Actions of pointed Hopf algebras on quantum torus
by
V. A. Artamonov
Moscow State University

Let Ø_q the associative algebra with a unit element over a field k generated by elements X₁^±1, ¼, X_r^±1, X_r+1, ¼, X_n subject to defining relations X_iX_i^-1=X_i^-1X_i=1,     1\leqslant i \leqslant r and X_iX_j=q_ijX_jX_i,     1\leqslant i, j\leqslant n. Here q_ij are element of k such that q_ii=q_ijq_ji=1 for all i, j. The algebra Ø_q is an algebra of quantum polynomials. It is a generic algebra of quantum polynomials if all multiparameters q_ij with 1\leqslant i < j \leqslant n, are independent in the multiplicative abelian group k^* of the field k. The algebra Ø_q can be considered as a coordinate algebra of product of a quantum torus and a quantum plane [, ].

In non-commutative algebraic geometry an action of a "finite quantum group" on a quantum space means an action of some finite dimensional Hopf algebra H on Ø_q. In my talk I shall consider the case when H is a pointed Hopf algebra. It is shown that there exists a class of standard cocommutative pointed finite dimensional Hopf algebra acting on Ø_q. An action of H is a composition of Hopf algebra homomorphism from H onto some standard algebra and an action of the standard one on Ø_q. It follows that an action of H on Ø_q is reduced to action of some automorphism group and some skew derivations of Ø_q. Moreover the subalgebra of invariants of this action is left and right Noetherian and Ø_q is finitely generated left and right module over the subalgebra of invariants.

In the case when the number n of variables is at least three was considered in []. It is interesting to mention that in the case n=r=2 a classification of automorphism group of Ø_q is similar to a classification of two-dimensional crystallographic groups.

References

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Artamonov V. A., Pointed Hopf algebras acting on quantum polynomials, J. Algebra 259(2003), N 2, 323-352.

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Brown K. A., Goodearl K. R., Lecture on algebraic quantum groups. Birkhäuser, Basel, Boston, 2002.

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Montgomery, S.: Hopf Algebras and Their Actions on Rings, Regional Conf. Ser. Math. Amer. Math. Soc., Providence RI, 1993.

Date received: December 5, 2005