Algebra Isomorphism Research Articles

On the category 𝒪 of a hybrid quantum group

We study the representation theory of a hybrid quantum group at root of unity ζ \zeta introduced by Gaitsgory. After discussing some basic properties of its category O \mathcal {O} , we study deformations of the category O \mathcal {O} . For subgeneric deformations, we construct the endomorphism algebra of big projective object and compute it explicitly. Our main result is an algebra isomorphism between the center of deformed category O \mathcal {O} and the equivariant cohomology of ζ \zeta -fixed locus on the affine Grassmannian attached to the Langlands dual group.

Gradings on the algebra of triangular matrices as a Lie algebra: Revisited

We investigate the group gradings on the algebras of upper triangular matrices over an arbitrary field, viewed as Lie algebras. Classification results were obtained in 2017 by the same authors when the base field has characteristic different from 2. In this paper we provide streamlined proofs of these results. Moreover we present a complete classification of isomorphism classes of the group gradings on these algebras over an arbitrary field. Recall that two graded Lie algebras L1 and L2 are practically-isomorphic if there exists an (ungraded) algebra isomorphism L1→L2 that induces a graded-algebra isomorphism L1/z(L1)→L2/z(L2). We provide a classification of the practically-isomorphism classes of the group gradings on the Lie algebra of upper triangular matrices. The latter classification is a better alternative way to consider these gradings up to being essentially the same object. Finally, we investigate in details the case where the characteristic of the base field is 2, a topic that was neglected in previous works.

Higher-dimensional Heegaard Floer homology and Hecke algebras

Given a closed oriented surface \Sigma of genus greater than 0, we construct a map \mathcal{F} from the wrapped higher-dimensional Heegaard Floer homology of the cotangent fibers of T^{*}\Sigma to the Hecke algebra associated to \Sigma and show that \mathcal{F} is an isomorphism of algebras. We also establish analogous results for punctured surfaces.

Lie algebras arising from two-periodic projective complex and derived categories

Let A be a finite-dimensional C-algebra of finite global dimension and A be the category of finitely generated right A-modules. By using of the category of two-periodic projective complexes C2(P), we construct the motivic Bridgeland's Hall algebra for A, where structure constants are given by Poincaré polynomials in t, then construct a C-Lie subalgebra g=n⊕h at t=−1, where n is constructed by stack functions about indecomposable radical complexes, and h is by contractible complexes. For the stable category K2(P) of C2(P), we construct its moduli spaces and a C-Lie algebra g˜=n˜⊕h˜, where n˜ is constructed by support-indecomposable constructible functions, and h˜ is by the Grothendieck group of K2(P). We prove that the natural functor C2(P)→K2(P) together with the natural isomorphism between Grothendieck groups of A and K2(P) induces a Lie algebra isomorphism g≅g˜. This makes clear that the structure constants at t=−1 provided by Bridgeland in [5] in terms of exact structure of C2(P) precisely equal to that given in [30] in terms of triangulated category structure of K2(P).

The isomorphism problem for rational group algebras of finite metacyclic nilpotent groups

. We prove that if G and H are finite metacyclic groups with isomorphic rational group algebras and one of them is nilpotent, then G and H are isomorphic.

Quasi-free Isomorphisms of Second Quantization Algebras and Modular Theory

Using Araki–Yamagami’s characterization of quasi-equivalence for quasi-free representations of the CCRs, we provide an abstract criterion for the existence of isomorphisms of second quantization local von Neumann algebras induced by Bogolubov transformations in terms of the respective one particle modular operators. We discuss possible applications to the problem of local normality of vacua of Klein-Gordon fields with different masses.

LLT polynomials in the Schiffmann algebra

Abstract We identify certain combinatorially defined rational functions which, under the shuffle to Schiffmann algebra isomorphism, map to LLT polynomials in any of the distinguished copies Λ ⁢ ( X m , n ) ⊂ E \Lambda(X^{m{,}n})\subset\mathcal{E} of the algebra of symmetric functions embedded in the elliptic Hall algebra ℰ of Burban and Schiffmann. As a corollary, we deduce an explicit raising operator formula for the ∇ operator applied to any LLT polynomial. In particular, we obtain a formula for ∇ m s λ \nabla^{m}s_{\lambda} which serves as a starting point for our proof of the Loehr–Warrington conjecture in a companion paper to this one.

On the symmetry algebra of a one-dimensional quantum-mechanical oscillator on a hyperbola

The quantum-mechanical problem of a harmonic oscillator on a hyperbola as a one-dimensional space of constant negative curvature is considered in this article. A generalization to the singular oscillator model in the context of one-dimensional Cayley – Klein geometries is given by the factorization method. The energy spectrum and wave functions of stationary states are found having the curvature of space as a parameter. For the energy levels of the singular oscillator, the effect of non-zero curvature is clearly manifested through a positive or negative term, depending on the sign of the curvature, which is quadratic in the level number. The results obtained are consistent with those previously published. The dynamical symmetry of the problem is shown explicitly as a quadratic Hahn algebra QH(3) or its isomorphic Higgs algebra.

