Papers by Michael Shapiro
∗ Dept. of Math., University of Stockholm, S-10691, Stockholm, [email protected] † Higher Coll... more ∗ Dept. of Math., University of Stockholm, S-10691, Stockholm, [email protected] † Higher College of Math., Independent University of Moscow, and Institute for System Research RAS, [email protected] ‡ Department of Mathematics, Royal Institute of Technology, S-10044, Stockholm, [email protected] ♮ Dept. of Math. and Dept. of Computer Science, University of Haifa, Haifa 31905, [email protected]
We use the Darboux coordinate representation found by two of the authors (L.Ch. and M.Sh.) for en... more We use the Darboux coordinate representation found by two of the authors (L.Ch. and M.Sh.) for entries of general symplectic leaves of the An-groupoid of upper-triangular matrices to express roots of the characteristic equation det(A − λA) = 0, with A ∈ An, in terms of Casimirs of this Darboux coordinate representation, which is based on cluster variables of Fock–Goncharov higher Teichmüller spaces for the algebra sln. We show that roots of the characteristic equation are simple monomials of cluster Casimir elements. This statement remains valid in the quantum case as well.
American Mathematical Society Translations: Series 2, 1999
arXiv (Cornell University), Mar 16, 2020
Using Fock-Goncharov higher Teichmüller space variables we derive Darboux coordinate representati... more Using Fock-Goncharov higher Teichmüller space variables we derive Darboux coordinate representation for entries of general symplectic leaves of the A n groupoid of uppertriangular matrices and, in a more general setting, of higher-dimensional symplectic leaves for algebras governed by the reflection equation with the trigonometric R-matrix. The obtained results are in a perfect agreement with the previously obtained Poisson and quantum representations of groupoid variables for A 3 and A 4 in terms of geodesic functions for Riemann surfaces with holes. We realize braid-group transformations for A n via sequences of cluster mutations in the special A n-quiver. We prove the groupoid relations for quantum transport matrices and, as a byproduct, obtain the Goldman bracket in the semiclassical limit. We prove the quantum algebraic relations of transport matrices for arbitrary (cyclic or acyclic) directed planar network.
International Mathematics Research Notices, 2020
As is well known, cluster transformations in cluster structures of geometric type are often model... more As is well known, cluster transformations in cluster structures of geometric type are often modeled on determinant identities, such as short Plücker relations, Desnanot–Jacobi identities, and their generalizations. We present a construction that plays a similar role in a description of generalized cluster transformations and discuss its applications to generalized cluster structures in $GL_n$ compatible with a certain subclass of Belavin–Drinfeld Poisson–Lie brackets, in the Drinfeld double of $GL_n$, and in spaces of periodic difference operators.
arXiv (Cornell University), Aug 6, 2020
In this paper a double quasi Poisson bracket in the sense of Van den Bergh is constructed on the ... more In this paper a double quasi Poisson bracket in the sense of Van den Bergh is constructed on the space of noncommutative weights of arcs of a directed graph embedded in a disk or cylinder Σ, which gives rise to the quasi Poisson bracket of G.Massuyeau and V.Turaev on the group algebra kπ 1 (Σ, p) of the fundamental group of a surface based at p ∈ ∂Σ. This bracket also induces a noncommutative Goldman Poisson bracket on the cyclic space C ♮ , which is a k-linear space of unbased loops. We show that the induced double quasi Poisson bracket between boundary measurements can be described via noncommutative r-matrix formalism. This gives a more conceptual proof of the result of [Ove20] that traces of powers of Lax operator form an infinite collection of noncommutative Hamiltonians in involution with respect to noncommutative Goldman bracket on C ♮. 1 Note that concatenation of paths is commonly written in the opposite order to the composition of morphisms in C, i.e. f g = g • f. 2 One can further assume that g = 0 for every g ∈ C which is not a loop. We do not need this convention in our text, however, it is worth noting that such convention allows one to define a map C → C ♮ , f → f which is commonly referred to as "universal trace". 3 More details can be found in [Art18, AR].
