arXiv subjects

Jacob Greenstein

Publications and source records attributed to Jacob Greenstein.

At least 19 recordsLinked to original sources

arXiv2026

Magnonic Combinatorial Memory based on a network of coupled active ring circuits

Magnonic Combinatorial Memory (MCM) is a type of memory where the bits of information are encoded in the signal propagation paths in the network. In this work, we consider MCM based on the network of a coupled active ring circuit (ARC). Each circuit includes a broadband amplifier, a magnonic delay line, an adjustable frequency filter, an adjustable phase shifter, and a power detector. The coupling between the circuits is via spin waves propagating in the common delay line - ferrite film. There may or may not be auto-oscillations in the active ring circuits, depending on the combination of circuit parameters and circuit coupling. The address of MCM is defined as the combination of the states of the phase shifters and frequency filters, while the MCM state is defined as the presence/absence of the auto-oscillations. The coupling between the circuits is achieved by placing micromagnets on top of the ferrite film. The number of bits that can be encoded in the network increases quadratically with the number of coupled circuits. This scaling provides a fundamental advantage over conventional memory. We present experimental data obtained for three magnonic ARCs connected via a single-crystal yttrium iron garnet Y3Fe2(FeO4)3 (YIG) film. The data illustrate an example of encoding a 27-bit binary response pattern, corresponding to the 27 experimentally accessible phase combinations. The results demonstrate a robust operation of MCM with an On/Off ratio exceeding 30 dB at room temperature. The advantages and shortcomings of the proposed approach are discussed.

cond-mat.other↗

arXiv2026

Artin monoids, their homomorphisms and twins

Motivated by the twin homomorphism problem for Coxeter groups and the corresponding Hecke monoids, we find a large class of its solutions originating from standard homomorphisms of Artin monoids and their compositions. These homomorphisms are expected to be injective when they are optimal and injective on generators, which generalizes the homogeneous homomorphisms and the famous Tits conjecture settled by Crisp and Paris. We classify disjoint standard homomorphisms and conjecture the complete classification when the domain is of rank two.

math.QA↗

arXiv2026

Hecke monoids, their homomorphisms and parabolicity

We study homomorphisms of Hecke monoids, notably parabolic homomorphisms, which map parabolic elements to parabolic elements, and injective ones. The importance of the first class stems from the fact that parabolic elements form a rather mysterious submonoid of the Hecke monoid, and we found a plethora of parabolic homomorphisms. Concerning injective ones, as a first step towards their classification, we classified all locally injective connected homomorphisms between Hecke monoids of classical types and expect all of them to be injective. As a surprising byproduct of our study of parabolic and injective homomorphisms we described, to some extent, all homomorphisms between Hecke monoids.

math.RT↗

arXiv2026

Monomial bialgebras

Starting from a single solution of QYBE (or CYBE) we produce an infinite family of solutions of QYBE (or CYBE) parametrized by transitive arrays and, in particular, by signed permutations. We are especially interested in cases when such solutions yield quasi-triangular structures on direct powers of Lie bialgebras and tensor powers of Hopf algebras. We obtain infinite families of such structures as well and study the corresponding Poisson-Lie structures and co-quasi-triangular algebras.

math.QA↗

arXiv2024

Hecke and Artin monoids and their homomorphisms

The aim of the present work is to systematically study homomorphisms of Hecke and Artin monoids and thus to develop their comprehensive theory. Our original motivation was the striking observation that parabolic projections of Hecke monoids respect all parabolic elements. We found other classes of homomorphisms of Hecke monoids with the same property and discovered that many of them lift to homomorphisms of covering Artin monoids with a similar property. It turned out that they belong to a much larger class (in fact, a category) of homomorphisms of Artin monoids, most of which appear to be new.

