Algebra Homomorphism Research Articles

Equivariant algebraic concordance of strongly invertible knots

AbstractBy considering a particular type of invariant Seifert surfaces we define a homomorphism from the (topological) equivariant concordance group of directed strongly invertible knots to a new equivariant algebraic concordance group . We prove that lifts both Miller and Powell's equivariant algebraic concordance homomorphism (J. Lond. Math. Soc. (2023), no. 107, 2025‐2053) and Alfieri and Boyle's equivariant signature (Michigan Math. J. 1 (2023), no. 1, 1–17). Moreover, we provide a partial result on the isomorphism type of and obtain a new obstruction to equivariant sliceness, which can be viewed as an equivariant Fox–Milnor condition. We define new equivariant signatures and using these we obtain novel lower bounds on the equivariant slice genus. Finally, we show that can obstruct equivariant sliceness for knots with Alexander polynomial one.

On Some Topological Properties of Ch(X) and of the Dual Operator

The hypo-topology on the algebra C(X) of real-valued continuous functions defined on a Tychonoff space 2X f ∈ C(X) with its hypograph, hypof = {(x,t) ∈ X x R:f(x) ≥ t}. This topology is very useful in the calculus of variations and in optimization theory (e.g. Maximization problems). We denote C(X) with the hypo-topology by Ch(X). Our study deals with fundamental properties of these function spaces, and with the linear operators on them and as well as the characterization of the topological properties of Ch(X) in terms of topological properties of the base space X. We are studying the linear operator between the functional algebras Ch(X) and the Ch(Y). We are primarily concerned with the continuity of the evaluation functional, the general evaluation and the continuity of the characters of Ch(X) before the investigation of the properties of the dual operator f* of a continuous function f:X→Y. This operator is defined by f*:Ch (Y)→Ch (X), where f*(g)=gof for all g ∈ C(Y). The continuity of f* enables us to characterize the continuity of an algebra homomorphism of the type φ:Ch (Y)→Ch (X) for a realcompact space Y. For such a space, we present a type of Riesz theorm with states that an algebra homomorphism φ: Ch(Y)→Ch(X) is continuous if and only if there exists a unique hypo-function f: X→Y such that φ = f*. There after, we give the equivalence between the properties of f and those of f*. The study of the continuous linear functional on Ch(X) helps us to compute the topological dual space of this algebra. We show here that this dual space is useful only when the set of isolated points of X is dense.

On Lie Homomorphisms of Complex IntuitionisticFuzzy Lie Algebras

In this paper, we investigate the properties of complex intuitionistic fuzzy Lie subalgebras and ideals under Lie algebra homomorphisms, focusing on both images and inverse images. We provide detailed proofs for several new results concerning the preservation of complex intuitionistic fuzzy structures through homomorphisms, and we introduce additional homomorphism-related properties for these structures. This work extends known results to the context of complex intuitionistic fuzzy Lie algebras, contributing new insights into their behavior under algebraic mappings.

Transferring algebra structures on complexes

With the goal of transferring dg algebra structures on complexes along contractions, we introduce a new condition on the associated homotopy, namely a generalized version of the Leibniz rule. We prove that, with this condition, the transfer works to yield a dg algebra (with vanishing descended higher A∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_\\infty $$\\end{document} products) and prove that it works also after an application of the Perturbation Lemma even though the new homotopy may no longer satisfy that condition. We also extend these results to the setting of A∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_\\infty $$\\end{document} algebras. Then we return to our original motivation from commutative algebra. We apply these methods to find a new method for building a dg algebra structure on a well-known resolution, obtaining one that is both concrete and permutation invariant. The naturality of the construction enables us to find dg algebra homomorphisms between these as well, enabling them to be used as inputs for constructing bar resolutions.

Twisted tensor products of quantum affine vertex algebras and coproducts

Let g be a symmetrizable Kac-Moody Lie algebra, and let Vgˆ,ħℓ, Lgˆ,ħℓ be the quantum affine vertex algebras constructed in [11]. For any complex numbers ℓ and ℓ′, we present an ħ-adic quantum vertex algebra homomorphism Δ from Vgˆ,ħℓ+ℓ′ to the twisted tensor product ħ-adic quantum vertex algebra Vgˆ,ħℓ⊗ˆVgˆ,ħℓ′. In addition, if both ℓ and ℓ′ are positive integers, we show that Δ induces an ħ-adic quantum vertex algebra homomorphism from Lgˆ,ħℓ+ℓ′ to the twisted tensor product ħ-adic quantum vertex algebra Lgˆ,ħℓ⊗ˆLgˆ,ħℓ′. Moreover, we prove the coassociativity of Δ.

