Abstract :
We define a family of graded quadratic algebras Aσ (on 4 generators) depending on a fixed nonsingular quadric Q in image3, a fixed line L in image3 and an automorphism σ set membership, variant Aut(Q union or logical sum L). This family contains imageq(M2(image)), the coordinate ring of quantum 2 × 2 matrices. Many of the algebraic properties of Aσ are shown to be determined by the geometric properties of {Q union or logical sum L, σ}. For instance, when Aσ = imageq(M2(image)), then the quantum determinant is the unique (up to a scalar multiple) homogeneous element of degree 2 in imageq(M2(image)) that vanishes on the graph in image3 × image3 of σQ but not on the graph of σL. Following results of M. Artin, J. Tate, and M. Van den Bergh ("The Grothendieck Festschrift," Birkhäuser, Basel, 1990; and Invent. Math.106, 1991, 335-388), we study point and line modules over the algebras Aσ, and find that their algebraic properties are consequences of the geometric data. In particular, the point modules are in one-to-one correspondence with the points of Q or L, and the line modules are in bijection with the lines in image3 that either lie on Q or meet L. In the case of imageq(M2(image)), when q is not a root of unity, the quantum determinant annihilates all the line modules M(l) corresponding to lines l subset of Q; the determinant generates the whole annihilator for such l subset of Q if and only if l and L = empty set︀.
