a k-vector space A together with a bilinear map m : A ⊗ A → A which is associative. The associativity is expressed by the commutativity of the following diagram:
The algebra A is unital if there is a unit 1 in it. This is expressed by the commutativity of the following diagram:
where u is the map from k to A defined by u(λ) = λ1. The algebra A is commutative if
A subspace J ⊂ A is called a subalgebra (respectively a left ideal, right ideal and two-sided ideal) of A if m(J ⊗ J) (respectively m(A ⊗ J), m(J ⊗ A), m(J ⊗ A + A ⊗ J)) is included in J.
With any vector space V, we can associate its tensor algebra T(V). As a vector space, it is defined by:
with V⊗0 = k and V⊗ k+1 := V ⊗ V⊗k. The product is given by the concatenation:
The embedding of k = V⊗0 into T(V) gives the unit map u. The tensor algebra T(V) is also called the free (unital) algebra generated by V. This algebra is characterized by the following universal property: for any linear map φ from V to a unital algebra A, there is a unique unital algebra morphism
Let A and B be the unital k-algebras. We put a unital algebra structure on A ⊗ B in the following way:
The unit element 1A⊗B is given by 1A ⊗ 1B, and the associativity is clear. This multiplication is thus given by:
where
1.2.2. Coalgebras
Coalgebras are the objects which are somehow dual to algebras: axioms for coalgebras are derived from axioms for algebras by reversing the arrows of the corresponding diagrams:
A k-coalgebra is by definition a k-vector space C together with a bilinear map Δ : C → C ⊗ C, which is coassociative. The coassociativity is expressed by the commutativity of the following diagram:
Coalgebra C is counital if there is a counit ε : C → k, such that the following diagram commutes:
A subspace J ⊂ C is called a subcoalgebra (respectively a left coideal, right coideal and two-sided coideal) of C if Δ(J) is contained in J ⊗ J (respectively C ⊗ J, J ⊗ C, J ⊗ C + C ⊗ J) is included in J. The duality alluded to above can be made more precise:
PROPOSITION 1.1.–
1 1) The linear dual C* of a counital coalgebra C is a unital algebra, with product (respectively unit map) the transpose of the coproduct (respectively of the counit).
2 2) Let J be a linear subspace of C. Denote by J⊥ the orthogonal of J in C*.Then:– J is a two-sided coideal if and only if J⊥ is a subalgebra of C*.– J is a left coideal if and only if J⊥ is a left ideal of C*.– J is a right coideal if and only if J⊥ is a right ideal of C*.– J is a subcoalgebra if and only if J⊥ is a two-sided ideal of C*.
PROOF.– For any subspace K of C*, we will denote by K⊥ the subspace of those elements of C on which any element of K vanishes. It coincides with the intersection of the orthogonal of K with C, via the canonical embedding C ↪ C**. Therefore, for any linear subspaces J ⊂ C and K ⊂ C* we have:
Suppose that J is a two-sided coideal. Take any ξ, η in J⊥. For any x ∈ J, we have:
as Δx ⊂ J ⊗ C + C ⊗ J. Therefore, J⊥ is a subalgebra of C*. Conversely if J⊥ is a subalgebra, then:
which proves the first assertion. We leave the reader to prove the three other assertions along the same lines. □
Dually, we have the following:
PROPOSITION 1.2.– Let K be a linear subspace of C*. Then:
– K⊥ is a two-sided coideal if and only if K is a subalgebra of C*.
– K⊥ is a left coideal if and only if K is a left ideal of C*.
– K⊥ is a right coideal if and only if K is a right ideal of C*.
– K⊥ is a subcoalgebra if and only if K is a two-sided ideal of C*.
PROOF.– The linear dual (C ⊗ C)* naturally contains the tensor product C* ⊗ C*. Take as a multiplication the restriction of tΔ to C* ⊗ C*:
and put u = tε : k → C*. It is easily seen, by just reverting the arrows of the corresponding diagrams, that the coassociativity of Δ implies the associativity of m, and that the counit property for ε implies that u is a unit. □
Note that the duality property is not perfect: