nLab
quadratic algebra

Definition

A (graded) quadratic algebra is an $\mathbb{N}$-graded algebra $A$ which as a graded algebra admits a presentation

$$A\cong T(V)/I$$A \cong T(V)/I

where $T(V)$ is the tensor algebra of a finite-dimensional vector space $V$ of degree 1, and $I$ is a ideal generated by a space $R\subseteq V\otimes V$ of homogeneous elements of degree 2 in $T(V)$. Observe that $V$ and $R$ are uniquely determined by $A$: $V=A_1$, and if $I$ is the kernel of the canonical algebra map $T(A_1)\to A$, then $R=I_2$. It is often convenient to identify quadratic algebras with such pairs $(V,R)$.

A morphism of quadratic algebras is just a morphism as graded algebras. Alternatively, a morphism of quadratic algebras $f:A\to B$ determines and is determined by a linear map $f_1:A_1\to B_1$ such that $(f\otimes f)(R_A)\subseteq R_B$.

Examples

1. The tensor algebra $T(V)$ ($V$ finite-dimensional) is of course quadratic.

2. The symmetric algebra $S(V)$ is quadratic.

3. The Grassmann algebra $\Lambda (V)$ is quadratic.

4. Extrapolating from the first three examples, a Koszul algebra is quadratic.

5. For many examples of quantum groups, for example quantum ${\mathrm{GL}}_2$, the underlying algebra is quadratic. See the reference by Manin for further examples.

Quadratic dual and Manin’s monoidal products

If $(V,i:R\hookrightarrow V\otimes V)$ defines a quadratic algebra, its quadratic dual is defined by the pair $(V^{*},R^{\perp })$, where $R^{\perp }$ is the kernel of the composite

$$V^{*}\otimes V^{*}\cong (V\otimes V{)}^{*}\stackrel{i^{*}}{\to }{R}^{*}$$V^* \otimes V^* \cong (V \otimes V)^* \stackrel{i^*}{\to} R^*

In the literature where it commonly appears, the dual of a quadratic algebra $A$ is usually denoted $A^{!}$. There is a canonical isomorphism $A\cong A^{!!}$. 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 $(V,R_A)$ and $(W,R_B)$ is defined by $(V\otimes W,\sigma (R_A\otimes R_B)$ where $\sigma$ refers to the canonical interchange isomorphism

  $$V\otimes V\otimes W\otimes W\cong V\otimes W\otimes V\otimes W$$V \otimes V \otimes W \otimes W \cong V \otimes W \otimes V \otimes W

  (writing as if the tensor product were strict, as justified by Mac Lane’s coherence theorem). Manin’s notation for this is $A•B$. The monoidal unit is the free algebra on one generator in degree 1.

• The dual monoidal product, denoted $A\circ B$, may be defined by the formula

  $$A\circ B=(A^{!}•B^{!}{)}^{!}$$A \circ B = (A^! \bullet B^!)^!

  Up to coherent isomorphism, this may be more explicitly defined by the pair $(V\otimes W,\sigma (R_A\otimes 1_{W\otimes W})+\sigma (1_{V\otimes V}\otimes R_B))$.

(To see this last more clearly, observe that for finite-dimensional $V$, the mapping

$$(-{)}^{\perp }:\mathrm{Sub}(V)\to \mathrm{Sub}(V^{*})$$(-)^\perp: Sub(V) \to Sub(V^*)

is a Galois correspondence, and hence a bijection that takes meets to joins and joins to meets. Now the meet of $\sigma (R_A^{\perp }\otimes 1_{W^{*}\otimes W^{*}})$ and $\sigma (1_{V^{*}\otimes V^{*}}\otimes R_B^{\perp })$ is $\sigma (R_A^{\perp }\otimes R_B^{\perp })$. Applying $(-{)}^{\perp }$ to this, one obtains the join of $\sigma (R_A\otimes 1_{W\otimes W})$ and $\sigma (1_{V\otimes V}\otimes R_B$ which is $\sigma (R_A\otimes 1_{W\otimes W})+\sigma (1_{V\otimes V}\otimes R_B$.)

Theorem

There is a natural isomorphism $\mathrm{QAlg}(A•B,C)\cong \mathrm{QAlg}(A,B^{!}\circ C)$.

Proof

A preliminary comment is that the aforementioned Galois correspondence is induced by the equivalence

$$\frac{X\subseteq Y^{\perp }}{⟨X,Y⟩=0}$$\frac{X \subseteq Y^\perp}{\langle X, Y \rangle = 0}

where $X\subseteq V^{*}$, $Y\subseteq V$ are subspaces and $⟨-,-⟩:V^{*}\otimes V\to k$ is the usual pairing to the ground field $k$.

Let $(U,R_A)$, $(V,R_B)$, $(W,R_C)$ define the quadratic algebras, and suppose that $f:U\otimes V\to W$ and $g:U\to V^{*}\otimes W$ correspond to one another under the adjunction

$${\mathrm{Vect}}_k(U\otimes V,W)\cong {\mathrm{Vect}}_k(U,V^{*}\otimes W)$$Vect_k(U \otimes V, W) \cong Vect_k(U, V^* \otimes W)

Now $f$ induces a (unique) graded algebra map $A•B\to C$ iff $f(R_A\otimes R_B)\subseteq R_C$, which is true iff $⟨f(R_A\otimes R_B),R_C^{\perp }⟩=0$ iff $⟨g(R_A),R_B\otimes R_C^{\perp }⟩=0$ iff $g(R_A)\subseteq (R_B\otimes R_C^{\perp }{)}^{\perp }$. This is true iff $g$ induces a (unique) graded algebra map $A\to B^{!}\circ C$.

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 $D$, i.e., an object for which the double dual embedding ${\delta }_A:A\to [[A,D],D]$ is a natural isomorphism. The monoidal unit is the polynomial algebra $k[x]$, and the dualizing object $D$ is $k[x{]}^{!}=k[\epsilon ]/({\epsilon }^2)$, the algebra of Grassmann numbers. We then have $A^{!}\cong [A,D]$ for any quadratic algebra $D$.

References

Y. Manin, Some remarks on Koszul algebras and quantum groups, Annales de l’institut Fourier, 37 no. 4 (1987), p. 191-205 (pdf)