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A morphism ϕ of unital algebras obeys
ϕ
=
ϕ
ϕ
and
ϕ
=
= η
Alternatively, we can characterize associativity by the following commutative diagram
while unitality reads
A ⊗ A ⊗ A μ⊗id A ⊗ A
id⊗μ
A ⊗ A
μ
K ⊗ A η⊗id A ⊗ A
id⊗η
μ
A
A ⊗ K
μ
A A A
Examples 2.1.3.
1. We give another important example of a K-algebra: let V be a K-vector space. The
tensor algebra over V is the associative unital K-algebra
T (V ) = ⊕ r≥0
with the tensor product as multiplication:
V ⊗r .
(v 1 ⊗ v 2 ⊗ ∙ ∙ ∙ ⊗ v r ) ∙ (w 1 ⊗ ∙ ∙ ∙ ⊗ w t ) := v 1 ⊗ ∙ ∙ ∙ ⊗ v r ⊗ w 1 ⊗ ∙ ∙ ∙ ⊗ w t .
The tensor algebra is a Z + -graded algebra: with the homogeneous component T (r) := V ⊗r
we have
T (r) ∙ T (s) ⊂ T (r+s) .
The tensor algebra is infinite-dimensional, even if V is finite-dimensional. In this case,
obviously
dim T (r) = dim V ⊗r = (dim V ) r .
On the homogenous subspace V ⊗r , it carries an action of the symmetric group S r .
2. Denote by I + (V ) the two-sided ideal of T (V ) that is generated by all elements of the form
x ⊗ y − y ⊗ x with x, y ∈ V . The quotient
S(V ) := T (V )/I + (V )
with its natural algebra structure is called the symmetric algebra over V . The symmetric
algebra is a Z + -graded algebra, as well. It is infinite-dimensional, even if V is finitedimensional.
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