Actions of finite quantum groups on quantum polynomials
by
V. A. Artamonov
Moscow State University

A noncommutative analog of a polynomial algebra over a field k is a quantum polynomial algebra \Lambda generated by elements

X1, X1-1, ... , Xr, Xr-1, Xr+1, ... , Xn

subject to defining relations X_iX_i^-1=X_i^-1X_i=1 if 1 <= i <= r and X_iX_j=q_ijX_jX_i if 1 <= i, j <= n. Elements q_ij in k^* satisfy the relations q_ii=q_ijq_ji=1 for all 1 <= i, j <= n, and r is an integer such that 0 <= r <= n. The algebra \Lambda can be considered as a coordinate algebra of a direct product of quantum torus T^r and a quantum affine space A^n-r [].

The aim of this study is a classification of actions of finite quantum groups on T^r×A^n-r in terms of H-module structure on the algebra \Lambda where H is a finite dimensional pointed Hopf algebra [].

There are some examples of actions of some special pointed finite dimensional Hopf algebras on \Lambda. Namely let U be a subgroup of a finite index in Zⁿ and [k(Zⁿ/U)]^* the dual Hopf algebra of the group algebra k(Zⁿ/U) of the finite factorgroup Zⁿ/U. For any f in [k(Zⁿ/U)]^* we put f o X^v = f(v+U)X^v for any monomial X^v, where v in Zⁿ is a multi-index. Hopf algebra [k(Zⁿ/U)]^* is provided with an involutive automorphism f --> [f\tilde] where [f\tilde](v)=f(-v). So we can form a smash product [k(Zⁿ/U)]^*\sharp k<\xi> with the cyclic group <\xi> of order 2. If r=n then the action of [k(Zⁿ/U)]^* on \Lambda can be extended to an action of the smash product where \xi o X^v=X^-v for all v in Zⁿ. Suppose that q_ij,   1 <= i < j <= n, are independent in the multiplicative group k^* of the field k and H is a pointed finite dimensional Hopf algebra H such that if r=n=2 then dimH is not divisible neither by 4 nor by 3. Let \Lambda be a left H-module algebra. Then there exists a subgroup U in Zⁿ of a finite index and a Hopf algebra homomorphism \Psi: H --> [k(Zⁿ/U)]^*\sharp k<\xi> such that the action of H in \Lambda is a product of \Psi and the mentioned action of [k(Zⁿ/U)]^*\sharp k<\xi > on \Lambda. If \Psi is not surjective then its image is equal to [k(Zⁿ/U)]^*. It is always the case if r < n.

References

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

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Montgomery S. Hopf algebras and their actions on rings // Regional Conference Series in Mathematics. v. 82 Amer. Math. Soc. Providence R. I., 1993.

Date received: January 4, 2003