Searcharxiv⌕ Search

arXiv subjects

Milen Yakimov

Publications and source records attributed to Milen Yakimov.

At least 55 records · Page 3Linked to original sources
arXiv2013

Rigidity of quantum tori and the Andruskiewitsch-Dumas conjecture

We prove the Andruskiewitsch-Dumas conjecture that the automorphism group of the positive part of the quantized universal enveloping algebra $U_q({\mathfrak{g}})$ of an arbitrary finite dimensional simple Lie algebra g is isomorphic to the semidirect product of the automorphism group of the Dynkin diagram of g and a torus of rank equal to the rank of g. The key step in our proof is a rigidity theorem for quantum tori. It has a broad range of applications. It allows one to control the (full) automorphism groups of large classes of associative algebras, for instance quantum cluster algebras.

math.RA↗
arXiv2013

The Launois-Lenagan conjecture

In this note we prove the Launois-Lenagan conjecture on the classification of the automorphism groups of the algebras of quantum matrices R_q[M_n] of square shape for all positive integers n, base fields K, and deformation parameters q \in K^* which are not roots of unity.

math.RA↗
arXiv2012

Spectra and catenarity of multiparameter quantum Schubert cells

We study the ring theory of the multiparameter deformations of the quantum Schubert cell algebras obtained from 2-cocycle twists. This is a large family, which extends the Artin-Schelter-Tate algebras of twisted quantum matrices. We classify set theoretically the spectra of all such multiparameter quantum Schubert cell algebras, construct each of their prime ideals by contracting from explicit normal localizations, and prove formulas for the dimensions of their Goodearl-Letzter strata for base fields of arbitrary characteristic and all deformation parameters that are not roots of unity. Furthermore, we prove that the spectra of these algebras are normally separated and that all such algebras are catenary.

math.RA↗
arXiv2012

A proof of the Goodearl-Lenagan polynormality conjecture

The quantum nilpotent algebras U^w_-(g), defined by De Concini-Kac-Procesi and Lusztig, are large classes of iterated skew polynomial rings with rich ring theoretic structure. In this paper, we prove in an explicit way that all torus invariant prime ideals of the algebras U^w_-(g) are polynormal. In the special case of the algebras of quantum matrices, this construction yields explicit polynormal generating sets consisting of quantum minors for all of their torus invariant prime ideals. This gives a constructive proof of the Goodearl-Lenagan polynormality conjecture. Furthermore we prove that Spec U^w_-(g) is normally separated for all simple Lie algebras g and Weyl group elements w, and deduce from it that all algebras U^w_-(g) are catenary.

math.QA↗
arXiv2011

Strata of prime ideals of De Concini-Kac-Procesi algebras and Poisson geometry

To each simple Lie algebra g and an element w of the corresponding Weyl group De Concini, Kac and Procesi associated a subalgebra U^w_- of the quantized universal enveloping algebra U_q(g), which is a deformation of the universal enveloping algebra U(n_- \cap w(n_+)) and a quantization of the coordinate ring of the Schubert cell corresponding to w. The torus invariant prime ideals of these algebras were classified by Mériaux and Cauchon [25], and the author [30]. These ideals were also explicitly described in [30]. They index the the Goodearl-Letzter strata of the stratification of the spectra of U^w_- into tori. In this paper we derive a formula for the dimensions of these strata and the transcendence degree of the field of rational Casimirs on any open Richardson variety with respect to the standard Poisson structure [15].

math.QA↗
arXiv2009

A classification of H-primes of quantum partial flag varieties

We classify the invariant prime ideals of a quantum partial flag variety under the action of the related maximal torus. As a result we construct a bijection between them and the torus orbits of symplectic leaves of the standard Poisson structure on the corresponding flag variety. It was previously shown by K. Goodearl and the author that the latter are precisely the Lusztig strata of the partial flag variety.

math.QA↗
arXiv2009

Invariant prime ideals in quantizations of nilpotent Lie algebras

De Concini, Kac and Procesi defined a family of subalgebras U^w_+ of a quantized universal enveloping algebra U_q(g), associated to the elements of the corresponding Weyl group W. They are deformations of the universal enveloping algebras U(n_+ \cap Ad_w(n_-)) where n_\pm are the nilradicals of a pair of dual Borel subalgebras. Based on results of Gorelik and Joseph and an interpretation of U^w_+ as quantized algebras of functions on Schubert cells, we construct explicitly the H invariant prime ideals of each U^w_+ and show that the corresponding poset is isomorphic to W^{\leq w}, where H is the group of group-like elements of U_q(g). Moreover, for each H-prime of U^w_+ we construct a generating set in terms of Demazure modules related to fundamental representations.

math.QA↗
arXiv2009

Cyclicity of Lusztig's stratification of Grassmannians and Poisson geometry

We prove that the standard Poisson structure on the Grassmannian Gr(k, n) is invariant under the action of the Coxeter element c =(1 2 ... n). In particular, its symplectic foliation is invariant under c. As a corollary, we obtain a second, Poisson geometric proof of the result of Knutson, Lam, and Speyer that the Coxeter element c interchanges the Lusztig strata of Gr(k, n). We also relate the main result to known anti-invariance properties of the standard Poisson structures on cominuscule flag varieties.

math.CO↗
arXiv2008

Completeness of determinantal Hamiltonian flows on the matrix affine Poisson space

