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Quantized Heisenberg space. (English) Zbl 0980.17006

Let \(N\) be an integer and let \(q\) be a primitive root of unity in the field \(\mathbb{C}\) of complex numbers. The authors study the algebra \(F_q(N)\) introduced by L. D. Faddeev, N. Yu. Reshetikhin and L. A. Takhtadzhan [Algebraic Analysis 1, 129-139 (1989; Zbl 0677.17010)]. The Poisson structure defined by \(F_q (N)\) on \(\mathbb{C}^N\), as explained by C. De Concini and C. Procesi [in Lect. Notes Math. 1565, 31-140 (1993; Zbl 0795.17005)], is described. The theory developed by De Concini and Procesi is then applied to determine the degree and the center of the algebra \(F_q(N)\), when \(q\) is of odd order; these considerations rely on the relationship between \(F_q(N)\) and an associated quasipolynomial algebra \(\overline{F_q(N)}\).
The irreducible representations of the algebra \(F_q(N)\) are studied and it is shown that it satisfies the De Concini-Kac-Procesi conjecture [C. De Concini, V. G. Kac and C. Procesi, J. Am. Math. Soc. 5, 151-189 (1992; Zbl 0747.17018)], which relates the representation theory of \(F_q(N)\) with the Poisson structure already mentioned.
The authors explain how the representation theory and symplectic leaves of the algebra \(F_q(N)\) are related to those of the algebra considered by S.-Q. Oh [in J. Algebra 174, 531-552 (1995; Zbl 0828.17011)]; they conclude that this algebra satisfies as well the De Concini-Kac-Procesi conjecture.
The last section studies annihilators of simple modules in the case where \(q\) is generic. It is shown that, for generic \(q\), a description of the primitive ideals of \(F_q(N)\) can be given in a way similar to Oh’s description by means of admissible sets.

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
16R20 Semiprime p.i. rings, rings embeddable in matrices over commutative rings
37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
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