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23 pages, 314 KiB

Open AccessArticle

Constructing and Analyzing BiHom-(Pre-)Poisson Conformal Algebras

by Sania Asif and Yao Wang

Symmetry 2024, 16(11), 1533; https://doi.org/10.3390/sym16111533 - 15 Nov 2024

Cited by 1 | Viewed by 974

This study introduces the notions of BiHom-Poisson conformal algebra, BiHom-pre-Poisson conformal algebra, and their related structures. We show that many new BiHom-Poisson conformal algebras can be constructed from a BiHom-Poisson conformal algebra. In particular, the direct product of two BiHom-Poisson conformal algebras is [...] Read more.

This study introduces the notions of BiHom-Poisson conformal algebra, BiHom-pre-Poisson conformal algebra, and their related structures. We show that many new BiHom-Poisson conformal algebras can be constructed from a BiHom-Poisson conformal algebra. In particular, the direct product of two BiHom-Poisson conformal algebras is also a BiHom-Poisson conformal algebra. We further describe the conformal bimodule and representation theory of the BiHom-Poisson conformal algebra. In addition, we define BiHom-pre-Poisson conformal algebra as the combination of BiHom-pre-Lie conformal algebra and BiHom-dendriform conformal algebra under some compatibility conditions. We further demonstrate a way to construct BiHom-Poisson conformal algebra from BiHom-pre-Poisson conformal algebra and provide the representation theory for BiHom-pre-Poisson conformal algebra. Finally, a detailed description of

O

-operators and Rota–Baxter operators on BiHom-Poisson conformal algebra is provided. Full article

(This article belongs to the Section Mathematics)

17 pages, 310 KiB

Open AccessArticle

Cohomology and Crossed Modules of Modified Rota–Baxter Pre-Lie Algebras

by Fuyang Zhu and Wen Teng

Mathematics 2024, 12(14), 2260; https://doi.org/10.3390/math12142260 - 19 Jul 2024

Viewed by 1146

Abstract

The goal of the present paper is to provide a cohomology theory and crossed modules of modified Rota–Baxter pre-Lie algebras. We introduce the notion of a modified Rota–Baxter pre-Lie algebra and its bimodule. We define a cohomology of modified Rota–Baxter pre-Lie algebras with [...] Read more.

The goal of the present paper is to provide a cohomology theory and crossed modules of modified Rota–Baxter pre-Lie algebras. We introduce the notion of a modified Rota–Baxter pre-Lie algebra and its bimodule. We define a cohomology of modified Rota–Baxter pre-Lie algebras with coefficients in a suitable bimodule. Furthermore, we study the infinitesimal deformations and abelian extensions of modified Rota–Baxter pre-Lie algebras and relate them with the second cohomology groups. Finally, we investigate skeletal and strict modified Rota–Baxter pre-Lie 2-algebras. We show that skeletal modified Rota–Baxter pre-Lie 2-algebras can be classified into the third cohomology group, and strict modified Rota–Baxter pre-Lie 2-algebras are equivalent to the crossed modules of modified Rota–Baxter pre-Lie algebras. Full article

(This article belongs to the Special Issue Advanced Research in Pure and Applied Algebra)

16 pages, 289 KiB

Open AccessArticle

Lie Modules of Banach Space Nest Algebras

by Pedro Capitão and Lina Oliveira

Mathematics 2024, 12(8), 1251; https://doi.org/10.3390/math12081251 - 20 Apr 2024

Viewed by 991

Abstract

In the present work, we extend to Lie modules of Banach space nest algebras a well-known characterisation of Lie ideals of (Hilbert space) nest algebras. Let

A

be a Banach space nest algebra and

L

be a weakly closed Lie

A

-module. We [...] Read more.

In the present work, we extend to Lie modules of Banach space nest algebras a well-known characterisation of Lie ideals of (Hilbert space) nest algebras. Let

A

be a Banach space nest algebra and

L

be a weakly closed Lie

A

-module. We show that there exist a weakly closed

A

-bimodule

K

, a weakly closed subalgebra

D_{K}

of

A

, and a largest weakly closed

A

-bimodule

J contained in L

, such that J \subseteq L \subseteq K + D_{K}

,

with [K, A] \subseteq L

. The first inclusion holds in general, whilst the second is shown to be valid in a class of nest algebras. Full article

(This article belongs to the Special Issue Advances on Nonlinear Functional Analysis)

10 pages, 216 KiB

Open AccessArticle

Frobenius Modules Associated to Algebra Automorphisms

by Ji-Wei He and Chenglong Rong

Mathematics 2024, 12(4), 531; https://doi.org/10.3390/math12040531 - 8 Feb 2024

Viewed by 1006

Abstract

Here, we study Frobenius bimodules associated with a pair of automorphisms of an algebra and discuss their basic properties. In particular, some equivalent conditions for a finite-dimensional bimodule are proved to be Frobenius and some isomorphisms between Ext-groups and Tor-groups of Frobenius modules [...] Read more.

