General quantum polynomials
by
V. A. Artamonov
Moscow State University

General quantum polynomials

Vyacheslav A. Artamonov

Department of Algebra, Faculty of Mechanics and Mathematics Moscow State University, Moscow, 119899, RUSSIA

Let k be a field with a fixed matrix Q=(q_ij) in \Mat(n, k), n >= 2, whose entries satisfy the relations q_ii=q_ijq_ji=1 for all i, j. Let also r be an integer such that 0 <= r <= n. Denote by \Lambda the associative k-algebra with a unit generated by elements X₁ ^{+/- 1}, ... , X_r ^{+/- 1}, X_r+1, ... , X_n subject to defining relations X_iX_i^-1=X_i^-1X_i=1 for all 1 <= i <= r and X_iX_j=q_ijX_jX_i, for all 1 <= i, j <= n. The algebra \Lambda is a left and right Noetherian domain. Therefore it satisfies Ore condition and it has a division ring of fractions F.

The algebra \Lambda is an algebra of quantum polynomials and the elements q_ij are multiparameters. The algebra \Lambda is of a considerable interest in noncommutative algebraic geometry []. If r=0 then \Lambda is a coordinate algebra of affine quantum space Aⁿ_Q. If r=n then \Lambda is a coordinate algebra of a quantum torus []. An approach to an identification of points of of Aⁿ_Q is related to the study of prime ideals of \Lambda []. In this paper it is assumed that r=0 and the multiparameters generate a torsion-free subgroup in the multiplicative group k^* of the field k. An algorithm for a calculation of Krull dimension of \Lambda in the case r=n is presented in []. Another characterization of Krull dimension of \Lambda with r=n in terms of commuting monomials is found [].

In what follows we shall assume that the algebra \Lambda is a general algebra of quantum polynomials that is all multiparameters q_ij, 1 <= i < j <= n, are independent in the multiplicative group k^* of the field k [].

We show that all valuations on the division ring of fractions F are Abelian and all of them are classified by linear orders on Zⁿ []. In some sense this is another classification of points of Aⁿ_Q.

With respect to a lexicographic order (and corresponding valuation \nu) we can correctly define the completion F_\nu which is a division ring containing F and which is known as Malcev-Neumann power series ring.

In our talk we shall show that all finitely generated projective modules over \Lambda of a rank at least 2 are free []. It will also be shown that if \Lambda,  \Lambda' are two Morita-equivalent quantum polynomial algebras and one of them is a general one, then \Lambda =~ \Lambda' [].

Theorem 1 If n >= 3 then any injective endomorphism of \Lambda is an automorphism. If r < n, then the automorphism group of \Lambda is Abelian. If r=n >= 3 then the automorphism group of \Lambda is Abelian-by-cyclic of order 2. If r=n=2 then the automorphism group of \Lambda is Abelian-by-\SL(2, Z). If r=n=2 then a finite automorphism group of \Lambda is Abelian-by-cyclic of order 1, 2, 3, 4, 6.

The case n >= 3 in Theorem * was considered in [].

Theorem 2 Any (injective) endomorphism of F_\nu is an automorphism. It is a composition of an automorphism of multiplication of variables by scalars and a conjugation by an element (1-z), where \nu(z) > 0. Any finite subgroup of automorphisms of F_\nu is conjugate to the subgroup consisting of automorphisms of multiplications of variables by scalars.

The case n >= 3 in Theorem * was considered in [].

The subalgebra \Derint\Lambda (respectively, \Derint F_\nu) of inner derivations is always an ideal in the the Lie algebra \Der\Lambda (respectively, \Der F_\nu) of derivations of \Lambda (resp. F_\nu). We have derivation \partial₁, ... , \partial_n such that \partial_j(X_i)=\delta_ijX_i. The span L of \partial₁, ... , \partial_n is an Abelian Lie algebra of dimension n.

Theorem 3 Let \Lambda be a general quantum polynomial algebra. There is a direct decomposition of vector spaces \Der\Lambda = \Derint \Lambda \oplusL. Similarly \DerF_\nu = \Derint F_\nu \oplusL. Any finite dimensional Lie subalgebra in \Derint \Lambda and in \DerF_\nu is Abelian. Let \ch k=0 and \partial a derivation of \Lambda or of F_\nu. Suppose that there exists a nonzero polynomial f(T) in k[T] such that f(\partial)=0. Then \partial = 0.

Generalizing these results we prove

Theorem 4 Let H be a pointed Hopf algebra acting of \Lambda with n >= 3 and the field k contain a primitive root of 1 of degree dim H. Suppose that \ch k coprime with dimH. Then there is a finite automorphism group \Omega of \Lambda and a Hopf algebra epimorphism \pi:H --> k\Omega such that the action of H on \Lambda is a composition of \pi and the action of \Omega on \Lambda Moreover \pi(G(H))=\Omega.

Similar theorem holds for an action of H on F_\nu. A classification of cocommutative Hopf algebras acting on \Lambda,  n >= 3 is given in []. All these results can be viewed as a determination of finite and commutative quantum groups acting on quantum space A_Qⁿ.

References

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Artamonov V. A. Quantum Serre's conjecture. // Uspehi mat. nauk. 53(1998), N 4., p. 3-76.

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Artamonov V. A. General quantum polynomials: irreducible modules and Morita-equivalence.// Izv. RAN, ser. math. 63(1999), N 5, p. 3-36.

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Artamonov V. A. Valuations on quantum fields. // Commun. Algebra, 29(2001), N 9.

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Artamonov V. A. Actions of Hopf algebras on quantum polynomials. In the book: Representation of algebras. Coelho/Merklen Ed. Marcel Dekker Inc., NY., 2001, p. 11-20.

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Alev J., Chamarie M., Dérivations et automorphismes de quelques algèbres quantiques, Comm. Algebra 20(6) (1992), 1787-1802.

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Artamonov V. A., Wisbauer R. Homological properties of quantum polynomials. // Algebras and representation theory. - 2001. - 4, N 3, - p. 219 - 247.

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Brookes C. J. B. Crossed products and finitely presented groups. // J. Group Theory. - 2000. - 3. - c. 433-444.

[]

Demidov E. E., Quantum group, Moscow: Factorial, 1998, 127P.

[]

Goodearl K.R., Letzter E.S., Prime factor of the coordinate ring of quantum matrices, Proc. Amer. Math. Soc. 121(4) (1994), p. 1017- 1025.

[]

McConnell J.C., Pettit J.J., Crossed products and multiplicative analogues of Weyl algebras.// J. London Math. Soc. 38(1) (1988), p. 47-55.

[]

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

Date received: December 3, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caig-05.