Characteristic Polynomials Of Quantized Sl(2,c)and Its Representations

The notion of projective spectrum was defined by Professor Rongwei Yang.For a tuple A=(A1,...,An)of elements in a unitl algebra B,its projective spectrum is defined as P(A)={z=(z1,...,zn)∈Cn:A(z)＝z1A1+…+ znAn is not invertible}.On this basis,Prof.Hu Zhiguang and others studied the characteristic polynomials of finite generated groups and finite-dimensional Lie algebras,and obtained a series of profound results.Dr.Liu Shoumin and others proved that the characteristic polynomials of sl(2,C)correspond to their representations one-to-one relationship,and endowed the characteristic polynomial of sl(2,C)with a monoid structure.This paper mainly studies the characteristic polynomial and its representation of the quantum envelope algebra Uq(sl(2,C))of sl(2,C),and establishes a one-to-one correspondence between its characteristic polynomial and the linearized characteristic polynomial.It is proved that there is a monoid structure on the characteristic polynomial of linearized Uq(sl(2,C))and proved that there is also a monoid structure on the characteristic polynomial of Uq(sl(2,C)),and the representation of Uq(sl(2,C))is isomorphic if and only if the characteristic polynomial of the corresponding representation are equal.