The matrix affine Poisson space (M_{m,n}, pi_{m,n}) is the space of complex rectangular matrices equipped with a canonical quadratic Poisson structure which in the square case m=n reduces to the standard Poisson structure on GL_n(C). We prove that the Hamiltonian flows of all minors are complete. As a corollary we obtain that all Kogan-Zelevinsky integrable systems on M_{n,n} are complete and thus induce (analytic) Hamiltonian actions of C^{n(n-1)/2} on (M_{n,n}, pi_{n,n}) (as well as on GL_n(C) and on SL_n(C)). We define Gelfand-Zeitlin integrable systems on (M_{n,n}, pi_{n,n}) from chains of Poisson projections and prove that their flows are also complete. This is an analog for the quadratic Poisson structure pi_{n,n} of the recent result of Kostant and Wallach [KW] that the flows of the complexified classical Gelfand-Zeitlin integrable systems are complete.

math.SG↗
arXiv2007

The real loci of Calogero-Moser spaces, representations of rational Cherednik algebras and the Shapiro conjecture

We prove a criterion for the reality of irreducible representations of the rational Cherednik algebras H_{0,1}(S_n). This is shown to imply a criterion for the real loci of the Calogero-Moser spaces C_n in terms of the Etingof-Ginzburg finite maps Υ\colon C_n \to C^n/S_n \times C^n/S_n, recovering a result of Mikhin, Tarasov, and Varchenko [MTV2]. As a consequence we obtain a criterion for the real locus of the Wilson's adelic Grassmannian of rank one bispectral solutions of the KP hierarchy. Using Wilson's first parametrisation of the adelic Grassmannian, we give a new proof of a result of [MTV2] on real bases of spaces of quasi polynomials. The Shapiro Conjecture for Grassmannians is equivalent to a special case of our result for Calogero-Moser spaces, namely for the fibres of Υover C^n/S_n \times 0.

math.RT↗
arXiv2007

A Deodhar type stratification on the double flag variety

We describe a stratification on the double flag variety $G/B^+\times G/B^-$ of a complex semisimple algebraic group $G$ analogous to the Deodhar stratification on the flag variety $G/B^+$, which is a refinement of the stratification into orbits both for $B^+\times B^-$ and for the diagonal action of $G$, just as Deodhar's stratification refines the orbits of $B^+$ and $B^-$. We give a coordinate system on each stratum, and show that all strata are coisotropic subvarieties. Also, we discuss possible connections to the positive and cluster geometry of $G/B^+\times G/B^-$, which would generalize results of Fomin and Zelevinsky on double Bruhat cells and Marsh and Rietsch on double Schubert cells.

math.SG↗
arXiv2007

Modular classes of regular twisted Poisson structures on Lie algebroids

We derive a formula for the the modular class of a Lie algebroid with a regular twisted Poisson structure in terms of a canonical Lie algebroid representation of the image of the Poisson map. We use this formula to compute the modular classes of Lie algebras with a twisted triangular r-matrix. The special case of r-matrices associated to Frobenius Lie algebras is also studied.

math.SG↗
arXiv2007

Group orbits and regular partitions of Poisson manifolds

We study a large class of Poisson manifolds, derived from Manin triples, for which we construct explicit partitions into regular Poisson submanifolds by intersecting certain group orbits. Examples include all varieties ${\mathcal L}$ of Lagrangian subalgebras of reductive quadratic Lie algebras $\d$ with Poisson structures defined by Lagrangian splittings of $\d$. In the special case of $\g \oplus \g$, where $\g$ is a complex semi-simple Lie algebra, we explicitly compute the ranks of the Poisson structures on ${\mathcal L}$ defined by arbitrary Lagrangian splittings of ${\mathfrak g} \oplus {\mathfrak g}$. Such Lagrangian splittings have been classified by P. Delorme, and they contain the Belavin--Drinfeld splittings as special cases.

math.SG↗
arXiv2006

Partitions of the wonderful group compactification

We define and study a family of partitions of the wonderful compactification \bar{G} of a semi-simple algebraic group G of adjoint type. The partitions are obtained from subgroups of G \times G associated to triples (A_1, A_2, a), where A_1 and A_2 are subgraphs of the Dynkin graph Γof G and a : A_1 \to A_2 is an isomorphism. The partitions of \bar{G} of Springer and Lusztig correspond respectively to the triples (\emptyset, \emptyset, \id) and (Γ, Γ, \id).

math.RT↗
arXiv2005

On a class of double cosets in reductive algebraic groups

We study a class of double coset spaces R_A \backslash G_1 \times G_2 /R_C, where G_1 and G_2 are connected reductive algebraic groups, and R_A and R_C are certain spherical subgroups of G_1 \times G_2 obtained by ``identifying'' Levi factors of parabolic subgroups in G_1 and G_2. Such double cosets naturally appear in the symplectic leaf decompositions of Poisson homogeneous spaces of complex reductive groups with the Belavin-Drinfeld Poisson structures. They also appear in orbit decompositions of the De Concini-Procesi compactifications of semi-simple groups of adjoint type. We find explicit parametrizations of the double coset spaces and describe the double cosets as homogeneous spaces of R_A \times R_C. We further show that all such double cosets give rise to set-theoretical solutions to the quantum Yang-Baxter equation on unipotent algebraic groups.

math.RT↗
arXiv2004

Triangular Poisson structures on Lie groups and symplectic reduction

We show that each triangular Poisson Lie group can be decomposed into Poisson submanifolds each of which is a quotient of a symplectic manifold. The Marsden-Weinstein-Meyer symplectic reduction technique is then used to give a complete description of the symplectic foliation of all triangular Poisson structures on Lie groups. The results are illustrated in detail for the generalized Jordanian Poisson structures on SL(n).

math.SG↗