arXiv subjects

Milen Yakimov

Publications and source records attributed to Milen Yakimov.

At least 37 records · Page 2Linked to original sources

arXiv2020

Bivariate Continuous q-Hermite Polynomials and Deformed Quantum Serre Relations

We introduce bivariate versions of the continuous q-Hermite polynomials. We obtain algebraic properties for them (generating function, explicit expressions in terms of the univariate ones, backward difference equations and recurrence relations) and analytic properties (determining the orthogonality measure). We find a direct link between bivariate continuous q-Hermite polynomials and the star product method of [Kolb and Yakimov, Adv. Math. 2020] for quantum symmetric pairs to establish deformed quantum Serre relations for quasi-split quantum symmetric pairs of Kac-Moody type. We prove that these defining relations are obtained from the usual quantum Serre relations by replacing all monomials by multivariate orthogonal polynomials.

math.QA↗

arXiv2020

Reflective prolate-spheroidal operators and the adelic Grassmannian

Beginning with the work of Landau, Pollak and Slepian in the 1960s on time-band limiting, commuting pairs of integral and differential operators have played a key role in signal processing, random matrix theory and integrable systems. Previously, such pairs were constructed by ad hoc methods, which worked because a commuting operator of low order could be found by a direct calculation. We describe a general approach to these problems that proves that every point $W$ of Wilson's infinite dimensional adelic Grassmannian $\mathrm Gr^ad$ gives rise to an integral operator $T_W$, acting on $L^2(Γ)$ for a contour $Γ\subset\mathbb C$, which reflects a differential operator $R(z,\partial_z)$ in the sense that $R(-z,-\partial_z)\circ T_W=T_W\circ R(w,\partial_w)$ on a dense subset of $L^2(Γ)$. By using analytic methods and methods from integrable systems, we show that the reflected differential operator can be constructed from the Fourier algebra of the associated bispectral function $ψ_W(x,z)$. The size of this algebra with respect to a bifiltration is in turn determined using algebro-geometric methods. Intrinsic properties of four involutions of the adelic Grassmannian naturally lead us to consider the reflecting property in place of plain commutativity. Furthermore, we prove that the time-band limited operators of the generalized Laplace transforms with kernels given by all rank one bispectral functions $ψ_W(x,-z)$ reflect a differential operator. A $90^\circ$ rotation argument is used to prove that the time-band limited operators of the generalized Fourier transforms with kernels $ψ_W(x,iz)$ admit a commuting differential operator. These methods produce vast collections of integral operators with prolate-spheroidal properties, associated to the wave functions of all rational solutions of the KP hierarchy vanishing at infinity, introduced by Krichever in the late 1970s.

math.CA↗

arXiv2019

$c$-vectors of 2-Calabi--Yau categories and Borel subalgebras of ${\mathfrak{sl}}_{\infty}$

We develop a general framework for $c$-vectors of 2-Calabi--Yau categories, which deals with cluster tilting subcategories that are not reachable from each other and contain infinitely many indecomposable objects. It does not rely on iterative sequences of mutations. We prove a categorical (infinite-rank) generalization of the Nakanishi--Zelevinsky duality for $c$-vectors and establish two formulae for the effective computation of $c$-vectors -- one in terms of indices and the other in terms of dimension vectors for cluster tilted algebras. In this framework, we construct a correspondence between the $c$-vectors of the cluster categories ${\mathscr{C}}(A_{\infty})$ of type $A_{\infty}$ due to Igusa--Todorov and the roots of the Borel subalgebras of ${\mathfrak{sl}}_{\infty}$. Contrary to the finite dimensional case, the Borel subalgebras of ${\mathfrak{sl}}_{\infty}$ are not conjugate to each other. On the categorical side, the cluster tilting subcategories of ${\mathscr{C}}(A_{\infty})$ exhibit different homological properties. The correspondence builds a bridge between the two classes of objects.

math.RA↗

arXiv2019

Green groupoids of 2-Calabi--Yau categories, derived Picard actions, and hyperplane arrangements

