A (graded) quadratic algebra is an -graded algebra which as a graded algebra admits a presentation
where is the tensor algebra of a finite-dimensional vector space of degree 1, and is a ideal generated by a space of homogeneous elements of degree 2 in . Observe that and are uniquely determined by : , and if is the kernel of the canonical algebra map , then . It is often convenient to identify quadratic algebras with such pairs .
A morphism of quadratic algebras is just a morphism as graded algebras. Alternatively, a morphism of quadratic algebras determines and is determined by a linear map such that .
The tensor algebra ( finite-dimensional) is of course quadratic.
The symmetric algebra is quadratic.
The Grassmann algebra is quadratic.
Extrapolating from the first three examples, a Koszul algebra is quadratic.
For many examples of quantum groups, for example quantum , the underlying algebra is quadratic. See the reference by Manin for further examples.
If defines a quadratic algebra, its quadratic dual is defined by the pair , where is the kernel of the composite
In the literature where it commonly appears, the dual of a quadratic algebra is usually denoted . There is a canonical isomorphism . It was first observed by Yuri Manin that this is the duality operator for a $*$-autonomous structure on the category of quadratic algebras:
The monoidal product of and is defined by where refers to the canonical interchange isomorphism
(writing as if the tensor product were strict, as justified by Mac Lane’s coherence theorem). Manin’s notation for this is . The monoidal unit is the free algebra on one generator in degree 1.
The dual monoidal product, denoted , may be defined by the formula
Up to coherent isomorphism, this may be more explicitly defined by the pair .
(To see this last more clearly, observe that for finite-dimensional , the mapping
is a Galois correspondence, and hence a bijection that takes meets to joins and joins to meets. Now the meet of and is . Applying to this, one obtains the join of and which is .)
There is a natural isomorphism .
A preliminary comment is that the aforementioned Galois correspondence is induced by the equivalence
where , are subspaces and is the usual pairing to the ground field .
Let , , define the quadratic algebras, and suppose that and correspond to one another under the adjunction
Now induces a (unique) graded algebra map iff , which is true iff iff iff . This is true iff induces a (unique) graded algebra map .
This result may be effectively summarized by saying that the category of quadratic algebras carries a star-autonomous category structure, i.e., a closed symmetric monoidal category structure equipped with a dualizing object , i.e., an object for which the double dual embedding is a natural isomorphism. The monoidal unit is the polynomial algebra , and the dualizing object is , the algebra of Grassmann numbers. We then have for any quadratic algebra .
Y. Manin, Some remarks on Koszul algebras and quantum groups, Annales de l’institut Fourier, 37 no. 4 (1987), p. 191-205 (pdf)