nLab
quadratic algebra

Definition

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

AT(V)/IA \cong T(V)/I

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

A morphism of quadratic algebras is just a morphism as graded algebras. Alternatively, a morphism of quadratic algebras f:ABf: A \to B determines and is determined by a linear map f 1:A 1B 1f_1: A_1 \to B_1 such that (ff)(R A)R B(f \otimes f)(R_A) \subseteq R_B.

Examples

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

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

  3. The Grassmann algebra Λ(V)\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 GL 2GL_2, the underlying algebra is quadratic. See the reference by Manin for further examples.

Quadratic dual and Manin’s monoidal products

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

V *V *(VV) *i *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 AA is usually denoted A !A^!. There is a canonical isomorphism AA !!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:

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

() :Sub(V)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 σ(R A 1 W *W *)\sigma(R_{A}^\perp \otimes 1_{W^* \otimes W^*}) and σ(1 V *V *R B )\sigma(1_{V^* \otimes V^*} \otimes R_{B}^\perp) is σ(R A R B )\sigma(R_{A}^\perp \otimes R_{B}^\perp). Applying () (-)^\perp to this, one obtains the join of σ(R A1 WW)\sigma(R_A \otimes 1_{W \otimes W}) and σ(1 VVR B\sigma(1_{V \otimes V} \otimes R_B which is σ(R A1 WW)+σ(1 VVR B\sigma(R_A \otimes 1_{W \otimes W}) + \sigma(1_{V \otimes V} \otimes R_B.)

Theorem

There is a natural isomorphism QAlg(AB,C)QAlg(A,B !C)QAlg(A \bullet B, C) \cong QAlg(A, B^! \circ C).

Proof

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

XY X,Y=0\frac{X \subseteq Y^\perp}{\langle X, Y \rangle = 0}

where XV *X \subseteq V^*, YVY \subseteq V are subspaces and ,:V *Vk\langle -, - \rangle: V^* \otimes V \to k is the usual pairing to the ground field kk.

Let (U,R A)(U, R_A), (V,R B)(V, R_B), (W,R C)(W, R_C) define the quadratic algebras, and suppose that f:UVWf: U \otimes V \to W and g:UV *Wg: U \to V^* \otimes W correspond to one another under the adjunction

Vect k(UV,W)Vect k(U,V *W)Vect_k(U \otimes V, W) \cong Vect_k(U, V^* \otimes W)

Now ff induces a (unique) graded algebra map ABCA \bullet B \to C iff f(R AR B)R Cf(R_A \otimes R_B) \subseteq R_C, which is true iff f(R AR B),R C =0\langle f(R_A \otimes R_B), R_{C}^\perp \rangle = 0 iff g(R A),R BR C =0\langle g(R_A), R_B \otimes R_{C}^\perp \rangle = 0 iff g(R A)(R BR C ) g(R_A) \subseteq (R_B \otimes R_{C}^\perp)^\perp. This is true iff gg induces a (unique) graded algebra map AB !CA \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 DD, i.e., an object for which the double dual embedding δ A:A[[A,D],D]\delta_A: A \to [[A, D], D] is a natural isomorphism. The monoidal unit is the polynomial algebra k[x]k[x], and the dualizing object DD is k[x] !=k[ε]/(ε 2)k[x]^! = k[\varepsilon]/(\varepsilon^2), the algebra of Grassmann numbers. We then have A ![A,D]A^! \cong [A, D] for any quadratic algebra DD.

References

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