A Realization Of The Quantum Queer Superalgebra And Its Representations

This paper is focused on the quantum queer superalgebra,which is the quantized algebra of the enveloping superalgebra of the queer Lie superalgebra.We introduce the twisted quantum queer Schur superalgebra and achieve the multiplication formulas on the natural basis of it.With these formulas,we construct the BLM type realization of the queer superalgebra and the quantum queer superalgebra.Based on the BLM type realization of the queer superalgebra,we achieve the integral Schur-Weyl-Sergeev duality over the integer ring,which connects the queer superalgebra,the Sergeev superalgebra and the twisted queer Schur superalgebra,and extend the result of Sergeev from the complex field to the integer ring.Based on the BLM type realization of the quantum queer superalgebra.we achieve the quantum integral Schur-Weyl-Olshanski duality over the integral Laurent polynomial ring,which connects the quantum queer superalgebra,the Hecke-Clifford superalgebra and the twisted quantum queer Schur superalgebra,and extend the result of Olshanski from the complex rational function field to integral Laurent polynomial ring.In the meanwhile,we study the representation theory of the quantum queer superalgebra,and achieve some properties of the highest weight modules.We also obtain the decomposition the regular module of the twisted quantum queer Schur superalgebra as a direct sum of irreducible submodules.