The group of real 4 by 4 upper triangular matrices with 1s on the diagonal has a left-invariant s... more The group of real 4 by 4 upper triangular matrices with 1s on the diagonal has a left-invariant subRiemannian (or Carnot-Caratheodory) structure whose underlying distribution corresponds to the superdiagonal. We prove that the associated subRiemannian geodesic flow is not completely integrable. This provides the first example of a Carnot group (graded nilpotent Lie group with an invariant subRiemannian structure supported
In this short survey we recall some basic results and relations between the qualitative theory of... more In this short survey we recall some basic results and relations between the qualitative theory of linear ordinary differential equations with real time and the reality problems in Schubert calculus. We formulate a few relevant conjectures.
The Michigan Mathematical Journal, 2000
Bulletin of the American Mathematical Society, 1991
Annales de l’institut Fourier, 2007
We show that if a meromorphic function of degree at most four on a real algebraic curve of an arb... more We show that if a meromorphic function of degree at most four on a real algebraic curve of an arbitrary genus has only real critical points then it is conjugate to a real meromorphic function after a suitable projective automorphism of the image. 1991 Mathematics Subject Classification. 14P05,14P25. Key words and phrases. total reality, meromorphic functions, flattening points, K3-surfaces. M.S. is partially supported by NSF grants DMS-0401178, by the BSF grant 2002375 and by the Institute of Quantum Science, MSU. I.I. is partially supported by the ANR-network "Interactions et aspectś enumératifs des géométries réelle, tropicale et symplectique".
Communications on Pure and Applied Mathematics, 1999
We introduce a Poisson variety compatible with a cluster algebra structure and a compatible toric... more We introduce a Poisson variety compatible with a cluster algebra structure and a compatible toric action on this variety. We study Poisson and topological properties of the union of generic orbits of this toric action. In particular, we compute the number of connected components of the union of generic toric orbits for cluster algebras over real numbers. As a corollary we compute the number of connected components of refined open Bruhat cells in Grassmanians G(k, n) over R.
In this short note we give an explicit presentation of the algebra A n spanned by the curvature 2... more In this short note we give an explicit presentation of the algebra A n spanned by the curvature 2-forms of the standard Hermitian linear bundles over SL n /B as the quotient of the polynomial ring. The difference between A n and H * (SL n /B) reflects the fact that SL n /B is not a symmetric space. This question was raised by V. I. Arnold in [Ar]. Sur l'algèbre engendrée par les 2-formes de Chern-Bott sur SL n /B Résumé: Dans cette note nous donnerons une présentation explicite, en tant que quotient de l'anneau polynomial, de l'algèbre A n engendrée par les 2-formes de courbure des fibrés en droites hermitiens standard sur SL n /B. La différence entre A n et H * (SL n /B) reflète le fait que SL n /B n'est pas un espace symmétrique. Cette question aété posée par V. I. Arnold dans [Ar]. Version française abrégée. Considérons l'espace SL n /B des drapeaux complets dans C n , aussi bien que les n fibrés quotient L i = E i /E i−1 , où E i signifie le fibré tautologique standard de dimension i au-dessus de SL N /B. Quand on fixe une métrique hermitienne dans C n tous les L i deviennent des fibrés hermitiens. On notera ω i la 2-forme de courbure de L i. Il estévident que chaque ω i est U ninvariante sur SL n /B. Dans cette note nous allons présenter l'algèbre A n = A(ω 1 ,. .. , ω n), engendrée par ω 1 ,. .. , ω n , comme un quotient de l'anneau polynomial en n générateurs, et on la comparera ensuite avec H * (SL n /B). Nousétudions aussi l'algèbre A k,n engendrée par k parmi les formes ω i. (Sa structure ne dépend pas du choix particulier de k-tuple.) Le résultat principal est le suivant. Proposition. L'algèbre A k,n est une algèbre graduée, isomorpheà C[x 1 ,. .. , x k ]/I k,n , où I k,n est un idéal engendré par 2 k − 1 polynômes de la forme g (n) i 1 ,...,i j = (x i 1 + • • • + x i j) j(n−j)+1 .
