Explicit Isomorphism Research Articles

Cyclotomic quiver Hecke algebras and Hecke algebra of $G(r,p,n)$

Given a quiver automorphism with nice properties, we give a presentation of the fixed subalgebra of the associated cyclotomic quiver Hecke algebra. Generalising an isomorphism of Brundan and Kleshchev between the cyclotomic Hecke algebra of type G(r,1,n) and the cyclotomic quiver Hecke algebra of type A, we apply the previous result to find a presentation of the cyclotomic Hecke algebra of type G(r,p,n) which looks very similar to the one of a cyclotomic quiver Hecke algebra. In the meanwhile, we give an explicit isomorphism which realises a well-known Morita equivalence between Ariki-Koike algebras.

The Harish-Chandra isomorphism for quantum $GL_2$

We construct an explicit Harish-Chandra isomorphism, from the quantum Hamiltonian reduction of the algebra \mathcal D_q(GL_2) of quantum differential operators on GL_2 , to the spherical double affine Hecke algebra associated to GL_2 . The isomorphism holds for all deformation parameters q \in \mathbb C^{\times} and t \neq ± i , such that q is not a non-trivial root of unity.

Two-block Springer fibers of types C and D: a diagrammatic approach to Springer theory

We explain an elementary topological construction of the Springer representation on the homology of (topological) Springer fibers of types C and D in the case of nilpotent endomorphisms with two Jordan blocks. The Weyl group and component group actions admit a diagrammatic description in terms of cup diagrams which appear in the definition of arc algebras of types B and D. We determine the decomposition of the representations into irreducibles and relate our construction to classical Springer theory. As an application we obtain presentations of the cohomology rings of all two-block Springer fibers of types C and D. Moreover, we deduce explicit isomorphisms between the Kazhdan-Lusztig cell modules attached to the induced trivial module and the irreducible Specht modules in types C and D.

Applications of self-distributivity to Yang–Baxter operators and their cohomology

Self-distributive (SD) structures form an important class of solutions to the Yang–Baxter equation (YBE), which underlie spectacular knot-theoretic applications of self-distributivity (SD). It is less known that one can go the other way around, and construct an SD structure out of any left non-degenerate (LND) set-theoretic YBE solution. This structure captures important properties of the solution: invertibility, involutivity, biquandle-ness, the associated braid group actions. Surprisingly, the tools used to study these associated SD structures also apply to the cohomology of LND solutions, which generalizes SD cohomology. Namely, they yield an explicit isomorphism between two cohomology theories for these solutions, which until recently were studied independently. The whole story is full of open problems. One of them is the relation between the cohomologies of a YBE solution and its associated SD structure. These and related questions are covered in the present survey.

The Hecke algebras for the orthogonal group SO(2,3) and the paramodular group of degree 2

In this paper, we consider the integral orthogonal group with respect to the quadratic form of signature [Formula: see text] given by [Formula: see text] for square-free [Formula: see text]. The associated Hecke algebra is commutative and also the tensor product of its primary components, which turn out to be polynomial rings over [Formula: see text] in two algebraically independent elements. The integral orthogonal group is isomorphic to the paramodular group of degree [Formula: see text] and level [Formula: see text], more precisely to its maximal discrete normal extension. The results can be reformulated in the paramodular setting by virtue of an explicit isomorphism. The Hecke algebra of the non-maximal paramodular group inside [Formula: see text] fails to be commutative if [Formula: see text].

