Taken from Hopf Algebras by Abe. I am not doing this over a commutative ring as he is doing but just over the field of complex numbers, \mathbb{C} — therefore some of my constructions will be simplified. Some of them might even be wrong.

In this section the notion of algebras will be introduced; and we will present some examples of such algebras. An algebra over \mathbb{C} is defined by giving a structure map.

Algebras

Suppose A=\{e_\lambda,0\}_{\lambda\in\Lambda} is a ring and we are given a ring morphism \eta_A:\mathbb{C}\rightarrow A. Then A can be viewed as a complex vector space. If we have

(ax)y=x(ay)=a(xy) for all a\in\mathbb{C} and x,\,y\in A

then A is said to be an algebra. Now the map defined by f(x,y)=xy turns out to be bilinear. Thus we obtain a morphism

\nabla_A:A\otimes A\rightarrow A.

From this fact, we see that an algebra A can be defined in the following manner. For a complex vector space A and morphisms \eta_A:\mathbb{C}\rightarrow A\nabla_A:A\otimes A\rightarrow A, we have the following:

\nabla_A\circ(\nabla_A\otimes I_A)=\nabla_A\circ(I_A\otimes\nabla_A)

(the associative law)

\nabla_A\circ(\eta_A\otimes I_A)=\nabla_A\circ (I_A\otimes \eta_A)=I_A

(the unitary property)

Here, \nabla_A is said to be the multiplicative map of A\eta_A the unit map of A, and together we call \nabla_A\eta_A the structure maps of the algebra A.

Given algebras A and B, if a map \varphi:A\rightarrow B is a homomorphism as well as a linear map, then we call \varphi an algebra morphism. For algebras AB, if we let \nabla_A,\,\nabla_B,\,\eta_A,\,\eta_B be their respective structure maps, then the linear map is an algebra morphism if and only if we have

\nabla_B\circ(\varphi\otimes\varphi)=\varphi\circ \nabla_A

and

\eta_B=\varphi\circ \eta_A.

Given algebras A and B, the vector space A\otimes B becomes an algebra where the structure maps are given by

\nabla_{A\otimes B}=(\nabla_A\otimes\nabla_B)(1_A\otimes \tau\otimes1_B), \eta_{A\otimes B}=\eta_A\otimes \eta_B.

The map \tau above denotes the flip map: a vector space isomorphism A\otimes B\rightarrow B\otimes A defined by x\otimes y\mapsto y\otimes x. The algebra A\otimes B is called the tensor product of A and B. The multiplication in A\otimes B is given by

(a_1\otimes b_1)(a_2\otimes b_2)=a_1a_2\otimes b_1b_2, for a_i\in Ab_i\in B,

and the unit element is given by 1_A\otimes 1_B.

Example 1.3

Given a set of indeterminates  \{x_\lambda\}_{\lambda\in\Lambda}, the set of all polynomials in x_\lambda (\lambda\in\Lambda) with coefficients in \mathbb{C} is written \mathbb{C}[x_\lambda]_{\lambda\in\Lambda} is a commutative ring via the addition and multiplication defined naturally. Moreover, the canonical embedding \mathbb{C}\rightarrow\mathbb{C}[x_\lambda]_{\lambda\in\Lambda} makes \mathbb{C}[x_\lambda]_{\lambda\in\Lambda} an algebra (i.e. the embedding is the unit map).

Example 1.4

The set M_n(\mathbb{C}) of all n\times n square matrices with complex coefficients is a ring with respect to addition and multiplication of matrices and is a vector space with respect to scalar multiplication. Moreover, M_n(\mathbb{C}) becomes an algebra when we define \eta_{M_n(\mathbb{C})}:a\mapsto aI_{n\times n}.

Example 1.5

Let A=F(S) the set of all functions from a set S to \mathbb{C}. For f\,g\in A, we define the operations

(f+g)(x)=f(x)+g(x)fg(x)=f(x)g(x)x\in S,

which makes A a ring. Defining the action

(kf)(x)=k\cdot f(x)k\in\mathbb{C}x\in S,

A becomes a vector space. Notice that A becomes an algebra when we define a vector space morphism

\eta_A(k)(x)=kx\in A.

Example 1.6

Let G be a group and denote the the free complex vector space (I’m restricting to complex vector spaces rather than modules.) generated by G as \mathbb{C}G. Defining the multiplication on \mathbb{C}g by

\left(\sum_{x\in G}a_xx\right)\left(\sum_{y\in G}b_yy\right)=\sum_{z\in G}\left(\sum_{xy=z}a_xb_y\right)z,

\mathbb{C}G becomes a ring. The map \eta_{\mathbb{C}G}:\mathbb{C}\rightarrow\mathbb{C}G given by \eta_{\mathbb{C}G}(k)=ke_G, where e_G is the identity element of G, makes \mathbb{C}G into an algebra. The algebra \mathbb{C}G is said to be the group algebra of G.

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