Quaternionic Quasideterminants and Determinants

Quasideterminants of noncommutative matrices introduced in [GR, GR1] have proved to be a powerfull tool in basic problems of noncommutative algebra and geometry (see [GR, GR1-GR4, GKLLRT, GV, EGR, EGR1, ER,KL, KLT, LST, Mo, Mo1, P, RS, RRV, Rsh, Sch]). In general, the quasideterminants of matrix A = (aij) are rational functions in (aij)’s. The minimal number of successive inversions required to express an rational function is called the height of this function. The ”inversion height” is an important invariant showing a degree of ”noncommutativity”. In general, the height of the quasideterminants of matrices of order n equals n−1 (see [Re]). Quasideterminants are most useful when, as in commutative case, their height is less or equal to 1. Such examples include quantum matrices and their generalizations, matrices of differential operators, etc. (see [GR, GR1-GR4, ER]). Quasideterminants of quaternionic matrices A = (aij) provide a closely related example. They can be written as a ratio of polynomials in aij’s and their conjugates. In fact, our Theorem 3.3 shows that any quasideterminant of A is a sum of monomials in the aij ’s and the āij ’s with real coefficients. The theory of quasideterminants leads to a natural definition of determinants of square matrices over noncommutative rings. There is a long history of attempts to develop such a theory. These works have resulted in a number of useful generalizations of determinants to special classes of noncommutative rings, e.g. [B, C, Ca, Di, Dy1, F, H, KS, O, R, SK, W] a.o. to noncommutative determinants. In all these cases noncommutative determinants can be expressed as products of quasideterminants. It shows that quasideterminants tend to be a more fundamental notion in noncommutative algebra. In particular, studies of quaternionic linear algebra initiated by Hamilton has resulted in notions of determinants due to Study and Moore and their applications [Ad, Al, As, Dy, Dy1]. We show here that these definitions may be obtained from a general procedure as products of quasideterminants. A good expostion of quaternionic determinants can be found in [As].