Курс лекций по теме: «Дилогарафмические тождества» с участием профессора Кириллова А.Н.,Университет Киото, Япония
Тема:
Introduction to Dilogarithm Identities, Rigged Con gurations and Fomin-
Kirillov algebras
Abstract
The Dilogarithm function
Li 2(x) = Σ n_ 1 x 2 n 2= ∫ 0 ;xlog(1 z) z dz;0 < x <1;
had been introduced by L. Euler more than 250 years ago, and since
that time the Dilogarithm function has been extensively studies by
many mathematicians and physicists including N. Abel, E. Kummer,
L. Rogers, S. Ramanujan,..., L. Lewin, L.D. Faddeev, D. Zagier,
A. Goncharov, A.Al. Zamolodchikov, among many others. Diloga-
rithm and its quantum analogue have found numerous deep applica-
tions in Number Theory, Hyperbolic Geometry, Knot invariants, Al-
gebraic K-theory, Representation Theory, Mathematical Physics and
Applied Mathematics.
In my talk I'm planning to draw attention of the audience to some
remarkable identities among the values of the Rogers dilogarithm func-
tion at some very special families of algebraic numbers. These rela-
tions admit an interesting interpretation in algebraic K-theory and
CFT.
I'm planning to talk about Rigged Con guration Bijection (RC-
bijection), which originated from the analysis of the Bethe Ansatz
Equations for the XXX and XXZ Heisenberg models, and have big
variety of applications to Combinatorics, Representation Theory, Dis-
crete Integrable Systems among other interesting applications.
I'm also planning to talk about some families of quadratic alge-
bras (the so-called Fomin{Kirillov type algebras) with applications
to Schubert Calculus, Quantum (and Elliptic (?)) Cohomology and
K-theory of
ag varieties, and beyond.
Basic references
Lewin, L. Dilogarithms and Associated Functions. London: Macdonald,
1958.
D. Zagier, The Dilogarithm function in Geometry and Number Theory,
Number Theory and related topics, Tata Inst. Fund. Res. Stud. Math. 12
Bombay (1988), 231 - 249.
Kirillov, A. N. Dilogarithm Identities. Progr. Theor. Phys. Suppl. 118,
61-142,