We present a construction of (faithful) group actions via derived equivalences in the general categorical setting of algebraic 2-Calabi--Yau triangulated categories. To each algebraic 2-Calabi--Yau category $\mathscr{C}$ satisfying standard mild assumptions, we associate a groupoid $\mathscr{G}_{ \mathscr{C} }$, named the green groupoid of $\mathscr{C}$, defined in an intrinsic homological way. Its objects are given by a set of representatives $\operatorname{mrig} \mathscr{C}$ of the equivalence classes of basic maximal rigid objects of $\mathscr{C}$, arrows are given by mutation, and relations are given by equating monotone (green) paths in the silting order. In this generality we construct a homomorphsim from the green groupoid $\mathscr{G}_{ \mathscr{C} }$ to the derived Picard groupoid of the collection of endomorphism rings of representatives of $\operatorname{mrig} \mathscr{C}$ in a Frobenius model of $\mathscr{C}$; the latter canonically acts by triangle equivalences between the derived categories of the rings. We prove that the constructed representation of the green groupoid $\mathscr{G}_{ \mathscr{C} }$ is faithful if the index chamber decompositions of the split Grothendieck groups of basic maximal rigid objects of $\mathscr{C}$ come from hyperplane arrangements. If $Σ^2 \cong \operatorname{id}$ and $\mathscr{C}$ has finitely many equivalence classes of basic maximal rigid objects, we prove that $\mathscr{G}_{ \mathscr{C} }$ is isomorphic to a Deligne groupoid of a hyperplane arrangement and that the representation of this groupoid is faithful.

math.RT↗

arXiv2019

Reflective prolate-spheroidal operators and the KP/KdV equations

Commuting integral and differential operators connect the topics of Signal Processing, Random Matrix Theory, and Integrable Systems. Previously, the construction of such pairs was based on direct calculation and concerned concrete special cases, leaving behind important families such as the operators associated to the rational solutions of the KdV equation. We prove a general theorem that the integral operator associated to every wave function in the infinite dimensional Adelic Grassmannian Gr ad of Wilson always reflects a differential operator (in the sense of Definition 1 below). This intrinsic property is shown to follow from the symmetries of Grassmannians of KP wave functions, where the direct commutativity property holds for operators associated to wave functions fixed by Wilson's sign involution but is violated in general. Based on this result, we prove a second main theorem that the integral operators in the computation of the singular values of the truncated generalized Laplace transforms associated to all bispectral wave functions of rank 1 reflect a differential operator. A 90$°$ rotation argument is used to prove a third main theorem that the integral operators in the computation of the singular values of the truncated generalized Fourier transforms associated to all such KP wave functions commute with a differential operator. These methods produce vast collections of integral operators with prolate-spheroidal properties, including as special cases the integral operators associated to all rational solutions of the KdV and KP hierarchies considered by Airault-McKean-Moser and Krichever, respectively, in the late 70's. Many novel examples are presented.

math.CA↗

arXiv2019

Symmetric pairs for Nichols algebras of diagonal type via star products

We construct symmetric pairs for Drinfeld doubles of pre-Nichols algebras of diagonal type and determine when they possess an Iwasawa decomposition. This extends G. Letzter's theory of quantum symmetric pairs. Our results can be uniformly applied to Kac-Moody quantum groups for a generic quantum parameter, for roots of unity in respect to both big and small quantum groups, to quantum supergroups and to exotic quantum groups of ufo type. We give a second construction of symmetric pairs for Heisenberg doubles in the above generality and prove that they always admit an Iwasawa decomposition. For symmetric pair coideal subalgebras with Iwasawa decomposition in the above generality we then address two problems which are fundamental already in the setting of quantum groups. Firstly, we show that the symmetric pair coideal subalgebras are isomorphic to intrinsically defined deformations of partial bosonizations of the corresponding pre-Nichols algebras. To this end we develop a general notion of star products on N-graded connected algebras which provides an efficient tool to prove that two deformations of the partial bosonization are isomorphic. The new perspective also provides an effective algorithm for determining the defining relations of the coideal subalgebras. Secondly, for Nichols algebras of diagonal type, we use the linear isomorphism between the coideal subalgebra and the partial bosonization to give an explicit construction of quasi K-matrices as sums over dual bases. We show that the resulting quasi K-matrices give rise to weakly universal K-matrices in the above generality.

math.QA↗

arXiv2018

Poisson geometry of PI 3-dimensional Sklyanin algebras

We give the 3-dimensional Sklyanin algebras $S$ that are module-finite over their center $Z$ the structure of a Poisson $Z$-order (in the sense of Brown-Gordon). We show that the induced Poisson bracket on $Z$ is non-vanishing and is induced by an explicit potential. The ${\mathbb Z}_3 \times \Bbbk^\times$-orbits of symplectic cores of the Poisson structure are determined (where the group acts on $S$ by algebra automorphisms). In turn, this is used to analyze the finite-dimensional quotients of $S$ by central annihilators: there are 3 distinct isomorphism classes of such quotients in the case $(n,3) \neq 1$ and 2 in the case $(n,3)=1$, where $n$ is the order of the elliptic curve automorphism associated to $S$. The Azumaya locus of $S$ is determined, extending results of Walton for the case $(n,3)=1$.

