilon (v_{1}...v_{n}) = 0} for {n \geq 1} and {\epsilon(1) = 1}. The map {\epsilon: T(V) \rightarrow \mathbb{K}} is called the augmentation. For a homogenous element {x \in V^{\otimes n},} the integer n is called the weight of x. We say that T(V) is weight-graded. \paragraph{} The reduced tensor algebra {\bar{T}(V)} is the reduced tensor module {\bar{T}(V):=V \oplus ... \oplus V^{\otimes n} \oplus ...} equipped with the concatenation product. It is a nonunital associative algebra.

2.4. Tensor Products from a Pure Mathematics point of view

In this section we write a bit about tensor product of algebras We start first with commutative rings R, of course in modern mathematics noncommutative rings are extremely important, and supercommutative rings. Of course these sorts of concepts are beyond the scope of this article. By an R-algebra we mean a ring homomorphism {R \rightarrow A} into a ring A such that the image of R is contained in the center of A. Let A, B be R-algebras. We shall make {A \otimes B} into an R-aglebra.

Given (a,b) \in A\times B, we\;have\; an\; R\; bilinear\; map
M_{a,b}: A \times B \rightarrow A \otimes B
such that M_{a,b}(a’,b’) = aa’ \otimes bb’

Hence {M_{a,b}} induces an R-linear map {m_{a,b}:A \times B \rightarrow A \otimes B} such that {m_{a,b}(a',b')= aa' \otimes bb'}. But {m_{a,b}} depends bilinearly on a and b, so we obtain a unique R-bilinear map {A \otimes B \times A \otimes B \rightarrow A \otimes B} such that {(a \otimes b)(a' \otimes b') = aa' \otimes bb'} This map is obviously associative, and we have a natural ring homomorphism {R \rightarrow A \otimes B} given by {c \mapsto 1 \otimes c = c \otimes 1} Thus {A \otimes B} is an R-algebra, called the ordinary tensor product.

2.5. Multilinear Algebra over Rings

The tensor product {M\otimes_{R}N} of two modules M, N over a commutative ring R is defined as the quotient of the free R-module {R^{(M \times N)}} generated by M {\times} N – and thus made up by the combinations { \sum_{(x,y)\in M \times N} r_{(x,y)}e_{(x,y)}, } where only a finite number of coefficients {r_{(x,y)}\in R} are nonzero – by the R-submodule generated by the elements that ‘correspond to R-bilinearity’, i.e. by the elements that ‘correspond to R-bilinearity’, i.e. by the elements

-e_{(x+x’,xy)} + e_{(x,y)} + e_{(x’,y)}, -e_{(x,y+y’)}+e_{(x,y)}+e_{(x,y’)}, e_{(rx,y)} + re_{(x,y)}, -e_{(x,ry)}+re_{(x,y)}

This tensor product R-module together with the obvious R-bilinear map { \otimes: M \times N \ni (x,y) \Rightarrow x \otimes y = [e_{(x,y)}] \in M \otimes_{R} N } are universal. In the case of a noncommutative ring R, we consider a right R-module N and a left R-module N and define the tensor product {M \otimes_{R} N} as a {\mathbb{Z}-module}, i.e. as an abelian group, and more precisely as the quotient of the free {\mathbb{Z}-module} {\mathbb{Z}^{(M \times N)}} generated by {M \times N}, by the {\mathbb{Z}}-submodule generated by the elements that ‘correspond to weakened bilinearity’, i.e. by the elements

{ -e_{(x+x',y)} + e_{(x,y)} + e_{(x',y)}, -e_{(x,y+y')}+e_{(x,y)}+e_{(x,y')}}
{-e_{(xr,y)} + e_{(x,ry)} }