International Mathematics Research Notices, 2011
We complete classification of mutation-finite cluster algebras by extending the technique derived... more We complete classification of mutation-finite cluster algebras by extending the technique derived by Fomin, Shapiro, and Thurston to skew-symmetrizable case. We show that for every mutation-finite skew-symmetrizable matrix a diagram characterizing the matrix admits an unfolding which embeds its mutation class to the mutation class of some mutation-finite skew-symmetric matrix. In particular, this establishes a correspondence between a large class of skew-symmetrizable mutationfinite cluster algebras and triangulated marked bordered surfaces. Contents 14 6. Unfoldings of mutation-finite matrices and diagrams 18 7. Triangulations of bordered surfaces and s-decomposable diagrams 24 8. Minimal non-decomposable diagrams 30 Appendix A.
Discrete Mathematics, 1996
We consider intersections of Schubert cells ~. ~ and tr~tr-[3. ~ in the space of complete flags F... more We consider intersections of Schubert cells ~. ~ and tr~tr-[3. ~ in the space of complete flags F = SL/~, where ~ denotes the Borel subgroup of upper triangular matrices, while ct, /~ and tr belong to the Weyl group W (coinciding with the symmetric group). We obtain a special decomposition of F which subdivides all ~at • ~ N a~a-~ • ~ into strata of a simple form. It enables us to establish a new geometrical interpretation of the structure constants for the corresponding Hecke algebra and in particular of the so-called R-polynomials used in Kazhdan-Lusztig theory. Structure constants of the Hecke algebra appear to be the alternating sums of the Hodge numbers for the mixed Hodge structure in the cohomology with compact supports of the above intersections. We derive a new efficient combinatorial algorithm calculating the R-polynomials and structure constants in general.
Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 1999
In this paper we find an explicit formula for the number of topologically different ramified cove... more In this paper we find an explicit formula for the number of topologically different ramified coverings C → CP 1 (C is a compact Riemann surface of genus g) with only one complicated branching point in terms of Hodge integrals over the moduli space of genus g curves with marked points.
Discrete Mathematics, 1998
We give explicit formulas for the Kazhdan-Lusztig P-and Rpolynomials for permutations coming from... more We give explicit formulas for the Kazhdan-Lusztig P-and Rpolynomials for permutations coming from the variety F 1,n−1 of incomplete flags consisting of a line and a hyperplane.
Memoirs of the American Mathematical Society, 2017
This is the second paper in the series of papers dedicated to the study of natural cluster struct... more This is the second paper in the series of papers dedicated to the study of natural cluster structures in the rings of regular functions on simple complex Lie groups and Poisson-Lie structures compatible with these cluster structures. According to our main conjecture, each class in the Belavin-Drinfeld classification of Poisson-Lie structures on $\G$ corresponds to a cluster structure in $\O(\G)$. We have shown before that this conjecture holds for any $\G$ in the case of the standard Poisson-Lie structure and for all Belavin-Drinfeld classes in $SL_n$, $n<5$. In this paper we establish it for the Cremmer-Gervais Poisson-Lie structure on $SL_n$, which is the least similar to the standard one.
Advances in Mathematics, 2016
The pentagram map was introduced by R. Schwartz more than 20 years ago. In 2009, V. Ovsienko, R. ... more The pentagram map was introduced by R. Schwartz more than 20 years ago. In 2009, V. Ovsienko, R. Schwartz and S. Tabachnikov established Liouville complete integrability of this discrete dynamical system. In 2011, M. Glick interpreted the pentagram map as a sequence of cluster transformations associated with a special quiver. Using compatible Poisson structures in cluster algebras and Poisson geometry of directed networks on surfaces, we generalize Glick's construction to include the pentagram map into a family of discrete integrable maps and we give these maps geometric interpretations. Contents 1. Introduction 1 2. Generalized Glick's quivers and the (p, q)-dynamics 4 3. Weighted directed networks and the (x, y)-dynamics 7 3.1. Weighted directed networks on surfaces 7 3.2. The (x, y)-dynamics 10 4. Poisson structure and complete integrability 16 4.1. Cuts, rims, and conjugate networks 16 4.2. Poisson structure 20 4.3. Conserved quantities 23 4.4. Lax representations 25 4.5. Complete integrability 27 4.6. Spectral curve 29 5. Geometric interpretation 30 5.1. The case k ≥ 3 30 5.2. The case k = 2: leapfrog map and circle pattern 38 References 44
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Papers by Michael Shapiro