math.RT↗

arXiv2018

Prime spectra of abelian 2-categories and categorifications of Richardson varieties

We describe a general framework for prime, completely prime, semiprime, and primitive ideals of an abelian 2-category. This provides a noncommutative version of Balmer's prime spectrum of a tensor triangulated category. These notions are based on containment conditions in terms of thick subcategories of an abelian category and thick ideals of an abelian 2-category. We prove categorical analogs of the main properties of noncommutative prime spectra. Similar notions, starting with Serre subcategories of an abelian category and Serre ideals of an abelian 2-category, are developed. They are linked to Serre prime spectra of $\mathbb{Z}_+$-rings. As an application, we construct a categorification of the quantized coordinate rings of open Richardson varieties for symmetric Kac-Moody groups, by constructing Serre completely prime ideals of monoidal categories of modules of the KLR algebras, and by taking Serre quotients with respect to them.

math.CT↗

arXiv2018

Noncommutative discriminants via Poisson primes

We present a general method for computing discriminants of noncommutative algebras. It builds a connection with Poisson geometry and expresses the discriminants as products of Poisson primes. The method is applicable to algebras obtained by specialization from families, such as quantum algebras at roots of unity. It is illustrated with the specializations of the algebras of quantum matrices at roots of unity and more generally all quantum Schubert cell algebras.

math.RA↗

arXiv2018

The Matrix Bochner Problem

A long standing question in the theory of orthogonal matrix polynomials is the matrix Bochner problem, the classification of $N \times N$ weight matrices $W(x)$ whose associated orthogonal polynomials are eigenfunctions of a second order differential operator. Based on techniques from noncommutative algebra (semiprime PI algebras of Gelfand-Kirillov dimension one), we construct a framework for the systematic study of the structure of the algebra $\mathcal D(W)$ of matrix differential operators for which the orthogonal polynomials of the weight matrix $W(x)$ are eigenfunctions. The ingredients for this algebraic setting are derived from the analytic properties of the orthogonal matrix polynomials. We use the representation theory of the algebras $\mathcal D(W)$ to resolve the matrix Bochner problem under the two natural assumptions that the sum of the sizes of the matrix algebras in the central localization of $\mathcal D(W)$ equals $N$ (fullness of $\mathcal D(W)$) and the leading coefficient of the second order differential operator multiplied by the weight $W(x)$ is positive definite. In the case of $2\times 2$ weights, it is proved that fullness is satisfied as long as $\mathcal D(W)$ is noncommutative. The two conditions are natural in that without them the problem is equivalent to much more general ones by artificially increasing the size of the matrix $W(x)$.

math.RA↗

arXiv2017

Recovering the topology of surfaces from cluster algebras

We present an effective method for recovering the topology of a bordered oriented surface with marked points from its cluster algebra. The information is extracted from the maximal triangulations of the surface, those that have exchange quivers with maximal number of arrows in the mutation class. The method gives new proofs of the automorphism and isomorphism problems for the surface cluster algebras, as well as the uniqueness of the Fomin-Shapiro-Thurston block decompositions of the exchange quivers of the surface cluster algebras. The previous proofs of these results followed a different approach based on Gu's direct proof of the last result. The method also explains the exceptions to these results due to pathological problems with the maximal triangulations of several surfaces.

math.CO↗

arXiv2016

Rigidity of quadratic Poisson tori

We prove a rigidity theorem for the Poisson automorphisms of the function fields of tori with quadratic Poisson structures over fields of characteristic 0. It gives an effective method for classifying the full Poisson automorphism groups of ${\mathbb{N}}$-graded connected cluster algebras equipped with Gekhtman-Shapiro-Vainshtein Poisson structures. Based on this, we classify the groups of algebraic Poisson automorphisms of the open Schubert cells of the full flag varieties of semisimple algebraic groups over fields of characteristic 0, equipped with the standard Poisson structures. Their coordinate rings can be identified with the semiclassical limits of the positive parts $U_q({\mathfrak{n}}_+)$ of the quantized universal enveloping algebras of semisimple Lie algebras, and the last result establishes a Poisson version of the Andruskiewitsch-Dumas conjecture on ${\mathrm{Aut}} U_q({\mathfrak{n}}_+)$.

math.RA↗

arXiv2016

Separating Ore sets for prime ideals of quantum algebras

Brown and Goodearl stated a conjecture that provides an explicit description of the topology of the spectra of quantum algebras. The conjecture takes on a more explicit form if there exist separating Ore sets for all incident pairs of torus invariant prime ideals of the given algebra. We prove that this is the case for the two largest classes of algebras of finite Gelfand-Kirillov dimension that fit the setting of the conjecture: the quantized coordinate rings of all simple algebraic groups and the quantum Schubert cell algebras for all symmetrizable Kac-Moody algebras.