Here, we study Frobenius bimodules associated with a pair of automorphisms of an algebra and discuss their basic properties. In particular, some equivalent conditions for a finite-dimensional bimodule are proved to be Frobenius and some isomorphisms between Ext-groups and Tor-groups of Frobenius modules over finite dimensional algebras are established. Full article

17 pages, 834 KiB

Open AccessArticle

Differentiating the State Evaluation Map from Matrices to Functions on Projective Space

by Ghaliah Alhamzi and Edwin Beggs

Symmetry 2023, 15(2), 474; https://doi.org/10.3390/sym15020474 - 10 Feb 2023

Cited by 1 | Viewed by 1333

Abstract

The pure state evaluation map from

M_{n} (C)

to

C (C P^{n - 1})

is a completely positive map of

C^{*}

-algebras intertwining the

U_{n}

symmetries on the two algebras. We show that it extends [...] Read more.

The pure state evaluation map from

M_{n} (C)

to

C (C P^{n - 1})

is a completely positive map of

C^{*}

-algebras intertwining the

U_{n}

symmetries on the two algebras. We show that it extends to a cochain map from the universal calculus on

M_{n} (C)

to the holomorphic

\bar{\partial}

calculus on

C P^{n - 1}

. The method uses connections on Hilbert

C^{*}

-bimodules. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry: Differential Geometry and Its Applications)

18 pages, 378 KiB

Open AccessArticle

Natural Transformations between Induction and Restriction on Iterated Wreath Product of Symmetric Group of Order 2

by Mee Seong Im and Can Ozan Oğuz

Mathematics 2022, 10(20), 3761; https://doi.org/10.3390/math10203761 - 12 Oct 2022

Cited by 1 | Viewed by 1629

Abstract

Let

C A_{n} = C [S_{2} ≀ S_{2} ≀ \dots ≀ S_{2}]

be the group algebra of an n-step iterated wreath product. We prove some structural properties of

A_{n}

such as their centers, centralizers, and [...] Read more.

Let

C A_{n} = C [S_{2} ≀ S_{2} ≀ \dots ≀ S_{2}]

be the group algebra of an n-step iterated wreath product. We prove some structural properties of

A_{n}

such as their centers, centralizers, and right and double cosets. We apply these results to explicitly write down the Mackey theorem for groups

A_{n}

and give a partial description of the natural transformations between induction and restriction functors on the representations of the iterated wreath product tower by computing certain hom spaces of the category of

⨁_{m \geq 0} (A_{m}, A_{n}) -

bimodules. A complete description of the category is an open problem. Full article

(This article belongs to the Section E4: Mathematical Physics)

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16 pages, 300 KiB

Open AccessArticle

Nonlinear Lie Triple Higher Derivations on Triangular Algebras by Local Actions: A New Perspective

by Xinfeng Liang, Dandan Ren and Qingliu Li

Axioms 2022, 11(7), 328; https://doi.org/10.3390/axioms11070328 - 5 Jul 2022

Cited by 1 | Viewed by 1918

Abstract

Let

R

be a commutative ring with unity and

T

be a triangular algebra over

R

. Let a sequence

Δ = {δ_{n}}_{n \in N}

of nonlinear mappings

δ_{n} : T \to T

is a Lie triple higher [...] Read more.

Let

R

be a commutative ring with unity and

T

be a triangular algebra over

R

. Let a sequence

Δ = {δ_{n}}_{n \in N}

of nonlinear mappings

δ_{n} : T \to T

is a Lie triple higher derivation by local actions satisfying the equation. Under some mild conditions on

T

, we prove in this paper that every Lie triple higher derivation by local actions on the triangular algebras is proper. As an application, we shall give a characterization of Lie triple higher derivations by local actions on upper triangular matrix algebras and nest algebras, respectively. Full article

28 pages, 618 KiB

Open AccessArticle

Relative Gorenstein Dimensions over Triangular Matrix Rings

by Driss Bennis, Rachid El Maaouy, Juan Ramón García Rozas and Luis Oyonarte

Mathematics 2021, 9(21), 2676; https://doi.org/10.3390/math9212676 - 22 Oct 2021

Cited by 3 | Viewed by 2268

Abstract

Let A and B be rings, U a

(B, A)

-bimodule, and

T = (\begin{matrix} A & 0 \\ U & B \end{matrix})

the triangular matrix ring. In this paper, several notions in relative Gorenstein algebra over a triangular matrix ring are investigated. We first [...] Read more.