math.RT↗

arXiv2018

On cacti and crystals

In the present work we study actions of various groups generated by involutions on the category $\mathscr O^{int}_q(\mathfrak g)$ of integrable highest weight $U_q(\mathfrak g)$-modules and their crystal bases for any symmetrizable Kac-Moody algebra $\mathfrak g$. The most notable of them are the cactus group and (yet conjectural) Weyl group action on any highest weight integrable module and its lower and upper crystal bases. Surprisingly, some generators of cactus groups are anti-involutions of the Gelfand-Kirillov model for $\mathscr O^{int}_q(\mathfrak g)$ closely related to the remarkable quantum twists discovered by Kimura and Oya.

math.QA↗

arXiv2016

Canonical bases of quantum Schubert cells and their symmetries

The goal of this work is to provide an elementary construction of the canonical basis $\mathbf B(w)$ in each quantum Schubert cell~$U_q(w)$ and to establish its invariance under modified Lusztig's symmetries. To that effect, we obtain a direct characterization of the upper global basis $\mathbf B^{up}$ in terms of a suitable bilinear form and show that $\mathbf B(w)$ is contained in $\mathbf B^{up}$ and its large part is preserved by modified Lusztig's symmetries.

math.QA↗

arXiv2015

Koszul duality for semidirect products and generalized Takiff algebras

We obtain Koszul-type dualities for categories of graded modules over a graded associative algebra which can be realized as the semidirect product of a bialgebra coinciding with its degree zero part and a graded module algebra for the latter. In particular, this applies to graded representations of the universal enveloping algebra of the Takiff Lie algebra (or the truncated current algebra) and its (super)analogues, and also to semidirect products of quantum groups with braided symmetric and exterior module algebras in case the latter are flat deformations of classical ones.

math.RT↗

arXiv2015

Generalized Joseph's decompositions

We generalize the decomposition of $U_q(\mathfrak g)$ introduced by A. Joseph and relate it, for $\mathfrak g$ semisimple, to the celebrated computation of central elements due to V. Drinfeld. In that case we construct a natural basis in the center of $U_q(\mathfrak g)$ whose elements behave as Schur polynomials and thus explicitly identify the center with the ring of symmetric functions.

math.QA↗

arXiv2014

Integrable clusters

The goal of this note is to study quantum clusters in which cluster variables (not coefficients) commute which each other. It turns out that this property is preserved by mutations. Remarkably, this is equivalent to the celebrated sign coherence conjecture recently proved by M. Gross, P. Hacking, S. Keel and M. Kontsevich

math.QA↗

arXiv2014

Double canonical bases

We introduce a new class of bases for quantized universal enveloping algebras $U_q(\mathfrak g)$ and other doubles attached to semisimple and Kac-Moody Lie algebras. These bases contain dual canonical bases of upper and lower halves of $U_q(\mathfrak g)$ and are invariant under many symmetries including all Lusztig's symmetries if $\mathfrak g$ is semisimple. It also turns out that a part of a double canonical basis of $U_q(\mathfrak g)$ spans its center.

math.QA↗

arXiv2012

Primitively generated Hall algebras

In the present paper we show that Hall algebras of finitary exact categories behave like quantum groups in the sense that they are generated by indecomposable objects. Moreover, for a large class of such categories, Hall algebras are generated by their primitive elements, with respect to the natural comultiplication, even for non-hereditary categories. Finally, we introduce certain primitively generated subalgebras of Hall algebras and conjecture an analogue of "Lie correspondence" for those finitary categories.

math.QA↗

arXiv2012

Quantum Chevalley groups

The goal of this paper is to construct quantum analogues of Chevalley groups inside completions of quantum groups or, more precisely, inside completions of Hall algebras of finitary categories. In particular, we obtain pentagonal and other identities in the quantum Chevalley groups which generalize their classical counterparts and explain Faddeev-Volkov quantum dilogarithmic identities and their recent generalizations due to Keller

math.QA↗

arXiv2010

On Homomorphisms Between Global Weyl Modules

Global Weyl modules for generalized loop algebras $\lie g\tensor A$, where $\lie g$ is a simple finite dimensional Lie algebra and A is a commutative associative algebra were defined, for any dominant integral weight $λ$, by generators and relations. They are expected to play the role similar to that of Verma modules in the study of categories of representations of these algebras. One of the fundamental properties of Verma modules is that the space of morphisms between two Verma modules is either zero or one--dimensional and also that any non--zero morphism is injective. The aim of this paper is to establish an analogue of this property for the global Weyl modules. This is done under certain restrictions on the Lie algebra $\lie g$, $λ$ and $A$. A crucial tool is the construction of fundamental global Weyl modules in terms of fundamental local Weyl modules given in Section 3.

