Constructing the quantum queer supergroup using Hecke-Clifford superalgebras

In [14] , we use certain special elements and their commutation relations in the Hecke-Clifford algebras H_r,R^c to derive some fundamental multiplication formulas associated with the natural bases in queer q -Schur superalgebras Q_q(n,r;R) introduced in [17] . Here a natural basis element is defined by a special element T_A⋆ in H_r,R^c associated with a pair of certain n×n matrices A^⋆=(A^0¯|A^1¯) over N with entries sum to r . The definition of T_A⋆ consists of an element c_A⋆ in the Clifford superalgebra and an element T_A in the Hecke algebra, where A=A^0¯+A^1¯. Note that all T_A can be used to define the natural basis for the corresponding q -Schur algebra S_q(n,r). This paper is a continuation of [14] . We start with standardized queer υ -Schur superalgebras Q_υ^s(n,r), for R=Z[υ,υ⁻¹] and q=υ², and their natural bases. With the υ -Schur algebra S_υ(n,r) at the background, the first key ingredient is a standardization of the natural basis for Q_υ^s(n,r) and their associated standard multiplication formulas. By introducing some long elements of finite sums, we then extend the formulas to these long elements which allow us to explicitly define Q(υ)-superalgebra homomorphisms ξ_n,r from the quantum queer supergroup U_v(q_n) to queer q -Schur superalgebras Q_υ^s(n,r), for all r≥1. Finally, taking limits of long elements yields certain infinitely long elements as formal infinite series which eventually lead to a new construction for U_v(q_n).