Modular Representation Theory and Commutative Banach Algebras

In a recent paper of Benson and Symonds, a new invariant was introduced for modular representations of a finite group. An interpretation was given as a spectral radius with respect to a Banach algebra completion of the representation ring. Our purpose here is to take these notions further, and investigate the structure of the resulting Banach algebras. Some of the material in that paper is repeated here in greater generality, and for clarity of exposition. We give an axiomatic definition of an abstract representation ring, and representation ideal. The completion is then a commutative Banach algebra, and the techniques of Gelfand from the 1940s are applied in order to study the space of algebra homomorphisms to C \mathbb {C} . One surprising consequence of this investigation is that the Jacobson radical and the nil radical of a (complexified) representation ring always coincide. These notes are intended for representation theorists. So background material on commutative Banach algebras is given in detail, whereas representation theoretic background is more condensed.

Quasiregular curves and cohomology

AbstractLet be a closed, connected, and oriented Riemannian manifold, which admits a quasiregular ‐curve with infinite energy. We prove that, if the de Rham class of is nonzero and a finite sum of nontrivial products, then there exists a nontrivial graded algebra homomorphism from the de Rham algebra of to the exterior algebra . As an application, we give examples of pairs , where is a closed manifold and is a closed ‐form for , for which every quasiregular ‐curve is constant.

Inverse homomorphisms of finite groups

Abstract Replacing the equality in the condition of the homomorphism of a multi-based universal algebra with inequality leads to the definition of an inverse homomorphism of a multi-based universal algebra. In this paper methods for constructing nontrivial inverse homomorphisms of finite groups are studied.

Algebra environments II. Algebra homomorphisms and derivations

Algebra environments provide requisites for studying objects of interest in operator algebra theory, group representation theory, spin geometry, Clifford analysis, and several variable operator theory. The concept is analyzed by developing an algebraic geometry approach. Specific algebraic sets, called structure manifolds of algebra environments, and their Zariski tangent spaces are introduced and described by using as critical tools derivations on algebras. Structure manifolds of tensor environments in particular yield spaces of algebra homomorphisms. Consequently, such spaces could be investigated as algebraic manifolds. Related issues include characterizations of their Zariski tangent spaces and of derivations that preserve algebra homomorphisms.

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.

Cohomology and deformation theory of crossed homomorphisms of Leibniz algebras

In this paper, we construct a differential graded Lie algebra whose Maurer–Cartan elements are given by crossed homomorphisms on Leibniz algebras. This allows us to define cohomology for a crossed homomorphism. Finally, we study linear deformations, formal deformations and extendibility of finite order deformations of a crossed homomorphism in terms of the cohomology theory.

Construction of the Affine Super Yangian

In this paper, we define the affine super Yangian Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}}(m|n)) with a coproduct structure. We also obtain an evaluation homomorphism, that is, an algebra homomorphism from Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}}(m|n)) to the completion of the universal enveloping algebra of \widehat{\mathfrak{gl}}(m|n) .

On cohomological and K-theoretical Hall algebras of symmetric quivers

We give a brief review of the cohomological Hall algebra CoHA H and the K-theoretical Hall algebra KHA R associated to quivers. In the case of symmetric quivers, we show that there exists a homomorphism of algebras (obtained from a Chern character map) R→Hˆσ˜ where Hˆσ˜ is a Zhang twist of the completion of H. Moreover, we establish the equivalence of categories of “locally finite” graded modules H-Modlf≃RQ-Modlf. Examples of locally finite Hˆ-, resp. RQ-modules appear naturally as the cohomology, resp. K-theory, of framed moduli spaces of quivers.

Homomorphisms of L1 algebras and Fourier algebras

We investigate conditions for the extendibility of continuous algebra homomorphisms ϕ from the Fourier algebra A(F) of a locally compact group F to the Fourier-Stieltjes algebra B(G) of a locally compact group G to maps between the corresponding L∞ algebras which are weak* continuous. When ϕ is completely bounded and F is amenable, it is induced by a piecewise affine map α:Y→F where Y⊆G. We show that extendibility of ϕ is equivalent to α being an open map. We also study the dual problem for contractive homomorphisms ϕ:L1(F)→M(G). We show that ϕ induces a w* continuous homomorphism between the von Neumann algebras of the groups if and only if the naturally associated map θ (Greenleaf [1965], Stokke [2011]) is a proper map.