A transfer-operator-based relation between Laplace eigenfunctions and zeros of Selberg zeta functions

Over the last few years Pohl (partly jointly with coauthors) has developed dual ‘slow/fast’ transfer operator approaches to automorphic functions, resonances, and Selberg zeta functions for a certain class of hyperbolic surfaces$\unicode[STIX]{x1D6E4}\backslash \mathbb{H}$with cusps and all finite-dimensional unitary representations$\unicode[STIX]{x1D712}$of$\unicode[STIX]{x1D6E4}$. The eigenfunctions with eigenvalue 1 of the fast transfer operators determine the zeros of the Selberg zeta function for$(\unicode[STIX]{x1D6E4},\unicode[STIX]{x1D712})$. Further, if$\unicode[STIX]{x1D6E4}$is cofinite and$\unicode[STIX]{x1D712}$is the trivial one-dimensional representation then highly regular eigenfunctions with eigenvalue 1 of the slow transfer operators characterize Maass cusp forms for$\unicode[STIX]{x1D6E4}$. Conjecturally, this characterization extends to more general automorphic functions as well as to residues at resonances. In this article we study, without relying on Selberg theory, the relation between the eigenspaces of these two types of transfer operators for any Hecke triangle surface$\unicode[STIX]{x1D6E4}\backslash \mathbb{H}$of finite or infinite area and any finite-dimensional unitary representation$\unicode[STIX]{x1D712}$of the Hecke triangle group $\unicode[STIX]{x1D6E4}$. In particular, we provide explicit isomorphisms between relevant subspaces. This solves a conjecture by Möller and Pohl, characterizes some of the zeros of the Selberg zeta functions independently of the Selberg trace formula, and supports the previously mentioned conjectures.

Isomorphism Between the R-Matrix and Drinfeld Presentations of Yangian in Types B, C and D

It is well-known that the Gauss decomposition of the generator matrix in the R-matrix presentation of the Yangian in type A yields generators of its Drinfeld presentation. Defining relations between these generators are known in an explicit form, thus providing an isomorphism between the presentations. It has been an open problem since the pioneering work of Drinfeld to extend this result to the remaining types. We give a solution for the classical types B, C and D by constructing an explicit isomorphism between the R-matrix and Drinfeld presentations of the Yangian. It is based on an embedding theorem which allows us to consider the Yangian of rank $${n-1}$$ as a subalgebra of the Yangian of rank n of the same type.

Lie Algebras of Slow Growth and Klein-Gordon PDE

We discuss the notion of characteristic Lie algebra of a hyperbolic PDE. The integrability of a hyperbolic PDE is closely related to the properties of the corresponding characteristic Lie algebra χ. We establish two explicit isomorphisms: Hence the Lie algebras $\chi (\sinh {u})$ and χ(eu + e− 2u) are slowly linearly growing Lie algebras with average growth rates $\frac {3}{2}$ and $\frac {4}{3}$ respectively.

A propos de l'algèbre de Hopf des mots tassés WMat

In this article we study the packed words Hopf algebra WMat introduced by Duchamp, Hoang-Nghia et Tanasa. We start by explaining that WMat is not cofree, giving its antipode and describing its graded dual. We consider then a Hopf sub-algebra of permutations called SH. Its graded dual SH⊛ has a quadri-algebra structure, so it has a double dendriform algebra structure too. Thereafter, we introduce ISPW, a Hopf algebra of increasing strict packed words. It is graded, connected and cocommutative so is isomorphic to the enveloping algebra of its primitive elements. We describe some families of primitive elements. We prove that ISPW and non commutative symmetric functions are isomorphic. We define then an extended compositions Hopf algebra Ce. It is not cocommutative but its primitive elements and those from ISPW are linked. We give an interpretation of Ce in terms of a semi-direct coproduct Hopf algebra. By using this, we can define two actions groups. We finish by giving an explicit isomorphism between ISPW⊛ and QSym and another one between ISPW and NSym.

Counting points on the Fricke–Macbeath curve over finite fields

The Fricke-Macbeath curve is a smooth projective algebraic curve of genus 7 with automorphism group PSL₂(픽₈). We recall two models of it (introduced, respectively, by Maxim Hendriks and by Bradley Brock) defined over ℚ, and we establish an explicit isomorphism defined over ℚ( −7 ) between these models. Moreover, we decompose up to isogeny over ℚ the jacobian of one of these models. As a consequence we obtain a simple formula for the number of points over 픽q on (the reduction of) this model, in terms of the elliptic curve with equation y² = x³ + x² − 114x − 127. Moreover, twists by elements of PSL₂(픽₈) of the curve over finite fields are described. The curve leads to a number of new records as maintained on manYPoints of curves of genus 7 with many rational points over finite fields.

