Algebras of functions on quantum groups. Part I by Leonid I. Korogodski

By Leonid I. Korogodski

The e-book is dedicated to the examine of algebras of capabilities on quantum teams. The authors' method of the topic is predicated at the parallels with symplectic geometry, permitting the reader to exploit geometric instinct within the concept of quantum teams. The booklet contains the speculation of Poisson Lie teams (quasi-classical model of algebras of features on quantum groups), an outline of representations of algebras of features, and the idea of quantum Weyl teams. This booklet can function a textual content for an advent to the speculation of quantum teams

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Pr(n-l). But pm - 1 = (pr - l)t, so, in the cyclic group Zj(pnr - 1) raising elements to the power t is a surjective homomorphism onto the cyclic subgroup Zj(pr - 1). It follows that when F is a finite field any central simple division algebra D has a maximal subfield lK with cyclic Galois group. But we have already seen that, in this case, A(lK, T, K) :::: A(lK, T, N(k)K) so A(lK, T, K) :::: A(ll(, T, 1) = Mn(IB') and there are no non-commutative central simple F-algebras. On the other hand the discussion above shows that HJ(Gal(lKjF); lK e) corresponds to the central simple F algebras which contain lK as a maximal subfield, so HJ(Gal(lKjF); lK e) = 0 when F is a finite field.

Indeed, let [gr ][gr-Il· .. [gil E F(J) be any word. Then we have [glg2· .. gr rl [gr ][gr-Il· .. [gil = ([gl ... gr rl [gr ][gl ... gr-Il)([gl ... gr_Il- 1 [gr-Il[gl ... gr-2D ... [glg2D([glg2r l [g2] [gil) and this belongs to M. Hence the original word belongs to the coset [gl ... gr ]M. It remains to show that the Ig, g'l are free generators. To this end consider the space Y = VjEJ S] which is defined as the union of a collection of disjoint circles obtained by identifying all their base-points to a single point *.

Then a generator for Gal(lKjF) is the map x f-+ xpr. Hence the norm map has the form x f-+ xt where t = 1 + pr + p2r + ... pr(n-l). But pm - 1 = (pr - l)t, so, in the cyclic group Zj(pnr - 1) raising elements to the power t is a surjective homomorphism onto the cyclic subgroup Zj(pr - 1). It follows that when F is a finite field any central simple division algebra D has a maximal subfield lK with cyclic Galois group. But we have already seen that, in this case, A(lK, T, K) :::: A(lK, T, N(k)K) so A(lK, T, K) :::: A(ll(, T, 1) = Mn(IB') and there are no non-commutative central simple F-algebras.

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