*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, — 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 * *is defined by giving a structure map.

### Algebras

Suppose is a ring and we are given a ring morphism . Then can be viewed as a complex vector space. If we have

for all and

then is said to be an *algebra*. Now the map defined by turns out to be bilinear. Thus we obtain a morphism

.

From this fact, we see that an algebra can be defined in the following manner. For a complex vector space and morphisms , , we have the following:

(*the associative law*)

(*the unitary property*)

Here, is said to be the *multiplicative map *of , the *unit map *of , and together we call , the structure maps of the algebra .

Given algebras and , if a map is a homomorphism as well as a linear map, then we call an *algebra morphism. *For algebras , , if we let be their respective structure maps, then the linear map is an algebra morphism if and only if we have

and

.

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

, .

The map above denotes the *flip *map: a vector space isomorphism defined by . The algebra is called the *tensor product *of and . The multiplication in is given by

, for , ,

and the unit element is given by .

### Example 1.3

Given a set of indeterminates , the set of all polynomials in () with coefficients in is written is a commutative ring via the addition and multiplication defined naturally. Moreover, the canonical embedding makes an algebra (i.e. the embedding is the unit map).

### Example 1.4

The set of all 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, becomes an algebra when we define .

### Example 1.5

Let the set of all functions from a set to . For , we define the operations

, , ,

which makes a ring. Defining the action

, , ,

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

, .

### Example 1.6

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

,

becomes a ring. The map given by , where is the identity element of , makes into an algebra. The algebra is said to be the *group algebra *of .

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October 25, 2011 at 4:58 pm

Hopf Algebras: Bialgebras and Hopf Algebras « J.P. McCarthy: Math Page[…] Proof : The conditions under which is an algebra morphism are (see here): (a) (b) The conditions under which is an algebra morphism are (c) (d) . […]