The tensor product {\mathbb{Z}-module} {M \otimes_{R} N} and the naturally weakly bilinear mapping \otimes: M \times N \ni (x,y) \Rightarrow x \otimes y = [e_{(x,y)}] \in M \otimes_{R} N are universal. This means that functor {-\otimes_{R}N} from {\mathtt{Mod_{R}}} to \texttt{AbGrp} is the left adjoint of the functor{Hom_{\mathbb{Z}}(N, -)} , where the right module structure on {Hom_{\mathbb{Z}}(N, P)} is defined by (fr)(N) = f(rn), i.e. we have Hom_{\mathbb{Z}}(M \otimes_{R} N, P) \simeq Hom_{R}(M, Hom_{\mathbb{Z}}(N,P) functorially in M and P. In general it is not possible to define an R-module structure on {M \otimes_{R} N}

Now of course we can go on to define more exotic algebraic structures such as ‘superalgebras’, or tensor products over supercommutative rings. However we consider these beyond the scope of this essay.

2.6. Commutative Algebra

By definition a commutative algebra is a vector space A over {\mathbb{K}} equipped with a binary operation {\mu:A \otimes A \rightarrow A,\mu(a,b)=(a,b),} which is both associative and commutative (i.e. symmetric): { ab = ba } In terms of the switching map { \tau:A \otimes A \rightarrow A \otimes A } defined by {\tau(x,y)=(y,x)}, the commutaton condition read {\mu \circ \tau = \mu}. Sometimes one needs to work with algebras whose operation satisfies the symmetry condition but isn’t associative It is proposed in to call these commutative magmatic algebras. \phantom{xxxx}The free unital commutative algebra over the vector space V is the symmetric algebra { S(V) = \bigoplus_{n\geq0}S^{n}(V):= \bigoplus_{n\geq 0}(V^{\otimes n})_{\mathbb{S}_{n}} } There is a lot more detail that one can go into for Commutative Algebras This is outside of the scope of this essay, so the following reference is recommended

2.7. Modules

Modules are intuitively a ‘vector space over a ring’, but this is just words. Let us consider the definition of a Module. A left module M over an algebra A is a vector space equipped with a linear map \lambda: A \otimes M \rightarrow M, \lambda(a,m)=am, called the left action, which is compatible with the product and the unit of A. See \cite{AlgebraicOperads} and \cite{CategoryTheory} for diagrams representing this. There is a similar notion of right module involving a right action {\lambda'}: {M \otimes A \rightarrow A, \lambda'(m,a) = ma.} Finally, a bimodule M over the algebra A is a vector space which is both a left module and a right module and which satisfy (a’m)a=a'(ma) for any {a,a'\;\in\;A} and {m\;\in\;M.} \paragraph{}For any vector space V the free left A-module over V is {M:= A \otimes V} equipped with the obvious left operation. Similarly the free A-bimodule over V is M:={A \otimes V \otimes A.} Let {0 \rightarrow M \rightarrow A' \rightarrow A \rightarrow 0} be an exact sequence of associative algebras such that the product in M is 0. Then it is easy to check that M is a bimodule over A. Some examples of modules

  • If K is a field, then the concepts “K-vector space” (a vector space over K) and K-module are identical.
  • The concept of a Z-module agrees with the notion of an abelian group. That is, every abelian group is a module over the ring of integers {\mathbb{Z}} in a unique way. For n > 0, let nx = x + x + … + x (n summands), 0x = 0, and (−n)x = −(nx). Such a module need not have a basis—groups containing torsion elements do not. (However, a finite field, considered as a module over the same finite field taken as a ring, does have a basis.)
  • If R is any ring and n a natural number, then the cartesian product {R^{n}} is both a left and a right module over R if we use the component-wise operations. Hence when n = 1, R is an R-module, where the scalar multiplication is just ring multiplication. The case n = 0 yields the trivial R-module {{0}} consisting only of its identity element. Modules of this type are called free and if R has invariant basis number (e.g. any commutative ring or field) the number n is then the rank of the free module.
  • If S is a nonempty set, M is a left R-module, and {M^{S}} is the collection of all functions f : S {\rightarrow} M, then with addition and scalar multiplication in {M^{S}} defined by (f + g)(s) = f(s) + g(s) and (rf)(s) = rf(s), {M^{S}} is a left R-module. The right R-module case is analogous. In particular, if R is commutative then the collection of R-module homomorphisms h : M {\rightarrow} N (see below) is an R-module (and in fact a submodule of {N^{M}}).
  • If X is a smooth manifold, then the smooth functions from X to the real numbers form a ring {C^{\infty}(X)}. The set of all smooth vector fields defined on X form a module over {C^{\infty}(X)}, and so do the tensor fields and the differential forms on X. More generally, the sections of any vector bundle form a projective module over {C^{\infty}(X)}, and by Swan’s theorem, every projective module is isomorphic to the module of sections of some bundle; the category of {C^{\infty}(X)}-modules and the category of vector bundles over X are equivalent.

