Classification of noncommutative conics associated to symmetric regular superpotentials

Abstract

Let S be a 3-dimensional quantum polynomial algebra, and f ∈ S 2 a central regular element. The quotient algebra A = S/(f ) is called a noncommutative conic. For the conic A, there is a finite dimensional algebra C(A) which determines the singularity of A. Since we try to find some pattern of the conic A, calculating the algebra C(A) is an important step. In this paper, we mainly focus on the quantum polynomial algebras determined by symmetric regular superpotentials except Type EC, and calculate the algebras C(A). It turns out in our cases, the set of isomorphic classes of algebras C(A) coincides with the those of 4-dimensional commutative Frobenius algebras.

Noncommutative conics in Calabi-Yau quantum projective planes

“…By Lemma 2.8, we can check that (σ| E 0 ) 2 = id for some irreducible component E 0 of E if and only if σ 2 = id if and only if S is isomorphic to one of the algebras of Type P, S, S', NC, EC in Table 2 above. On the other hand, if S = k[x, y, z] (Type P), then clearly, Z(S) 2 = S 2 = 0, and if S is one of the algebras of Type S, S', NC, EC in Table 2 above, then Z(S) 2 = kx 2 + ky 2 + kz 2 = 0 by [8,Lemma 3.6].…”

“…In this section, we classify noncommutative conics A = S/(f ) up to isomorphism where S are 3-dimensional Calabi-Yau quantum polynomial algebras and 0 = f ∈ Z(S) 2 . It is not easy to check which S has a property that Z(S) 2 = 0 by algebraic calculations especially if S does not have a PBW basis (see [8]), so we use a geometric method in this paper. Lemma 3.1.…”

“…If #(K f ) = 1, then #(Spec C(A)) = #(K f ) = 1 by Proposition 4.4, so C(A) is isomorphic to either k[u, v]/(u 2 , v 2 ) or k[u]/(u 4 ) (cf. [8,Corollary 4.10]). We will show that this is not the case.…”