The Basis And The Center Of The Universal Q-analog Of The Bannai-Ito Algebra

Let C denote a complex field and fix a nonzero q?C such that q4?1.The universal q-analog of the Bannai-Ito algebra over C,denoted by ?,is an associative algebra defined by generators A,B,C and relations asserting that each of qBC+ q-1CB-A,qCA+ q-1AC-B,qAB + q-1BA-C is central in A.Let ?,? and ? denote the above central elements respectively.It is known that the universal q-analog of the Bannai-Ito algebra is tightly related to the Askey-Wilson algebra,the Bannai-Ito algebra and the q-analog of the Bannai-Ito algebra.In this paper,we mainly study the basis and the center of ?,and the main results are as follows.(i)The set {AiBjCk?r?s?t|i,j,k,r,s,t?N} gives the basis of ?.(ii)When q is not a root of unity,the center of ? can be generated by ?,?,?,?,where ?denotes the Casimir element of ?.(iii)When q is a primitive lth root of unity,the center of ? can be generated by Th(A),Th(B),Th(C),?,?',?',?'.where A=i(q2-q-2)A,B =i(g2-q-2)B,C= i(q2-q-2)C,?'= i(q2-q-2)?,?'=i(q2-q-2)?,?'=i(q2-q-2)?,i2=-1,Th(x)= ?j=0[h/2](-1)j((jh-j)+(j-1h-j-1)xh-2j,h=2l if l is odd;h =l/2 if l is even and l/2 is even;h = l if l is even and l/2 is odd.