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:

  • The monoidal product of (V,R A)(V, R_A) and (W,R B)(W, R_B) is defined by (VW,σ(R AR B)(V \otimes W, \sigma(R_A \otimes R_B) where σ\sigma refers to the canonical interchange isomorphism

    VVWWVWVWV \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 ABA \bullet B. The monoidal unit is the free algebra on one generator in degree 1.

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

    AB=(A !B !) !A \circ B = (A^! \bullet B^!)^!

    Up to coherent isomorphism, this may be more explicitly defined by the pair (VW,σ(R A1 WW)+σ(1 VVR B))(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 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)

Created on August 21, 2010 14:40:44 by Todd Trimble (69.118.58.208)