3. Lie Algebras and Coalgebras

Firstly one can’t overestimate the importance of Lie Algebras in Mathematical and Theoretical Physics. They are an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Subsequently they are of utmost importance in Quantum Mechanics, Particle Physics and of course in their own right. Some of this material was inspired by~\cite{FoundationsDiffMan,Ben-Aryeh2004}. Whilst those of us with the Abstract abilities of Grothendieck may wonder what is the point in including applications. For most readers some relevance to what Samuel Johnson termed ‘reality in all its kickability’ is necessary. The further subsections will introduce coalgebras, naively ‘coalgebras are the dual of an algebra’

3.1. Lie Algebras

Let us begin the definition: A Lie Algebra is a vector space {\mathfrak{g}} over {\mathbb{K}} equipped with a binary operation c: {\mathfrak{g}\otimes \mathfrak{g} \rightarrow \mathfrak{g}}, {c(x,y):=[x,y],}(c for ‘crochet’ in French) called bracket, which is anti-symmetric: { [x,y] = -[y,x], equivalently c \circ \tau = -c } and satisfies the Leibniz identity { [[x,y],z]=[x,[y,z]]+[[x,z],y], } equivalently {c \circ (c \otimes id) = c \circ (id \otimes c) + c \circ (id \otimes c)\circ (id \otimes \tau).}

In the literature the last relation is, most of the time replaced by the Jacobi identity { [[x,y],z]+[[y,z],x]+[[z,x],y]\;=\;0. } These are equivalent under the anti-symmetry condition, but not otherwise. The free Lie algebra over the vector space V is denoted Lie(V).

3.2. Universal enveloping algebra

Mathematically speaking the construction of a universal enveloping algebra allows one to move from a non-associative algebra to a more familiar associative algebra. Construct a functor: Lie-alg {\rightarrow} As-alg as follows. Let {\mathfrak{g}} and let T({\mathfrak{g}}) be the tensor algebra over the vector space {\mathfrak{g}}. By definition the universal algebra U({\mathfrak{g}}) is the quotient of T({\mathfrak{g}}) by the two-sided ideal generated by the elements { x \otimes y - y \otimes x -[x,y], for all x,y \in \mathfrak{g}. }

The functor U: Lie-alg {\rightarrow} As-alg is left adjoint to the functor {(-)_{Lie}:As-alg \rightarrow Lie-alg.}

3.3. Coassociative Coalgebras

Definition 1 A coassociative algebra is a vector space C with comultiplication {\Delta:C \rightarrow C \otimes C} and a counit map {\epsilon:C\;\rightarrow\;\mathbb{K}} satisfying (a) coassociativity {(id \otimes \delta)\delta = (\delta \otimes id)\delta} (b) counitary property {(id \otimes \epsilon)\delta = (\epsilon \otimes id)\delta = id_{C}}

Thus a coalgebra~\cite{CoalgebraRep, AlgebraicOperads} is obtained by dualizing the associative multiplication map {A \otimes A \rightarrow A} and unit map {\mathbb{K} \rightarrow A.} So a finite dimensional coalgebra is the linear dual of a finite dimensional algebra (and vice versa). While this duality might lead an air of redundancy to coalgebra theory, there are properties enjoyed by infinite dimensional coalgebras which are denied infinite dimensional algebras.

