| The theory of quantum groups and their representations is a new branch in math-ematics since 1980s.It has a rich background in physics and is closely related to other branches of mathematics,including Lie algebra,representation theory and so on.This dissertation primarily focuses on the connectedness of quantum general linear supergroup.The connectedness of an affine algebraic group is determined by its Zariski topology,which can be solely described in terms of its coordinates rings.Moreover,according to the works of Takeuchi and his collaborators,an affine algebraic group determines two Hopf algebras:the coordinate algebra and the hyperalgebra.They form a Hopf dual pair,which gives a canonical Hopf algebra homomorphism from the coordinate algebra to the finite dual of the hyperalgebra.Under this point of view,the connectedness of an affine algebraic group is equivalent to the injectivity of this canonical homomorphism.Quantum groups are generalizations of the algebraic structures of algebra groups,which can be characterized a Hopf pair of Hopf algebra,the quantized coordinate algebra and the quantized enveloping algebra of a Lie algebra.Since the lack of a Zariski topology on a quantum group,only a few geometric tools were emerged into the study of quantum groups.Nonetheless,the pure algebraic description of connectedness of an affine group can be generalized to a quantum group.Namely,a quantum group is said to be connected if the canonical homomorphism from the quantized coordinate algebra to the finite dual of quantized enveloping algebra is injective.With this viewpoint,Takeuchi showed that the quantum general linear group GL_q(n)is(absolutely)connected.We will generalize this result to the quantum general linear supergroup GL_q(m|n),which is characterized by the quantum coordinates superalge-bra A(GL_q(m|n))and the quantum enveloping superalgebra U(GL_q(m|n)).We intro-duce a triangular decomposition of GL_q(m|n).The positive part T_q~+(m|n)is determined by a subalgebra U(T_q~+(m|n))of U(GL_q(m|n))and the corresponding quotient algebra A(T_q~+(m|n))of A(GL_q(m|n)).We firstly prove the connectedness of T_q~+(m|n)by explic-itly computing the evaluation of the Hopf pairing on appropriate bases of A(T_q~+(m|n))and U(T_q~+(m|n)).The key idea is rearranging the generators of U(T_q~+(m|n))to find a ba-sis of U(T_q~+(m|n))that is dual to a fixed PBW basis of A(T_q~+(m|n)).Finally,we combine the connectedness of T_q~+(m|n)and T_q~-(m|n)via the comultiplication of A(GL_q(m|n))and conclude the connectedness of GL_q(m|n). |
