Abstract A Z -subalgebra U ₙ^ (n) (Z= Z, ^-1) for the i -quantum group {U}^ (n) over the field Q () is constructed by two of the authors ‘A new realisation of the i -quantum group U^ (n) ’, J. Pure Appl. Algebra 226 (1) (2022), Paper no. 106793, 27 pages, Theorem 6. 5, using a Beilinson–Lusztig–MacPherson (BLM) type realisation. In this paper, we construct bases for U ₙ^ (n), including the monomial basis conjectured in ‘A new realisation of the i -quantum group U^ (n) ’, J. Pure Appl. Algebra 226 (1) (2022), Paper no. 106793, 27 pages, Remark 6. 6 (4). This proves that the Z -algebra U ₙ^ (n) is a free Z -module. Hence, U ₙ^ (n) is in fact an integral form of Lusztig type. This construction is further extended to the i -quantum hyperalgebra over a field of any characteristic. By specialising to an l th primitive root of 1 with l odd, a realisation of the quotient of modulo the ideal generated by dᵢˡ-1, for all 1 i n+1, is also given as a by-product.
Du et al. (Fri,) studied this question.