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 is a vector space A equipped with a binary operation (linear map) \mu:A \otimes A \rightarrow A which is associative, i.e. *product*. Denoting ab:=*unital* if there is a map *algebra morphims* (or simply morphism) is a linear map

** 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:

**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 **E**. If **E** is finite dimensional and **E** and **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*

equipped with the *concatenation* product *augmentation*. For a homogenous element *weight* of x. We say that T(V) is *weight-graded*. \paragraph{} The *reduced tensor algebra* *reduced tensor module*

** 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

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

** 2.5. Multilinear Algebra over Rings **

The tensor product *commutative* ring R is defined as the quotient of the free R-module

-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 *noncommutative* ring R, we consider a right R-module N and a left R-module N and define the tensor product

The tensor product

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 *switching map* *commutative magmatic algebras*. \phantom{xxxx}The free unital commutative algebra over the vector space V is the *symmetric algebra*

** 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 *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 *free left A-module* over V is *free A-bimodule* over V is M:=

- 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
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
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 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
is the collection of all functions f : S M, then with addition and scalar multiplication in defined by (f + g)(s) = f(s) + g(s) and (rf)(s) = rf(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 N (see below) is an R-module (and in fact a*submodule*of). - If X is a smooth manifold, then the smooth functions from X to the real numbers form a ring
. The set of all smooth vector fields defined on X form a module over , 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 , and by Swan’s theorem, every projective module is isomorphic to the module of sections of some bundle; the category of -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 *bracket*, which is anti-symmetric: *Leibniz identity*

In the literature the last relation is, most of the time replaced by the *Jacobi identity* *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 *As*-alg as follows. Let *universal algebra* U(

The functor U: Lie-alg

** 3.3. Coassociative Coalgebras **

Definition 1Acoassociative algebrais a vector space C with comultiplicationand a counit map satisfying (a) coassociativity (b) counitary property

Thus a coalgebra~\cite{CoalgebraRep, AlgebraicOperads} is obtained by dualizing the associative multiplication map

** 3.4. From algebra to coalgebra and vice-versa **

Let

** 3.5. Coderivation **

Let C = *coderivation* is a linear map

Proposition 2If C is cofree, i.e.for some vector space V, then a coderivation d 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 *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 *cocommutative* if the coproduct

** 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 3Abialgebraover is a vector space equipped with an algebra structure and a coalgebra structure related by the Hopf compatability relation\bigtriangleup(xy) = \bigtriangleup(x)\bigtriangleup(y), where xy :=and the product on (in the non-graded case) by It is often better to write the compatability relation in terms of and . 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

** 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

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

Definition 4AHopf algebraover a fieldis a quadruple , where (i) H is a unital algebra over ; (ii) are algebra homo- morphisms such that ( = and ; (iii) is a linear map such that , 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

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

For example, Haar integration is a linear map

**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.

I really appreciate your job! Helped me a lot 🙂