Generators of the quantized function algebra of FRT-type, for example the quantum linear groups , are forming a matrix of generators with entries in and satisfying the matrix identity for the coproduct ; in fact, is a matrix Hopf algebra with basis . Using quasideterminants, one can decompose the matrix as where is a permutation matrix, upper triangualr unidiagonal and lower triangular; for this one needs to enlarge to include the inverses which are needed to find the solution. This can be done for example in the quotient skewfield of , but this is suboptimal. Nevertheless in some cases (and some super-analogues) the simple formulas for this decomposition in terms of quantum minors have been studied by Tolstoy, Kulish, Dobrev and others and many times rediscovered by many people and sometimes called quantum Gauss decomposition.
However, as in the classical case, more true meaning can be given to the decomposition . The following should be true for most matrix Hopf algebras (except that the Ore localizations should be replaced by Cohn localizations for big examples) and is proved for quantum linear groups. Instead working in the quotient field one can define Ore sets generated by certain “flag” quantum minors, so that the equation has a solution in the localized algebra . One defines the quotient Hopf algebra from by letting the Hopf ideal generated by the entries of above diagonal to zero; let the projection be . This projection induces a right -coaction on (“quantum subgroup acts on a quantum group”) which has a compatibility property that it uniquely extends to as an algebra map (this geometrically means that is a -invariant subset). The coinvariants of this extended coaction are so-called localized coinvariants and they provide a patch on the noncommutative coset space; namely the category of modules over the algebra of localized coinvariants is a localization of the category of quasicoherent sheaves on the quantum coset space; the algebra of localized coinvariants is the smallest invariant subalgebra of containing the entries of matrix from , here the invariance is with respect to the natural action of provided in a natural way using the Gauss decomposition as sketched below. The quantum Gauss decomposition provides the local trivialization of the quantum fiber bundle which could be symbolically denoted . Namely, is isomorphic as a right -comodule algebra to a smash product algebra with the canonical -coaction, where the isomorphism is induced by the unique homomorphism of algebras (it is nontrivial that such exists!) such that which is the projection of the -entry of in maps to the -th entry of the matrix defined by the decomposition ; the left action of on is given by where is the antipode of . Thus the true meaning of quantum Gauss decomposition is the local trivialization (trivial bundle=smash product) of quantum principal bundle realized by a right -comodule algebra . Locality is in the sense of the cover by coaction-compatible local trivializations. This result is sketched in
and the Ore condition for not only , but any (multiplicative set) of quantum minors is proved in
In the case of the local trivialization has been applied to compute the resolution of unity formula for the -coherent states in
The general picture of actions, coset spaces, compatibility with localizations, localized coinvariants and equivariance in noncommutative algebraic geometry is outlined in
The analogous results for the case of multiparametric can be reduced to the one-parametric case using the twisting due Artin and Tate, and reinterpreted for a usage in quantum minor calculations in
See also Gauss decomposition.