The algebra Uq+ and its alternating central extension 𝒰q+

Let [Formula: see text] denote the positive part of the quantized enveloping algebra [Formula: see text]. The algebra [Formula: see text] has a presentation involving two generators [Formula: see text], [Formula: see text] and two relations, called the [Formula: see text]-Serre relations. In 1993 Damiani obtained a PBW basis for [Formula: see text], consisting of some elements [Formula: see text], [Formula: see text], [Formula: see text]. In 2019, we introduced the alternating central extension [Formula: see text] of [Formula: see text]. We defined [Formula: see text] by generators and relations. The generators, said to be alternating, are denoted [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text]. Let [Formula: see text] denote the subalgebra of [Formula: see text] generated by [Formula: see text], [Formula: see text]. It is known that there exists an algebra isomorphism [Formula: see text] that sends [Formula: see text] and [Formula: see text]. Via this isomorphism we identify [Formula: see text] with [Formula: see text]. In our main result, we express the Damiani PBW basis elements in terms of the alternating generators. We give the answer in terms of generating functions. A key step is a factorization of the generating function for the elements that generate the center of [Formula: see text].

A NOTE ON PROJECTIONS IN ÉTALE GROUPOID ALGEBRAS AND DIAGONAL-PRESERVING HOMOMORPHISMS

Abstract Carlsen [‘ $\ast $ -isomorphism of Leavitt path algebras over $\Bbb Z$ ’, Adv. Math.324 (2018), 326–335] showed that any $\ast $ -homomorphism between Leavitt path algebras over $\mathbb Z$ is automatically diagonal preserving and hence induces an isomorphism of boundary path groupoids. His result works over conjugation-closed subrings of $\mathbb C$ enjoying certain properties. In this paper, we characterise the rings considered by Carlsen as precisely those rings for which every $\ast $ -homomorphism of algebras of Hausdorff ample groupoids is automatically diagonal preserving. Moreover, the more general groupoid result has a simpler proof.

Quantization of Lorentzian free BV theories: factorization algebra vs algebraic quantum field theory

We construct and compare two alternative quantizations, as a time-orderable prefactorization algebra and as an algebraic quantum field theory valued in cochain complexes, of a natural collection of free BV theories on the category of m-dimensional globally hyperbolic Lorentzian manifolds. Our comparison is realized as an explicit isomorphism of time-orderable prefactorization algebras. The key ingredients of our approach are the retarded and advanced Green’s homotopies associated with free BV theories, which generalize retarded and advanced Green’s operators to cochain complexes of linear differential operators.

On twisted group ring isomorphism problem for p-groups

Abstract In this article, we explore the problem of determining isomorphisms between the twisted complex group algebras of finite $p$ -groups. This problem bears similarity to the classical group algebra isomorphism problem and has been recently examined by Margolis-Schnabel. Our focus lies on a specific invariant, referred to as the generalized corank, which relates to the twisted complex group algebra isomorphism problem. We provide a solution for non-abelian $p$ -groups with generalized corank at most three.

Lie structure of the Heisenberg-Weyl algebra

As an associative algebra, the Heisenberg--Weyl algebra $\HWeyl$ is generated by two elements $A$, $B$ subject to the relation $AB-BA=1$. As a Lie algebra, however, where the usual commutator serves as Lie bracket, the elements $A$ and $B$ are not able to generate the whole space $\HWeyl$. We identify a non-nilpotent but solvable Lie subalgebra $\coreLie$ of $\HWeyl$, for which, using some facts from the theory of bases for free Lie algebras, we give a presentation by generators and relations. Under this presentation, we show that, for some algebra isomorphism $\isoH:\HWeyl\into\HWeyl$, the Lie algebra $\HWeyl$ is generated by the generators of $\coreLie$, together with their images under $\isoH$, and that $\HWeyl$ is the sum of $\coreLie$, $\isoH(\coreLie)$ and $\lbrak \coreLie,\isoH(\coreLie)\rbrak$.

Primitive decompositions of idempotents of the group algebras of dihedral groups and generalized quaternion groups

<p>In this paper, we introduce a method for computing the primitive decomposition of idempotents in any semisimple finite group algebra, utilizing its matrix representations and Wedderburn decomposition. Particularly, we use this method to calculate the examples of the dihedral group algebras $ \mathbb{C}[D_{2n}] $ and generalized quaternion group algebras $ \mathbb{C}[Q_{4m}] $. Inspired by the orthogonality relations of the character tables of these two families of groups, we obtain two sets of trigonometric identities. Furthermore, a group algebra isomorphism between $ \mathbb{C}[D_{8}] $ and $ \mathbb{C}[Q_{8}] $ is described, under which the two complete sets of primitive orthogonal idempotents of these group algebras correspond bijectively.</p>

On The Diagonalization Problem of Weak Fuzzy Complex Matrices Based On a Special Isomorphism

In this paper, we study the diagonalization problem of weak fuzzy complex matrices. To solve this problem we build a special algebraic isomorphism between the ring of weak fuzzy complex matrices and the direct product of the classical ring of real-entries matrices with itself, then we use it to solve the diagonalization problem by using the classical diagonalization problem for real matrices with the inverse isomorphism formula. Also, we illustrate many examples to explain the validity of our method.