Representation of operators using fusion frames

To solve operator equations numerically, matrix representations are employing bases or more recently frames. For finding the numerical solution of operator equations a decomposition in subspaces is needed in many applications. To combine those two approaches, it is necessary to extend the known methods of matrix representation to the utilization of fusion frames. In this paper, we investigate this representation of operators on a Hilbert space H with Bessel fusion sequences, fusion frames and fusion Riesz bases. Fusion frames can be considered as a frame-like family of subspaces. Taking the particular property of the duality of fusion frames into account, this allows us to define a matrix representation in a canonical as well as an alternate way, the later being more efficient and well behaved in respect to inversion. We will give the basic definitions and show some structural results, like that the functions assigning the alternate representation to an operator is an algebra homomorphism. We give formulas for pseudo-inverses and the inverses (if existing) of such matrix representations. We apply this idea to Schatten p-class operators. Consequently, we show that tensor products of fusion frames are fusion frames in the space of Hilbert-Schmidt operators. We will show how this can be used for the solution of operator equations and link our approach to the additive Schwarz algorithm. Consequently, we propose some methods for solving an operator equation by iterative methods on the subspaces. Moreover, we implement the alternate Schwarz algorithms employing our perspective and provide small proof-of-concept numerical experiments. Finally, we show the application of this concept to overlapped convolution and the non-standard wavelet representation of operators.

Projective relative unification through duality

AbstractUnification problems can be formulated and investigated in an algebraic setting, by identifying substitutions to modal algebra homomorphisms. This opens the door to applications of the notorious duality between Heyting or modal algebras and descriptive frames. Through substantial use of this correspondence, we give a necessary and sufficient condition for formulas to be projective. A close inspection of this characterization will motivate a generalization of standard unification, which we dub relative unification. Applying this result to a number of different logics, we then obtain new proofs of their projective—or non-projective—character. Aside from reproving known results, we show that the projective extensions of $\textbf{K5}$ are exactly the extensions of $\textbf{K45}$. This resolves the open question of whether $\textbf{K5}$ is projective.

Hybrid near Algebra

The objective of this paper is to study the hybrid near algebra. It has been summarized with the proper definitions and theorems of hybrid near algebra, hybrid near algebra homomorphism and direct product of hybrid near algebra. It has been proved that a homomorphic image of a hybrid near algebra is a hybrid near algebra. It also investigated the intersection of two hybrid near algebras is a hybrid near algebra.

Quasi-Whittaker modules for the n-th Schrödinger algebra

The n-th Schrödinger algebra schn defined in [14] is the semi-direct product of the Lie algebra sl2 with the n-th Heisenberg Lie algebra hn, which generalizes the Schrödinger algebra sl2⋉h1. Let ϕ:hn→C be a nonzero Lie algebra homomorphism. A schn-module V is called quasi-Whittaker of type ϕ if V=U(schn)v, where U(schn) is the universal enveloping algebra of schn, v is a nonzero vector such that xv=ϕ(x)v for any x∈hn. In this paper, we prove that a simple schn-module V is a quasi-Whittaker module if and only if V is a locally finite hn-module. Then we classify the simple quasi-Whittaker modules of ϕ, according to the rank of ϕ. Furthermore, we characterize arbitrary quasi-Whittaker modules through the rank of ϕ.

A connection behind the Terwilliger algebras of H(D,2) and [formula omitted

The universal enveloping algebra U(sl2) of sl2 is a unital associative algebra over C generated by E,F,H subject to the relations[H,E]=2E,[H,F]=−2F,[E,F]=H. The distinguished central elementΛ=EF+FE+H22 is called the Casimir element of U(sl2). The universal Hahn algebra H is a unital associative algebra over C with generators A,B,C and the relations assert that [A,B]=C and each ofα=[C,A]+2A2+B,β=[B,C]+4BA+2C is central in H. The distinguished central elementΩ=4ABA+B2−C2−2βA+2(1−α)B is called the Casimir element of H. By investigating the relationship between the Terwilliger algebras of the hypercube and its halved graph, we discover the algebra homomorphism ♮:H→U(sl2) that sendsA↦H4,B↦E2+F2+Λ−14−H28,C↦E2−F24. We determine the image of ♮ and show that the kernel of ♮ is the two-sided ideal of H generated by β and 16Ω−24α+3. By pulling back via ♮ each U(sl2)-module can be regarded as an H-module. For each integer n≥0 there exists a unique (n+1)-dimensional irreducible U(sl2)-module Ln up to isomorphism. We show that the H-module Ln (n≥1) is a direct sum of two non-isomorphic irreducible H-modules.

Algebra extensions and derived-discrete algebras

Let be an algebra homomorphism between finite dimensional algebras. We prove that if is split, the derived-discreteness of A implies the derived-discreteness of B; if is separable and the right A-module BA is projective, the converse holds. We prove an analogous statement for piecewise hereditary algebras.