math.RT↗

arXiv2010

Quantum folding

In the present paper we introduce a quantum analogue of the classical folding of a simply-laced Lie algebra g to the non-simply-laced algebra g^sigma along a Dynkin diagram automorphism sigma of g For each quantum folding we replace g^sigma by its Langlands dual g^sigma^v and construct a nilpotent Lie algebra n which interpolates between the nilpotnent parts of g and (g^sigma)^v, together with its quantized enveloping algebra U_q(n) and a Poisson structure on S(n). Remarkably, for the pair (g, (g^sigma)^v)=(so_{2n+2},sp_{2n}), the algebra U_q(n) admits an action of the Artin braid group Br_n and contains a new algebra of quantum n x n matrices with an adjoint action of U_q(sl_n), which generalizes the algebras constructed by K. Goodearl and M. Yakimov in [12]. The hardest case of quantum folding is, quite expectably, the pair (so_8,G_2) for which the PBW presentation of U_q(n) and the corresponding Poisson bracket on S(n) contain more than 700 terms each.

math.QA↗

arXiv2010

Minimal affinizations as projective objects

We prove that the specialization to q=1 of a Kirillov-Reshetikhin module for an untwisted quantum affine algebra of classical type is projective in a suitable category. This yields a uniform character formula for the Kirillov-Reshetikhin modules. We conjecture that these results holds for specializations of minimal affinization with some restriction on the corresponding highest weight. We discuss the connection with the conjecture of Nakai and Nakanishi on q-characters of minimal affinizations. We establish this conjecture in some special cases. This also leads us to conjecture an alternating sum formula for Jacobi-Trudi determinants.

math.QA↗

arXiv2008

Quivers with relations arising from Koszul algebras of $\mathfrak g$-invariants

Let $\mathfrak g$ be a complex simple Lie algebra and let $Ψ$ be an extremal set of positive roots. One associates with $Ψ$ an infinite dimensional Koszul algebra $\bold S_Ψ^{\lie g}$ which is a graded subalgebra of the locally finite part of $((\bold V)^{op}\tensor S(\lie g))^{\lie g}$, where $\bold V$ is the direct sum of all simple finite dimensional $\lie g$-modules. We describe the structure of the algebra $\bold S_Ψ^{\lie g}$ explicitly in terms of an infinite quiver with relations for $\lie g$ of types $A$ and $C$. We also describe several infinite families of quivers and finite dimensional algebras arising from this construction.

math.RT↗

arXiv2008

A family of Koszul algebras arising from finite-dimensional representations of simple Lie algebras

Let $\lie g$ be a simple Lie algebra and let $\bs^{\lie g}$ be the locally finite part of the algebra of invariants $(_\bc\bv\otimes S(\lie g))^{\lie g}$ where $\bv$ is the direct sum of all simple finite-dimensional modules for $\lie g$ and $S(\lie g)$ is the symmetric algebra of $\lie g$. Given an integral weight $ξ$, let $Ψ=Ψ(ξ)$ be the subset of roots which have maximal scalar product with $ξ$. Given a dominant integral weight $λ$ and $ξ$ such that $Ψ$ is a subset of the positive roots we construct a finite-dimensional subalgebra $\bs^{\lie g}_Ψ(\le_Ψλ)$ of $\bs^{\lie g}$ and prove that the algebra is Koszul of global dimension at most the cardinality of $Ψ$. Using this we then construct naturally an infinite-dimensional Koszul algebra of global dimension equal to the cardinality of $Ψ$. The results and the methods are motivated by the study of the category of finite-dimensional representations of the affine and quantum affine algebras.

math.RT↗