G-Algebra Structure on the Higher Order Hochschild Cohomology HS2∗(A,A)

We present a deformation theory associated to the higher Hochschild cohomology [Formula: see text]. We also study a [Formula: see text]-algebra structure associated to this deformation theory.

The holographic c-theorem and infinite-dimensional Lie algebras

We discuss a non-dynamical theory of gravity in three dimensions which is based on an infinite-dimensional Lie algebra that is closely related to an infinite-dimensional extended AdS algebra. We find an intriguing connection between on the one hand higher-derivative gravity theories that are consistent with the holographic c-theorem and on the other hand truncations of this infinite-dimensional Lie algebra that violate the Lie algebra structure. We show that in three dimensions different truncations reproduce, up to terms that do not contribute to the c-theorem, Chern-Simons-like gravity models describing extended 3D massive gravity theories. Performing the same procedure with similar truncations in dimensions larger than or equal to four reproduces higher derivative gravity models that are known in the literature to be consistent with the c-theorem but do not have an obvious connection to massive gravity like in three dimensions.

Hyers Stability and Multi-Fuzzy Banach Algebra

In this paper, we define multi-fuzzy Banach algebra and then prove the stability of involution on multi-fuzzy Banach algebra by fixed point method. That is, if f:A→A is an approximately involution on multi-fuzzy Banach algebra A, then there exists an involution H:A→A which is near to f. In addition, under some conditions on f, the algebra A has multi C*-algebra structure with involution H.

Poisson algebra structure on the invariants of pairs of matrices of degree three

We provide a table of multiplication of the Poisson algebra on the minimal set of generators of the invariants of pairs of matrices of degree three.

Generalized Reynolds operators on 3-Lie algebras and NS-3-Lie algebras

In this paper, first, we introduce the notion of a generalized Reynolds operator on a [Formula: see text]-Lie algebra [Formula: see text] with a representation on [Formula: see text]. We show that a generalized Reynolds operator induces a 3-Lie algebra structure on [Formula: see text], which represents on [Formula: see text]. By this fact, we define the cohomology of a generalized Reynolds operator and study infinitesimal deformations of a generalized Reynolds operator using the second cohomology group. Then we introduce the notion of an NS-[Formula: see text]-Lie algebra, which produces a 3-Lie algebra with a representation on itself. We show that a generalized Reynolds operator induces an NS-[Formula: see text]-Lie algebra naturally. Thus NS-[Formula: see text]-Lie algebras can be viewed as the underlying algebraic structures of generalized Reynolds operators on [Formula: see text]-Lie algebras. Finally, we show that a Nijenhuis operator on a 3-Lie algebra gives rise to a representation of the deformed 3-Lie algebra and a 2-cocycle. Consequently, the identity map will be a generalized Reynolds operator on the deformed 3-Lie algebra. We also introduce the notion of a Reynolds operator on a [Formula: see text]-Lie algebra, which can serve as a special case of generalized Reynolds operators on 3-Lie algebras.

ANALISIS KEMAMPUAN BERPIKIR KREATIF MAHASISWA KELOMPOK ATAS DALAM MENYELESAIKAN SOAL STRUKTUR ALJABAR RING MATERI IDEAL PRIMA DAN IDEAL MAKSIMAL

This study aims to describe the creative thinking ability of upper group students in solving Ring Algebra Structure questions on the prime ideal and maximum ideal material. This research uses descriptive qualitative research. The research subjects are students of Mathematics Education-2 Semester VI who have taken the Ring Algebra Structure course. The research subjects are three (3) students of Mathematics Education-2 Semester VI who have high mathematical abilities based on test results in the form of assignments on Ideal material. The instruments used in the study include: 1) Test; 2) Interview guide; and 3) Documentation. Based on the results of research and discussion, the findings resulted that the thinking ability of upper-class students had fulfilled all aspects of creative thinking, namely fluency, flexibility, and originality. The subject has been able to explain all the indicators that have been set very well and precisely. Subjects have also been able to prove prime ideals or maximum ideals or not both and can identify examples and non-examples of prime ideals and maximum ideals through high-level elaboration and analysis carried out by the subject. Subjects are also very able to describe Lattice diagrams on each question and can determine the subring of a ring. It's just that there is a uniqueness of the worksheet between MS, ZB, and HF. But overall the subject has fulfilled the creative thinking aspect. Thus, it can be concluded that the upper group students can be categorized at level 4, which is very creative.

Riemannian approximation in Carnot groups

We present self-contained proofs of the stability of the constants in the volume doubling property and the Poincaré and Sobolev inequalities for Riemannian approximations in Carnot groups. We use an explicit Riemannian approximation based on the Lie algebra structure that is suited for studying nonlinear subelliptic partial differential equations. Our approach is independent of the results obtained in [11].

Hopf Algebra Structure of Generalized Quasi-Symmetric Functions in Partially Commutative Variables

We introduce a coloured generalization $\mathrm{NSym}_A$ of the Hopf algebra of non-commutative symmetric functions described as a subalgebra of the of rooted ordered coloured trees Hopf algebra. Its natural basis can be identified with the set of sentences over alphabet $A$ (the set of colours). We present also its graded dual algebra $\mathrm{QSym}_A$ of coloured quasi-symmetric functions together with its realization in terms of power series in partially commutative variables. We provide formulas expressing multiplication, comultiplication and the antipode for these Hopf algebras in various bases — the corresponding generalizations of the complete homogeneous, elementary, ribbon Schur and power sum bases of $\mathrm{NSym}$, and the monomial and fundamental bases of $\mathrm{QSym}$. We study also certain distinguished series of trees in the setting of restricted duals to Hopf algebras.

Cumulant–Cumulant Relations in Free Probability Theory from Magnus’ Expansion

Relations between moments and cumulants play a central role in both classical and non-commutative probability theory. The latter allows for several distinct families of cumulants corresponding to different types of independences: free, Boolean and monotone. Relations among those cumulants have been studied recently. In this work, we focus on the problem of expressing with a closed formula multivariate monotone cumulants in terms of free and Boolean cumulants. In the process, we introduce various constructions and statistics on non-crossing partitions. Our approach is based on a pre-Lie algebra structure on cumulant functionals. Relations among cumulants are described in terms of the pre-Lie Magnus expansion combined with results on the continuous Baker–Campbell–Hausdorff formula due to A. Murua.