quantum Gauss decomposition

Generators of the quantized function algebra GG of FRT-type, for example the quantum linear groups GL q(n,k)GL_q(n,k), SL q(n,k)SL_q(n,k) are forming a matrix TT of generators with entries in GG and satisfying the matrix identity for the coproduct ΔT=TT\Delta T = T\otimes T; in fact, GG is a matrix Hopf algebra with basis TT. Using quasideterminants, one can decompose the matrix TT as wUAwUA where ww is a permutation matrix, UU upper triangualr unidiagonal and AA lower triangular; for this one needs to enlarge GG to include the inverses which are needed to find the solution. This can be done for example in the quotient skewfield of GG, 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 T=wUAT = wUA. 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 n!n! Ore sets S wS_w generated by certain “flag” quantum minors, so that the equation T=wUAT=wUA has a solution in the localized algebra G w:=S w 1GG_w :=S_w^{-1}G. One defines the quotient Hopf algebra BB from GG by letting the Hopf ideal generated by the entries of TT above diagonal to zero; let the projection be p:GBp:G\to B. This projection induces a right BB-coaction (idp)Δ G:GGB(id\otimes p)\circ\Delta_G:G\to G\otimes B on GG (“quantum subgroup acts on a quantum group”) which has a compatibility property that it uniquely extends to G wG_w as an algebra map (this geometrically means that SpecG wSpec G_w is a SpecBSpec B-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 G wG_w containing the entries of matrix UU from g=wUAg=wUA, here the invariance is with respect to the natural action of BB 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 Spec(G)Spec(G)/Spec(B)Spec(G)\to Spec(G)/Spec(B). Namely, G wG_w is isomorphic as a right BB-comodule algebra to a smash product algebra (G w) coHB(G_w)^{co H}\sharp B with the canonical BB-coaction, where the isomorphism is induced by the unique homomorphism of algebras (it is nontrivial that such exists!) γ w:BG w\gamma_w:B\hookrightarrow G_w such that b j ib^i_j which is the projection of the (i,j)(i,j)-entry of TT in BB maps to the (i,j)(i,j)-th entry a j ia^i_j of the matrix A=A wA=A_w defined by the decomposition T=wUAT=wUA; the left action of BB on (G w) coB(G_w)^{co B} is given by bu=γ(b (1))uγ(S Bb (2))b\triangleright u = \sum\gamma(b_{(1)})u\gamma(S_B b_{(2)}) where S B:BB op copS_B:B\to B^{cop}_{op} is the antipode of BB. Thus the true meaning of quantum Gauss decomposition is the local trivialization (trivial bundle=smash product) of quantum principal bundle realized by a right BB-comodule algebra GG. Locality is in the sense of the cover by coaction-compatible local trivializations. This result is sketched in

  • Zoran Škoda, Localizations for construction of quantum coset spaces, math.QA/0301090, in “Noncommutative geometry and Quantum groups”, W.Pusz, P.M. Hajac, eds. Banach Center Publications vol.61, pp. 265–298, Warszawa 2003.

and the Ore condition for not only S wS_w, but any (multiplicative set) of quantum minors is proved in

  • Zoran Škoda, Every quantum minor generates an Ore set, International Math. Res. Notices 2008, rnn063-8; math.QA/0604610.

In the case of SL q(2)SL_q(2) the local trivialization has been applied to compute the resolution of unity formula for the SU q(2)SU_q(2)-coherent states in

  • Zoran Škoda, Coherent states for Hopf algebras, Letters in Mathematical Physics 81, N.1, pp. 1-17, July 2007. (earlier arXiv version: math.QA/0303357 )

The general picture of actions, coset spaces, compatibility with localizations, localized coinvariants and equivariance in noncommutative algebraic geometry is outlined in

  • Zoran Škoda, Some equivariant constructions in noncommutative algebraic geometry, Georgian Mathematical Journal 16 (2009), No. 1, 183–202, arXiv:0811.4770

The analogous results for the case of multiparametric GL P,Q(n)GL_{P,Q}(n) 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

  • Zoran Škoda, A simple algorithm for extending the identities for quantum minors to the multiparametric case arXiv;0801.4965.

See also Gauss decomposition.

Revised on December 16, 2009 19:44:05 by Toby Bartels (