In some noncommutative algebras there are elements which can be in terms of a distinguished generators (resembling matrix entries of a generic matrix) written by formulas analogous to the formula for the usual determinant.

In some cases they have special properties, for example being central (e.g. in the case of Yangian).

In the case of quantum linear groups, as the usual determinant is related to the exterior algebra, the corresponding quantum determinant is related to a quantum exterior algebra. In some cases, quantum determinants may be written as products of quasideterminants, similarly to the usual determinant.

Historically, quantum determinants first appeared in the works of Leningrad school of quantum integrable systems.

Literature

D. I. Gurevich, P. A. Saponov, Determinants in quantum matrix algebras and integrable systems, Theor Math Phys 207, 626–639 (2021) doi

E. E. Demidov, Multiparameter quantum deformations of the group GL(n), (Russian) Uspehi Mat. Nauk 46 (1991), no. 4 (280) 147–148; translation in Russian Math. Surveys 46 (1991) no. 4, 169–171.

M. Hashimoto, T. Hayashi, Quantum multilinear algebra, Tohoku Math. J. 44 (1992) 471–521 doi

Yu. I. Manin, Quantum groups and non-commutative geometry, CRM, Montreal 1988.

B. Parshall, J. Wang, Quantum linear groups, Mem. Amer. Math. Soc. 89 (1991), No. 439, vi+157 pp.

For RTT algebras:

Dmitry V. Talalaev, RE-algebras, quasi-determinants and the full Toda system, arXiv:2406.07434