3.4. From algebra to coalgebra and vice-versa

Let {V^{*}:=Hom(V,\mathbb{K})} be the linear dual of the space V. There is a canonical map {\omega:V^{*} \otimes V^{*} \rightarrow (V \otimes V)^{*}} given by {\omega(f \otimes g)(x \otimes y)=f(x) g(y)} (we work in the non-graded framework). When V is finite dimensional {\omega} is an isomorphism. \paragraph{}\phantom{kkk}If {(C,\Delta)} is a coalgebra, then {(C^{*},\Delta^{*} \circ \omega)} is an algebra. \phantom{kkk}If ({A,\mu}) is an algebra which is finite dimensional, then {(A^{*},\omega^{-1}\circ \mu^{*})} is a coalgebra.

3.5. Coderivation

Let C = {(C,\Delta)} be a conilpotent coalgebra. By a definition a coderivation is a linear map {d:C \rightarrow C} such that {d(1) = 0} and \Delta \circ d = (d \otimes id)\circ \Delta + (id \otimes d) \circ \Delta. We denote by Coder(C) the space of coderivations of C.

Proposition 2 If C is cofree, i.e. {C = T^{c}(V)} for some vector space V, then a coderivation d {\in} Coder(C) is completely determined by its weight 1 component C

4. Comodule

Just as there is a dual to an algebra, there is also a dual to a module. A left comodule N over a coalgebra C is a vector space endowed with a linear map \Delta^{l}:N \rightarrow C \otimes N, which is compatible with the coproduct and the counit. It is called a coaction map. The notion of a right comodule using {\Delta^{r}:N \rightarrow N \otimes C} is analogous. A co-bimodule is a left and right comodule such that the two coaction maps satisfy the coassociativity condition: (id_{C} \otimes \Delta^{r})\circ \Delta^{l}=(\Delta^{l} \otimes id_{C})\circ \Delta^{r}. For instance, any coalgebra is a co-bimodule over itself.

4.1. Cocommutative coalgebra

An associative algebra {(C,\Delta)} is said to be cocommutative if the coproduct {\Delta} satisfies the following symmetry condition: {\Delta = \Delta \circ \tau.} In other words we assume that the image of {\Delta} lies in the invariant space {(C \otimes C)^{\mathbb{S}_{2}}} where the generator of {\mathbb{S}_{2}} acts via {\tau}. The cofree commutative coalgebra over the space V (taken in the category of conilpotent commutative coalgebras of course) can be identified to the symmetric module S(V) equipped with the following coproduct: {\Delta(x_{1}...x_{n})=\sum_{\delta \in Sh(i,j),i+j=n} x_{\delta^{-1}(1)}....x_{\delta^{-1}(i)}\otimes x_{\delta^{-1}(i+1)}...x_{\delta^{-1}(i+j)},} where Sh(i,j) is the set of (i,j)-shuffles.

4.2. Bialgebra

~\cite{Lang,AlgebraicOperads} We introduce the classical notions of bialgebra and Hopf Algebra, which are characterized by the Hopf compatibility relation.

Definition 3 A bialgebra {\mathcal{H}=(\mathcal{H},\mu,\bigtriangleup,u,c)} over {\mathbb{K}} is a vector space {\mathcal{H}} equipped with an algebra structure {\mathcal{H} = (\mathcal{H},\mu,u)} and a coalgebra structure {\mathcal{H}=(\mathcal{H},\bigtriangleup,c)} related by the Hopf compatability relation \bigtriangleup(xy) = \bigtriangleup(x)\bigtriangleup(y), where xy := {\mu(x\otimes y)} and the product on {\mathcal{H} \otimes \mathcal{H}} (in the non-graded case) by {(x \otimes y)(x' \otimes y') = xx' \otimes yy'.} It is often better to write the compatability relation in terms of {\bigtriangleup} and {\mu}. Then one has to introduce the switching map \tau:\mathcal{H} \otimes \mathcal{H} \rightarrow \mathcal{H} \otimes \mathcal{H}, \tau(x \otimes y):= y \otimes x. With this notation the compatability relation reads: \boxed{\bigtriangleup \circ \mu = (\mu \otimes \mu)\circ(is \otimes \tau \otimes id)}\circ(\bigtriangleup \otimes \bigtriangleup): \mathcal{H} \otimes \mathcal{H} \rightarrow \mathcal{H} \otimes \mathcal{H}. The boxed formula is {\mu_{\mathcal{H}\otimes \mathcal{H}}}