Let A and B be rings, U a

(B, A)

-bimodule, and

T = (\begin{matrix} A & 0 \\ U & B \end{matrix})

the triangular matrix ring. In this paper, several notions in relative Gorenstein algebra over a triangular matrix ring are investigated. We first study how to construct w-tilting (tilting, semidualizing) over T using the corresponding ones over A and B. We show that when U is relative (weakly) compatible, we are able to describe the structure of

G_{C}

-projective modules over T. As an application, we study when a morphism in T-Mod is a special

G_{C} P (T)

-precover and when the class

G_{C} P (T)

is a special precovering class. In addition, we study the relative global dimension of T. In some cases, we show that it can be computed from the relative global dimensions of A and B. We end the paper with a counterexample to a result that characterizes when a T-module has a finite projective dimension. Full article

(This article belongs to the Special Issue New Advances in Algebra, Ring Theory and Homological Algebra)

11 pages, 291 KiB

Open AccessArticle

Entropy as a Topological Operad Derivation

by Tai-Danae Bradley

Entropy 2021, 23(9), 1195; https://doi.org/10.3390/e23091195 - 9 Sep 2021

Cited by 7 | Viewed by 11418

Abstract

We share a small connection between information theory, algebra, and topology—namely, a correspondence between Shannon entropy and derivations of the operad of topological simplices. We begin with a brief review of operads and their representations with topological simplices and the real line as [...] Read more.

We share a small connection between information theory, algebra, and topology—namely, a correspondence between Shannon entropy and derivations of the operad of topological simplices. We begin with a brief review of operads and their representations with topological simplices and the real line as the main example. We then give a general definition for a derivation of an operad in any category with values in an abelian bimodule over the operad. The main result is that Shannon entropy defines a derivation of the operad of topological simplices, and that for every derivation of this operad there exists a point at which it is given by a constant multiple of Shannon entropy. We show this is compatible with, and relies heavily on, a well-known characterization of entropy given by Faddeev in 1956 and a recent variation given by Leinster. Full article

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11 pages, 288 KiB

Open AccessArticle

K-Groups of Trivial Extensions and Gluings of Abelian Categories

by Qinghua Chen and Min Zheng

Mathematics 2021, 9(16), 1864; https://doi.org/10.3390/math9161864 - 6 Aug 2021

Viewed by 1751

Abstract

This paper focuses on the

K_{i}

-groups of two types of extensions of abelian categories, which are the trivial extension and the gluing of abelian categories. We prove that, under some conditions,

K_{i}

-groups of a certian subcategory of the trivial [...] Read more.

This paper focuses on the

K_{i}

-groups of two types of extensions of abelian categories, which are the trivial extension and the gluing of abelian categories. We prove that, under some conditions,

K_{i}

-groups of a certian subcategory of the trivial extension category is isomorphic to

K_{i}

-groups of the similar subcategory of the original category. Moreover, under some conditions, we show that the

K_{i}

-groups of a left (right) gluing of two abelian categories are isomorphic to the direct sum of

K_{i}

-groups of two abelian categories. As their applications, we obtain some results of the

K_{i}

-groups of the trivial extension of a ring by a bimodule (

i \in N

). Full article

(This article belongs to the Section A: Algebra and Logic)

11 pages, 233 KiB

Open AccessArticle

Finitistic Homological Dimensions Relative to Subcategories

by Yuntao Huang, Xia Wu and Weiling Song

Mathematics 2020, 8(12), 2228; https://doi.org/10.3390/math8122228 - 15 Dec 2020

Viewed by 1692

Abstract

Let

C \subseteq T

be subcategories of an abelian category

A

. Under some certain conditions, we show that the

C

-finitistic and

T

-finitistic global dimensions of

A

are identical. Some applications are given; in particular, some known results are obtained as [...] Read more.

Let

C \subseteq T

be subcategories of an abelian category

A

. Under some certain conditions, we show that the

C

-finitistic and

T

-finitistic global dimensions of

A

are identical. Some applications are given; in particular, some known results are obtained as corollaries. Full article

(This article belongs to the Section A: Algebra and Logic)

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