# Download e-book for iPad: Communications in Mathematical Physics - Volume 191 by A. Jaffe (Chief Editor)

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4. The right currents. Let us stress that there is a major ideological difference between our discussion of lattice current algebras and the work in [4]. e. from the variables Jn . Here the Jn ’s represent only the left-currents, and we expect also right- currents to be present in the theory. They are constructed from the variables Jn and the elements ξn ∈ Gn and hence give a physical meaning to the graph algebras. We define a family of new variables JnR ∈ Ga ⊗ KN on the lattice by setting −1 −1 Jn Nn−1,+ .

As predicted by the general theory, the elements N r ∈ Zq generate the algebra Zq when p is odd. 3. Definition of KN . Next, we turn to the definition of the lattice current algebras KN associated to a fixed modular Hopf algebra. Before entering the abstract formalism, it is useful to analyse the classical geometry of the discrete model. Our classical continuum theory contains two Lie-algebra valued fields, namely η(x) and j(x). To describe a configuration of η, for instance, we have to place a copy of the Lie-algebra at every point x on the circle.

This observation will enable us to use some of the general properties established there. The most important one among such general properties is the existence of a *operation. On the copies Gn , a *-operation comes from the structure of the modular Hopf-*-algebra G. 8) 42 A. Y. Alekseev, L. D. Faddeev, J. Fr¨ohlich, V. Schomerus where Sn = (id⊗ιn )[ (κ−1 )(κ⊗κ)R−1 ] ∈ Ga ⊗Gn ⊂ Ga ⊗KN . e. κ2 = v (cp. [4] for details). 8) can be rewritten using the elements Nn,± ∈ Ga ⊗ Gn constructed from the R-element according to Nn,+ ≡ (id ⊗ ιn )(R ) , Nn,− ≡ (id ⊗ ιn )(R−1 ) .