4.3. Hopf Algebras and Quantum Groups

\paragraph{}The motivation behind studying Hopf Algebras is their links to Quantum Groups. The following section is inspired by both ~\cite{AlgebraicOperads} and Shahn Majid’s article on Quantum Groups in the Princeton Companion to Mathematics\cite{Princeton}. There are at least three different paths that lead to the objects known today as quantum groups. They could be summarized as quantum geometry, quantum symmetry and self duality. Each of these are important, and each could be a reason to invent quantum groups. This article will only consider the ‘quantum geometric’ reason for inventing Quantum Groups. \subsubsection{Quantum Geometry} Anyone who has ever had the pleasure to study Physics, knows that one of the biggest leaps in 20th century thought was the discovery that Classical Mechanics could be replaced by a process of ‘quantization’ with quantum mechanics. In quantum mechanics, in which the space of possible positions and momenta of a par- ticle is replaced by the formulation of position and momentum as mutually noncommuting operators. This noncommutativity underlies Heisenberg’s “uncertainty principle,” but it also suggests the need for a more gen- eral notion of geometry in which coordinates need not commute. One approach to noncommutative geometry is discussed in operator algebras~\cite{Princeton} (this is the approach that Alain Connes is famed for). However, another approach is to note that geometry really grew out of examples such as spheres, tori, and so forth, which are lie groups or objects closely related to Lie groups. See the following for introductions to Lie Groups \cite{GeometryPhysicsTopology,GeometricPhasesinClassicalandQuantum}. \paragraph{}\phantom{xxx}If one wants to “quantize” geometry, one should first think about how to generalize basic examples like this: in other words, one should try to define “quantum Lie groups” and associated “quantum” homogeneous spaces. The first step is to consider geometrical structures not so much in terms of their points but in terms of corresponding algebras. This duality between geometry and algebra is a very profound method in Mathematical Physics. For example, the group {SL_{2} }({\mathbb{C}}) is defined as the set of 2 {\times} 2 matrices of com- plex numbers such that {\alpha\delta}{\beta\gamma} = 1. We can think of this as a subset of {\mathbb{C}^{4}} , and indeed not just a subset but a variety The natural class of functions associated with this variety is the set of polynomials in four variables (which are defined on {\mathbb{C}^{4}}) restricted to the variety. However, if two polynomials take equal values on the variety, then we identify them. In other words, we take the algebra of polynomials in four vari- ables a, b, c, and d and quotient by the ideal generated by all polynomials of the form ad − bc − 1. Let us call the resulting algebra {\mathbb{C}[SL_{2} ].