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Riemann surfaces, Lie algebras and mathematical physics
December 9, 2011 17:00, Moscow, Independent University of Moscow, room 309

From Braided Geometry to Integrable systems

D. I. Gurevich

Université de Valenciennes et du Hainaut-Cambrésis

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Abstract: By Braided Geometry I mean a theory dealing with braidings (i.e. solutions of the Quantum Yang-Baxter Equation) playing the role of usual flips (or super-flips). The main object of Braided Geometry is the so-called Reflection Equation algebra associated to a given braiding. This algebra can be treated as an analog of the enveloping algebra U(gl(m|n)). Besides, for a matrix $L$ coming in its definition there is a version of the Cayley-Hamilton identity. This enables one to introduce “eigenvalues” of the matrix $L$. Also, a version of partial derivatives can be defined on this algebra via a deep generalization of the Woronowicz's differential calculus on a pseudogroup.
By assuming the initial braiding to be a deformation of a super-flip, and passing to the limit $q=1$ we get partial derivatives on the algebra $U(gl(m|n))$ (with a very surprising modification of the Leibniz rule). This enables one to define analogs of the Laplacian operator and its higher counterparts on the algebra $U(gl(m|n))$. Such operators can be also defined on the braided deformation of this algebra (i.e., the corresponding Reflection Equation algebra). By restricting these operators to the center of the algebra in question and by expressing them via the aforementioned “eigenvalues”; one can get a family of operators (hopefully, difference ones) in involution. They are two-parameter deformations of operators which are gauge equivalent to Calogero-Moser ones.
I plan to exhibit the simplest example in details.

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