Affine cellularity of affine Yokonuma–Hecke algebras

We establish an explicit algebra isomorphism between the affine Yokonuma–Hecke algebra Yˆr,n(q) and a direct sum of matrix algebras with coefficients in tensor products of affine Hecke algebras of type A. As an application of this result, we show that Yˆr,n(q) is affine cellular in the sense of Koenig and Xi, and further prove that it has finite global dimension when the parameter q is not a root of the Poincaré polynomial. As another application, we also recover the modular representation theory of Yˆr,n(q) previously obtained in [7].

A note on symmetry classes of tensors and Specht modules

We construct induced bases for critical orbital spaces indexed by standard tableaux. As a corollary we obtain an explicit isomorphism of C[Sm]-modules between Specht modules and critical orbital spaces which yields a realization of the complete set of complex irreducible representations of the symmetric group in terms of decomposable symmetrized tensors.

A constructive approach to a conjecture by Voskresenskii

Voskresenskii conjectured that stably rational tori are rational. Klyachko proved this assertion for a wide class of tori by general principles. We re-prove Klyachko’s result by providing simple explicit birational isomorphisms, and elaborate on some links to torus-based cryptography.

Isomorphisms between simple modules of degenerate cyclotomic Hecke algebras

We give explicit isomorphisms between simple modules of degenerate cyclotomic Hecke algebras defined via various cellular bases. A special case gives a generalized Mullineux involution in the degenerate case.

Cyclotomic Yokonuma–Hecke algebras are cyclotomic quiver Hecke algebras

We prove that cyclotomic Yokonuma–Hecke algebras of type A are cyclotomic quiver Hecke algebras and we give an explicit isomorphism with its inverse, using a similar result of Brundan and Kleshchev on cyclotomic Hecke algebras. The quiver we use is given by disjoint copies of cyclic quivers. We relate this work to an isomorphism of Lusztig.

A classical approach to smooth supermanifolds

This is an extension of the author's Master's thesis written under the supervision of Dr. Gregor Weingart at the National Autonomous University of Mexico. By extension we mean that some of the results have been rewritten and some others have been added to the original work. The purpose of this study is to rewrite di erential supergeometry in terms of classical di erential geometry. This rewriting from rst principles has two main motivations: 1. avoid using local (and usually not very well-de ned) odd coordinates; 2. use both the language and the tools (both highly developed) of classical di erential geometry to state and prove results in supergeometry. Although there is work in this direction (for instance [SV86]) this work's point of view might be useful to translate from the sheaf-theoretic language to one that is better suited for explicit calculations. We now give a summary of the contents. Chapter 1 is about superalgebras. In particular, we give explicit isomorphisms for the space of superderivations of an exterior algebra. The nal section of this chapter is devoted to the twisted action of the symmetric group on the supervector space of tensors with a xed rank; this allows us to construct the supersymmetric and superexterior algebras of a nite-dimensional supervector space. In chapter 2 we begin our study of smooth nite-dimensional supermanifolds. As the title of this monograph indicates, our approach will be rather classical in the following sense: our de nition of a supermanifold is not given in terms of local charts nor in terms of sheaves of superalgebras; we rather study superalgebra bundles over smooth manifolds. That is, supermanifolds in our sense are vector bundles such that the bre at each point is a free supercommutative superalgebra of nite rank. This approach allows the use of tools from di erential geometry which in many cases (e.g. Batchelor's theorem, which we prove as corollary 2.12) simpli es the proofs; furthermore, we prove (theorem 2.11) that both approaches are equivalent. A noteworthy feature of our approach is that supersmooth maps turn out to be generalizations of linear di erential operators (proposition 2.6). With our approach we prove special splittings of the tangent bundle (theorem 2.25) and study the tangential maps of a supersmooth map between two supermanifolds which give conditions for the supermanifold to be split (proposition 2.36). We also prove that the de Rham cohomology of a supermanifold is isomorphic to the de Rham cohomology of the underlying smooth manifold (theorem 2.47) by writing down the exterior derivative in a di erent way (2.45). We include three appendices. The rst one concerns linear di erential operators; there we develop the necessary tools to understand supersmooth maps with our approach. The two other appendices deal with algebraic results of independent interest. The second appendix is the statement and proof of the Cartan Poincare lemma, a result on the (co)homology of a complex that arises in our study. The third appendix deals with the proof of an algebraic fact (lemma 1.26) we use in order to prove a owbox coordinates theorem for supermanifolds (theorem 2.34) which is the foundation of the main results of chapter 2.

Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation

This paper deals with left non-degenerate set-theoretic solutions to the Yang–Baxter equation (= LND solutions), a vast class of algebraic structures encompassing groups, racks, and cycle sets. To each such solution there is associated a shelf (i.e., a self-distributive structure) which captures its major properties. We consider two (co)homology theories for LND solutions, one of which was previously known, in a reduced form, for biracks only. An explicit isomorphism between these theories is described. For groups and racks we recover their classical (co)homology, whereas for cycle sets we get new constructions. For a certain type of LND solutions, including quandles and non-degenerate cycle sets, the (co)homologies split into the degenerate and the normalized parts. We express 2-cocycles of our theories in terms of group cohomology, and, in the case of cycle sets, establish connexions with extensions. This leads to a construction of cycle sets with interesting properties.

A Uniform Model for Kirillov–Reshetikhin Crystals II. Alcove Model, Path Model, and $P=X$

We establish the equality of the specialization |$P_\lambda(x;q,0)$| of the Macdonald polynomial at |$t=0$| with the graded character |$X_\lambda(x;q)$| of a tensor product of “single-column” Kirillov–Reshetikhin modules for untwisted affine Lie algebras. This is achieved by constructing two uniform combinatorial models for the crystals associated with the mentioned tensor products: the quantum alcove model (which is naturally associated with Macdonald polynomials) and the quantum Lakshmibai–Seshadri path model. We provide an explicit affine crystal isomorphism between the two models, and realize the energy function in both models. In particular, this gives the first proof of the positivity of the |$t = 0$| limit of the symmetric Macdonald polynomial in the untwisted and non-simply laced cases, when it is expressed as a linear combination of the irreducible characters for a finite-dimensional simple Lie subalgebra, as well as a representation-theoretic meaning of the coefficients in this expression in terms of degree functions.

Analytic and topological index maps with values in the $K$-theory of mapping cones

Index maps taking values in the K -theory of a mapping cone are defined and discussed. The resulting index theorem can be viewed in analogy with the Freed–Melrose index theorem. The framework of geometric K -homology is used in a fundamental way. In particular, an explicit isomorphism from a geometric model for K -homology with coefficients in a mapping cone, C_{\phi} , to KK(C(X),C_{\phi}) is constructed.

Homological Description of Crystal Structures on Lusztig’s Quiver Varieties

Using methods of homological algebra, we obtain an explicit crystal isomorphism between two realizations of crystal bases of the lower part of the quantized enveloping algebra of (almost all) finite dimensional simply-laced Lie algebras. The first realization we consider is a geo- metric construction in terms of irreducible components of certain quiver varieties established by Kashiwara and Saito. The second is a realization in terms of isomorphism classes of quiver repre- sentations obtained by Reineke using Ringel's Hall algebra approach to quantum groups. We show that these two constructions are closely related by studyin g sufficiently generic representations of the preprojective algebra.