} \paragraph{}\phantom{xxx}We can do the same for any subset X {\subset \mathbb{C}^{n}} that is defined by polynomial relations. This gives us a precise one-to-one correspondence between subsets of this type and certain commutative algebras equipped with n generators. Let us write {\mathbb{C}[X]} for the algebra that corresponds to X. As with many similar construc- tions a suitable map from X to Y gives rise to a map from {\mathbb{C}[Y ]} to {\mathbb{C}[X]}. More precisely, the map {\phi} from X to Y has to be polynomial (in a suitable sense) and the resulting map from {\mathbb{C}[Y ]} to {\mathbb{C}[X]} is an algebra homomorphism {\phi^{*}} that satisfies the formula {\phi^{*}} (p)(x) = p({\phi}x) for every {x \in X}. Going back to our example, the set {SL_{2 }(\mathbb{C})} has a group structure {SL_{2 }(\mathbb{C}) \times SL_{2}(\mathbb{C}) \rightarrow SL_{2} (\mathbb{C})} defined by the matrix product. The set {SL_{2}(\mathbb{C} \times SL_{2}(\mathbb{C}} is a variety in {\mathbb{C}^{8}} and the matrix product depends in a polynomial way on the entries in the matrices, so we obtain an algebra homomorphism {\Delta : \mathbb{C}[SL_{2} ] → \mathbb{C}[SL_{2} ] \otimes \mathbb{C}[SL_{2} ]}, which is known as the coproduct. (The algebra {\mathbb{C}[SL_{2} ] \otimes \mathbb{C}[SL_{2} ]} is isomorphic to {\mathbb{C}[SL_{2} \times SL_{2} ].}) It turns out that {\delta} can be expressed by the formula \Delta a & b
c & d = a & b
c & d \otimes a & b
c & d This formula needs a word or two of explanation: the variables a, b, c, and d are the four generators of the algebra of polynomials in four variables (and hence of its quotient by ad − bc − 1), and the right-hand side is a shorthand way of saying that {\Delta a = a \otimes a + b \otimes c}, and so on. Thus, {\Delta} is defined on the generators by a sort of mixture of tensor products and matrix multiplication. One can then show that the associativity of matrix multiplication in SL2 is equivalent to the assertion that {(\Delta \otimes id)\Delta} = {(id \otimes \Delta)\Delta}. To understand what these expressions mean, bear in mind that {\Delta} takes elements of {\mathbb{C}[SL_{2} ]} to elements of C[SL2 ] ⊗ C[SL2 ]. Thus, when we apply the map {(\Delta \otimes id)\Delta}, for example, we begin by applying {\Delta}, and thereby creating an element of {\mathbb{C}[SL_{2} ] \otimes \mathbb{C}[SL_{2} ]}. This element will be a linear combi- nation of elements of the form {p \otimes q}, each of which will then be replaced by {\Delta p \otimes q}. Similarly, one can express the rest of the group struc- ture of {SL_{2 }(\mathbb{C})} equivalently in terms of the algebra {\mathbb{C}[SL_{2} ]}. There is a counit map {\epsilon:} {\mathbb{C}[SL_{2} ] \rightarrow k}, which corresponds to the group identity, and an antipode map {S : \mathbb{C}[SL_{2} ] \rightarrow \mathbb{C}[SL_{2} ]}, which corresponds to the group inversion. The group axioms appear as equivalent prop- erties of these maps, making {\mathbb{C}[SL_{2} ]} into a “Hopf algebra“ or ”quantum group“

