Stratification of prime spectrum of quantum solvable algebras

1 Introduction. We consider the class of Noetherian ring, appeared as a result of quantization of algebraic groups and their representations within framework of mathematical physics. One set up the problem of description of prime and primitive spectrum of these rings. This problem has been solved first for algebras of low dimension , then for the case GL q (n), later for general case of regular functions on quantum semisimple group [J]. Simultaneously some authors consider the examples of Quantum Weyl algebra, Quantum ortogonal and symplectic spaces, Quantum Heisenberg algebra. One can present new examples considering subalgebras, fartor algebras of the above algebras and also their deformations. It was noted that prime ideals in these algebras have common properties like: all prime ideals are comletely prime, there exists a stratification of spectrum by locally closed sets, the hypothesis on fields of fractions of prime factors is true (quantum Gelfand-Kirillov conjecture) and others. In the survey [G] all known examples are considered, the common properties are extracted and the problem of creating the general theory covering all these examples is setted up. The classical analog of this proposed theory is the theory of prime ideals in universal enveloping algebra of a solvable Lie algebra [D],[BGR]. In this paper the author presents his version of the system of axioms, covering the listed examples. First, all mentioned algebras are solvable (see Definition 1.2). The class of solvable algebras obeying Conditons Q1-Q3 covers " quantum " algebras, " classical " algebras (universal enveloping algebra of solvable Lie algebra) and there are " mixed " algebras [AD]. Considered by the author Condition Q4 may be treated as some condition for quantum algebra to be " algebraic ". The classical analog of this property is as follows. The Lie algebra of an algebraic group, defined over Z, admits p-structure after reduction modulo prime number p. Its universal enveloping algebra becomes finite dimensional over the center. Condition Q4 helps to isolate " quantum " algebras (Definition 2.8). All Conditions Q1-Q4 are easily checkable. The Condition Q4 in the examples is derived from the property that quantum algebras are finite over center at roots of 1. In the paper the finite stratification of spectrum of quantum solvable algebra, obeying Conditions Q1-Q4, is constructed (Theorems 3.4, 3.10) The hypothesis on