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

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 thought of quantum teams. The ebook comprises the speculation of Poisson Lie teams (quasi-classical model of algebras of capabilities on quantum groups), an outline of representations of algebras of capabilities, and the speculation of quantum Weyl teams. This publication can function a textual content for an creation to the idea of quantum teams

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Extra resources for Algebras of functions on quantum groups. Part I

Example text

So Ox is a local ring in the sense of commutative algebra with the maximal ideal mx consisting of all rational functions representable in the form fg with f (x) = 0 = g(x). 2 can be interpreted as k[X] = ∩x∈X Ox . 3) Informally speaking the local ring Ox describes what happens ‘near the point x’. 7 Rational functions 31 germs of rational functions at x. One can think of germs as equivalence classes of pairs (U, f ), where U is an open set containing x, f is a rational function regular at all points of U , and (U, f ) ∼ (V, g) if there is an open set W ⊂ U ∩ V and f |W = g|W .

More generally, let X be an algebraic set in Pn and Y be an algebraic set in Pm . By the product of X and Y we understand ϕ(X × Y ), which we show to be algebraic. Well, if X is given 36 Affine and Projective Algebraic Sets by the equations Fα (T0 , . . , Tn ) = 0 and Y is given by the equations Gβ (S0 , . . 5) together with Fα (w0j , . . , wnj ) for 1 ≤ j ≤ m and Gβ (wi0 , . . , wim ) for 1 ≤ i ≤ n. 11 Example: Grassmann varieties and flag varieties Let V be an n-dimensional vector space. As a set, the Grassmann variety Gr (V ) (or Gr (n)) is just the set of all r-dimensional (linear) subspaces in V .

Ir . ir are Pl¨ ukker coordinates of some rdimensional subspace L ⊂ V if and only if they are not simultaneously zero and if for all i1 , . . , ir+1 , j1 , . . jr−1 = 0. jr−1 = asik Ns , s=1 where Ns does not depend on k. 5) k=1 for all s. Add the sth row to A to obtain an (r + 1) × n matrix As . 5) is, up to a sign, the expansion of the determinant of the matrix formed by the columns of As with indices i1 , . . , ir+1 along the last row. But this determinant is zero. ir are not simultaneously zero and the Pl¨ ukker relations hold.