Definition 4 A Hopf algebra over a field {\mathbb{K}} is a quadruple {(H, \Delta,\epsilon , S)}, where (i) H is a unital algebra over {\mathbb{K}}; (ii) {\Delta : H \rightarrow H \otimes H, : H \rightarrow \mathbb{K} }are algebra homo- morphisms such that ({(\Delta \otimes id)\Delta} = {(id \otimes \Delta)\Delta} and {(\epsilon \otimes id)\Delta = (id \otimes \epsilon)\Delta = id}; (iii) {S : H \rightarrow H} is a linear map such that {m(id \otimes S)\Delta = m(S\otimes id)\Delta = 1\epsilon} , where m is the product operation on H.

There are two great things about this formulation. The first is that the notion of a Hopf algebra makes sense over any field. The second is that nowhere did we demand that H was commutative. Of course, if H is derived from a group, then it certainly is commutative (since multiplying two polynomials is commutative), so if we can find a noncommutative Hopf algebra, then we have obtained a strict generalization of the notion of a group. The great discovery of the past two decades is that there are indeed many natural noncommutative examples. For example, the quantum group {\mathbb{C}_{q} [SL_{2} ] }is defined as the free associative noncommutative algebra on symbols a, b, c, and d modulo the relations ba = qab, bc = cb, ca = qac, dc = qcd, db = qbd, da = ad + (q − q^{−1} )bc, ad − q^{−1} bc = 1. This forms a Hopf algebra with {\Delta} given by the same formula as it is for {\mathbb{C}_{q} [SL_{2} ] } and with suitable maps and S. Here q is a nonzero element of {\mathbb{C}}, and as q {\rightarrow} 1 one obtains {\mathbb{C}_{q} [SL_{2} ] }. This example generalizes to canonical examples {\mathbb{C}_{q}} [G] for all complex simple Lie groups G.

Remark 1 Much of group theory and Lie group theory can be generalized to quantum groups.

For example, Haar integration is a linear map {\int: H \rightarrow k} that is translation invariant in a certain sense that involves {\Delta}. If it exists, it is unique up to a scalar multiple, and it does indeed exist in most cases of interest, including all finite- dimensional Hopf algebras. Likewise, the notion of a complex of differential forms {(\Omega, d)} makes sense over any algebra H as a proxy for a differential structure. Here, {\Omega = \oplus_{n} \Omega ^{n}} is required to be an asso- ciative algebra generated by {\Omega^{0} = H} and {\Omega ^{1}} , but one does not assume that it is graded-commutative as in the classical case. When H is a Hopf algebra one canask that {\Omega} is translation invariant, again in a certain sense that involves the coproduct {\Delta}. In this case both {\Omega} and its cohomology as a complex are super (or graded) quantum groups. The axioms of a (graded) Hopf algebra were originally introduced by Heinz Hopf in 1947 precisely to express the structure of the cohomology ring of a group, so this result brings us back full circle to the origins of the subject. For most quantum groups, including all the {\mathbb{C}_{q} [G]}, one has a natural minimal complex {(\Omega, d)}. Thus, a “quantum group” is not merely a Hopf algebra but has additional structure analogous to that of a Lie group. This area of Mathematics, is rich and whilst it is comparatively modern – these Quantum Groups have occured in many areas of Physics and Mathematics. Background on this topic is found in the wikipedia entries, in book form an excellent introduction is provided in Shahn Majids book on Quantum Groups

5. Conclusion

The conclusion is personal, the author feels that when one struggles through a Mathematical text sometimes it feels as if ‘the human’ is forgotten about. What we all share is our humanity, and we all sometimes feel inadequate at the scope and complexity of the physical/mathematical world. Tomonaga the celebrated master of Quantum Field Theory once in his notebook wrote ‘Why, can’t nature be easier to understand’. Reminders like this are necessary for all Students. And one should remember that ‘we are all students’. \phantom{mm} We’ve started with commutative algebra, associative algebras and gone on to define and include examples of Modules, Multilinear Algebra, Lie Algberas, Coalgebras and Hopf Algebras. This is of course a rather quick journey through deep areas of Mathematics. However the aim of this ‘essay’ was to try to explain some of the most powerful techniques and concepts. Mathematics has many levels of understanding, one often struggles to define when one ‘understands’ a topic. For instance one can’t say one understands Quantum Groups unless one understands the alternative representations, and the details of this area already extend into a few monographs. Nonetheless Abstract Algebra is one of the most powerful languages in Theoretical Physics and Mathematics. Tensors and Tensor Products (over vector spaces or modules) are used in a lot of the sort of Mathematical Physics that excites the general public and the naively ambitious student (string theory, noncommutative geometry,Quantum Information) Obviously such topics are well beyond the abilities of the author currently. However one learns Mathematics because it is beautiful, and one also learns it to learn more complicated things. So an appeal to the reader is to not feel disheartened when struggling with ‘the easy stuff’. Einstein famously struggled with the Tensor Calculus, and very few of the notions introduced in this piece are included in the standard ‘bachelor’ or undergraduate curriculum. It is hoped that the student can now understand Coalgebras or the various areas of Theoretical Physics alluded to in the introduction.

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