Exact Borel subalgebras of path algebras of quivers of Dynkin type [formula omitted

Hereditary algebras are quasi-hereditary with respect to any adapted partial order on the indexing set of the isomorphism classes of their simple modules. For any adapted partial order on {1,…,n}, we compute the quiver and relations for the Ext-algebra of standard modules over the path algebra of a uniformly oriented linear quiver with n vertices. Such a path algebra always admits a regular exact Borel subalgebra in the sense of König and we show that there is always a regular exact Borel subalgebra containing the idempotents e1,…,en and find a minimal generating set for it. For a quiver Q and a deconcatenation Q=Q1⊔Q2 of Q at a sink or source v, we describe the Ext-algebra of standard modules over KQ, up to an isomorphism of associative algebras, in terms of that over KQ1 and KQ2. Moreover, we determine necessary and sufficient conditions for KQ to admit a regular exact Borel subalgebra, provided that KQ1 and KQ2 do. We use these results to obtain sufficient and necessary conditions for a path algebra of a linear quiver with arbitrary orientation to admit a regular exact Borel subalgebra.

Finitely presented isomorphisms of Cuntz-Krieger algebras and continuous orbit equivalence of one-sided topological Markov shifts

We introduce the notion of finitely presented isomorphism between Cuntz–Krieger algebras, and of finitely presented isomorphic Cuntz–Krieger algebras. We prove that there exists a finitely presented isomorphism between Cuntz–Krieger algebras $\mathcal{O}_A$ and $\mathcal{O}_B$ if and only if their one-sided topological Markov shifts $(X_A,\sigma_A)$ and $(X_B,\sigma_B)$ are continuously orbit equivalent. Hence the value $\det (I-A)$ is a complete invariant for the existence of a finitely presented isomorphism between isomorphic Cuntz–Krieger algebras, so that there exists a pair of Cuntz–Krieger algebras which are isomorphic but not finitely presented isomorphic.

Feigin–Semikhatov conjecture and related topics

Feigin–Semikhatov conjecture, now established, states algebraic isomorphisms between the cosets of the subregular [Formula: see text]-algebras and the principal [Formula: see text]-superalgebras of type A by their full Heisenberg subalgebras. It can be seen as a variant of Feigin–Frenkel duality between the [Formula: see text]-algebras and also as a generalization of the connection between the [Formula: see text] superconformal algebra and the affine algebra [Formula: see text]. We review the recent developments on the correspondence of the subregular [Formula: see text]-algebras and the principal [Formula: see text]-superalgebras of type A at the level of algebras, modules and intertwining operators, including fusion rules.

Stable Isomorphisms of Operator Algebras

Abstract Let ${{\mathcal{A}}}$ and ${{\mathcal{B}}}$ be operator algebras with $c_{0}$-isomorphic diagonals and let ${{\mathcal{K}}}$ denote the compact operators. We show that if ${{\mathcal{A}}}\otimes{{\mathcal{K}}}$ and ${{\mathcal{B}}}\otimes{{\mathcal{K}}}$ are isometrically isomorphic, then ${{\mathcal{A}}}$ and ${{\mathcal{B}}}$ are isometrically isomorphic. If the algebras ${{\mathcal{A}}}$ and ${{\mathcal{B}}}$ satisfy an extra analyticity condition a similar result holds with ${{\mathcal{K}}}$ being replaced by any operator algebra containing the compact operators. For nonselfadjoint graph algebras this implies that the graph is a complete invariant for various types of isomorphisms, including stable isomorphisms, thus strengthening a recent result of Dor-On, Eilers, and Geffen. Similar results are proven for algebras whose diagonals satisfy cancellation and have $K_{0}$-groups isomorphic to ${{\mathbb{Z}}}$. This has implications in the study of stable isomorphisms between various semicrossed products.

New Kind of Banach Algebra Via Proximit structure

This article introduces the concept of Banach proximit algebra and examines several results in this new context. We give new definitions for the terms "proximit algebra," "commutative proximit algebra," "identity proximit algebra," and "normed proximit algebra." However, we introduce the concepts of subproximit algebra, proximit homomorphism, and proximit isomorphism of Banach proximit algebras in a natural way. Finally, we offer a few intriguing possibilities to support our arguments.