Let $\mathfrak{g}$ be a finite-dimensional complex Lie algebra and $\textrm{HLie}_{m}(\mathfrak{g})$ be the affine variety of all multiplicative Hom-Lie algebras on $\mathfrak{g}$. We use a method of computational ideal theory to describe $\textrm{HLie}_{m}(\mathfrak{gl}_{n}(\mathbb{C}))$, showing that $\textrm{HLie}_{m}(\mathfrak{gl}_{2}(\mathbb{C}))$ consists of two 1-dimensional and one 3-dimensional irreducible components, and showing that $\textrm{HLie}_{m}(\mathfrak{gl}_{n}(\mathbb{C}))=\{\textrm{diag}\{δ,\dots,δ,a\}\mid δ=1\textrm{ or }0,a\in\mathbb{C}\}$ for $n\geqslant 3$. We construct a new family of multiplicative Hom-Lie algebras on the Heisenberg Lie algebra $\mathfrak{h}_{2n+1}(\mathbb{C})$ and characterize the affine varieties $\textrm{HLie}_{m}(\mathfrak{u}_{2}(\mathbb{C}))$ and $\textrm{HLie}_{m}(\mathfrak{u}_{3}(\mathbb{C}))$. We also study the derivation algebra $\textrm{Der}_{D}(\mathfrak{g})$ of a multiplicative Hom-Lie algebra $D$ on $\mathfrak{g}$ and under some hypotheses on $D$, we prove that the Hilbert series $\mathcal{H}(\textrm{Der}_{D}(\mathfrak{g}),t)$ is a rational function.
In this paper, we mainly study structure of multiplicative simple Hom-Jordan algebras. We talk about equivalent conditions for multiplicative Hom-Jordan algebras being solvable, simple and semi-simple. As an application, we give a theorem about classification of multiplicative simple Hom-Jordan algebras. Moreover, some propositions about bimodules of multiplicative Hom-Jordan algebras are also displayed.
In this paper, we prove that the multiplicative group of a unital non-commutative Leavitt path algebra $L_K(E)$ and Cohn path algebra $C_K(E)$ contain a non-cyclic free subgroup, provided $K$ is a non-absolute field. We also provide a description of the generators of free subgroups in term of the graph $E$. Finally, we determine multiplicative groups of Leavitt path algebras of some special types.
We introduce the notions of pre-morphism and pre-derivation for arbitrary non-associative algebras over a commutative ring $k$ with identity. These notions are applied to the study of pre-Lie $k$-algebras and, more generally, Lie-admissible $k$-algebras. Associating with any algebra $(A,\cdot)$ its sub-adjacent anticommutative algebra $(A,[-,-])$ is a functor from the category of $k$-algebras with pre-morphisms to the category of anticommutative $k$-algebras. We describe the commutator of two ideals of a pre-Lie algebra, showing that the condition (Huq=Smith) holds for pre-Lie algebras. This a…
The purpose of this paper is to generalize some results on $n$-Lie algebras and $n$-Hom-Lie algebras to $n$-Hom-Lie color algebras. Then we introduce and give some constructions of $n$-Hom-Lie color algebras.
The construction of HNN-extensions of involutive Hom-associative algebras and involutive Hom-Lie algebras is described. Then, as an application of HNN-extension, by using the validity of Poincaré-Birkhoff-Witt theorem for involutive Hom-Lie algebras, we provide an embedding theorem.
Recently, some concepts such as Hom-algebras, Hom-Lie algebras, Hom-Lie admissible algebras, Hom-coalgebras are studied and some of classical properties of algebras and some geometric objects are extended on them. In this paper by recall the concept of Hom-$ρ$-commutative algebras, we intend to develop some of the most classical results in Riemannian geometry such as metric, connection, torsion tensor, curvature tensor on it and also we discuss about differential operators and get some results of differential calculus using them. The notions of symplectic structures and Poisson structures are …
Over a field of characteristic zero, we show that the forgetful functor from the homotopy category of commutative dg algebras to the homotopy category of dg associative algebras is faithful. In fact, the induced map of derived mapping spaces gives an injection on all homotopy groups at any basepoint. We prove similar results both for unital and non-unital algebras, and also Koszul dually for the universal enveloping algebra functor from dg Lie algebras to dg associative algebras. An important ingredient is a natural model for these derived mapping spaces as Maurer-Cartan spaces of complete fil…
We introduce and study "norm-multiplicative" homomorphisms $φ: {\cal L}^1(F) \rightarrow {\cal M}_r(G)$ between group and measure algebras, and $φ: {\cal L}^1(ω_F) \rightarrow {\cal M}(ω_G)$ between Beurling group and measure algebras, where $F$ and $G$ are locally compact groups with continuous weights $ω_F$ and $ω_G$. Through a unified approach we recover, and sometimes strengthen, many of the main known results concerning homomorphisms and isomorphisms between these (Beurling) group and measure algebras. We provide a first description of all positive homomorphisms $φ: {\cal L}^1(F) \rightar…
Computing Small 1-Homological Models for Commutative Differential Graded Algebras
我们使用同调扰动机械具体的代数范畴[P。同调扰动理论与结合论。Homology，Homotopy and Applications第2卷，n. 5（2000）51-88]给出了计算交换微分分次代数（简称CDGA）的小1-同调模型的微分结构的算法。的过程的复杂性进行了研究，并在Mathematica的计算机软件包中描述用于确定这样的模型。