math.QA↗

arXiv2016

Quantized Weyl algebras at roots of unity

We classify the centers of the quantized Weyl algebras that are PI and derive explicit formulas for the discriminants of these algebras over a general class of polynomial central subalgebras. Two different approaches to these formulas are given: one based on Poisson geometry and deformation theory, and the other using techniques from quantum cluster algebras. Furthermore, we classify the PI quantized Weyl algebras that are free over their centers and prove that their discriminants are locally dominating and effective. This is applied to solve the automorphism and isomorphism problems for this family of algebras and their tensor products.

math.RA↗

arXiv2016

Noncommutative bispectral Darboux transformations

We prove a general theorem establishing the bispectrality of noncommutative Darboux transformations. It has a wide range of applications that establish bispectrality of such transformations for differential, difference and q-difference operators with values in all noncommutative algebras. All known bispectral Darboux transformations are special cases of the theorem. Using the methods of quasideterminants and the spectral theory of matrix polynomials, we explicitly classify the set of bispectral Darboux transformations from rank one differential operators and Airy operators with values in matrix algebras. These sets generalize the classical Calogero-Moser spaces and Wilson's adelic Grassmannian.

math.CA↗

arXiv2014

Weak splittings of quotients of Drinfeld and Heisenberg doubles

We investigate the fine structure of the simplectic foliations of Poisson homogeneous spaces. Two general results are proved for weak splittings of surjective Poisson submersions from Heisenberg and Drinfeld doubles. The implications of these results are that the torus orbits of symplectic leaves of the quotients can be explicitly realized as Poisson-Dirac submanifolds of the torus orbits of the doubles. The results have a wide range of applications to many families of real and complex Poisson structures on flag varieties. Their torus orbits of leaves recover important families of varieties such as the open Richardson varieties.

math.SG↗

arXiv2014

On the Spectra of Quantum Groups

Joseph and Hodges-Levasseur (in the A case) described the spectra of all quantum function algebras R_q[G] on simple algebraic groups in terms of the centers of certain localizations of quotients of R_q[G] by torus invariant prime ideals, or equivalently in terms of orbits of finite groups. These centers were only known up to finite extensions. We determine the centers explicitly under the general conditions that the deformation parameter is not a root of unity and without any restriction on the characteristic of the ground field. From it we deduce a more explicit description of all prime ideals of R_q[G] than the previously known ones and an explicit parametrization of Spec R_q[G]. We combine the latter with a result of Kogan and Zelevinsky to obtain in the complex case a torus equivariant Dixmier type map from the symplectic foliation of the group G to the primitive spectrum of R_q[G]. Furthermore, under the general assumptions on the ground field and deformation parameter, we prove a theorem for separation of variables for the De Concini-Kac-Procesi algebras U^w_\pm, and classify the sets of their homogeneous normal elements and primitive elements. We apply those results to obtain explicit formulas for the prime and especially the primitive ideals of U^w_\pm lying in the Goodearl-Letzter stratum over the 0-ideal. This is in turn used to prove that all Joseph's localizations of quotients of R_q[G] by torus invariant prime ideals are free modules over their subalgebras generated by Joseph's normal elements. From it we derive a classification of the maximal spectrum of R_q[G] and use it to resolve a question of Goodearl and Zhang, showing that all maximal ideals of R_q[G] have finite codimension. We then prove that all maximal chains in Spec R_q[G] have the same length equal to GKdim R_q[G]= dim G, i.e. R_q[G] satisfies the first chain condition for prime ideals in Nagata's terminology.

math.RA↗

arXiv2014

Quantum Schubert cells via representation theory and ring theory

We resolve two questions of Cauchon and Meriaux on the spectra of the quantum Schubert cell algebras U^-[w]. The treatment of the first one unifies two very different approaches to Spec U^-[w], a ring theoretic one via deleting derivations and a representation theoretic one via Demazure modules. The outcome is that now one can combine the strengths of both methods. As an application we solve the containment problem for the Cauchon-Meriaux classification of torus invariant prime ideals of U^-[w]. Furthermore, we construct explicit models in terms of quantum minors for the Cauchon quantum affine space algebras constructed via the procedure of deleting derivations from all quantum Schubert cell algebras U^-[w]. Finally, our methods also give a new, independent proof of the Cauchon-Meriaux classification.

math.RA↗