Automorphisms of Regular Algebras

Manin associated to a quadratic algebra (quantum space) the quantum matrix group of its automorphisms. This Talk aims to demonstrate that Manin’s construction can be extended for quantum spaces which are nonquadratic homogeneous algebras. Here given a regular Artin-Schelter algebra of dimension 3 we construct the quantum group of its symmetries, i.e., the Hopf algebra of its automorphisms. For quadratic Artin-Schelter algebras these quantum groups are contained in the the classification of the GL(3) quantum matrix groups due to Ewen and Ogievetsky. For cubic Artin-Schelter algebras we obtain new quantum groups which are automorphisms of cubic quantum spaces. All vector spaces and algebras are over a ground field K of characteristics 0. We adopt the Einstein convention of summing on repeated an upper and a lower indices except when these are in brackets, e.g. there is no summation in Q (i). 1 N -Homogeneous Algebras A N -homogeneous algebra is an algebra of the form [2], [3] A = A(E,R) = T (E)/(R) where E is finite dimensional vector space, T (E) is the tensor algebra of E and (R) is the two-sided ideal generated by a vector subspace R ⊂ E . Since the space R is homogeneous by ascribing the degree 1 to the generators in E one obtains that the algebra A is graded, A = ⊕n≥0An, generated in degree 1, A0 = K and such that the degrees An are finite-dimensional vector spaces. The dual A of A = A(E,R) is defined to be the N -homogeneous algebra A = A(E, R) where E is the dual vector space of E and R ⊂ E = (E ) is the annhilator of R, R(R) = 0 . One has (A) = A.