设$B$是表示有限的$\mathbb{C}$ -代数.与$B$相关的$\mathbb{Z}$ -李代数$L(B)$已由Ch. Riedtmann定义。若$B$是表示定向的，则存在另一个与C定义的$B$相关联的$\mathbb{Z}$ -李代数. M. Ringel，并由$\CK(B)$表示。我们证明了李代数$L(B)$和$\CK(B)$对于任何有向表示的$\mathbb{C}$ -代数$B$都是同构的。
英文摘要：
Let $B$ be a representation-finite $\mathbb{C}$-algebra.
The $\mathbb{Z}$-Lie algebra $L(B)$ associated with $B$ has been defined by Ch. Riedtmann.
If $B$ is representation-directed there is another $\mathbb{Z}$-Lie algebra associated with $B$ defined by C. M. Ringel and denoted by $\CK(B)$. We prove that the Lie algebras $L(B)$ and $\CK(B)$ are isomorphic for any representation-directed $\mathbb{C}$-algebra $B$.
Koenig, Külshammer and Ovsienko showed that Morita equivalence classes of quasi-hereditary algebras are in one-to-one correspondence with equivalence classes of the module categories over directed bocses. In this article, we extend this result to $Δ$-filtered algebras and $\overlineΔ$-filtered algebras.
We study the Frobenius-Perron dimension of representation-directed algebras and quotient algebras of canonical algebras of type ADE, prove that the Frobenius-Perron dimension of a representation-directed algebra is always zero and the Frobenius-Perron dimension of a quotient algebra of canonical algebras of type ADE is 0 or 1. Moreover, we give a sufficient and necessary condition for a quotient algebra of a canonical algebra of type ADE under which its Frobenius-Perron dimension is 0.
The theory of standard pentads is the theory aims to construct a graded Lie algebra whose local part consists of a given Lie algebra and its representation. In other words, using standard pentads, we can embed given Lie algebra and its representation into a larger graded Lie algebra. As special cases of Lie algebras associated with standard pentads, we have the notion of PC Lie algebras. Our aim of this paper is to show that the class of PC Lie algebras contains the class of Kac-Moody algebras, that is, to show that the notion of PC Lie algebras is an extension of Kac-Moody algebras.
In a previous paper by the authors, we obtain the first example of a finitely freely generated simple $\mathbb Z$-graded Lie conformal algebra of linear growth that cannot be embedded into any general Lie conformal algebra. In this paper, we obtain, as a byproduct, another class of such Lie conformal algebras by classifying $\mathbb Z$-graded simple Lie conformal algebras ${\cal G}=\oplus_{i=-1}^\infty{\cal G}_i$ satisfying the following, (1) ${\cal G}_0\cong{\rm Vir}$, the Virasoro conformal algebra; (2) Each ${\cal G}_i$ for $i\ge-1$ is a ${\rm Vir}$-module of rank one. These algebras includ…
The purpose of this paper is to introduce twisted $\mathcal{O}$-operators on $3$-Lie algebras. We define a cohomology of a twisted $\mathcal{O}$-operator $T$ as the Chevalley-Eilenberg cohomology of a certain $3$-Lie algebra induced by $T$ with coefficients in a suitable representation. Then we consider infinitesimal and formal deformations of twisted $\mathcal{O}$-operators from cohomological points of view. Furthermore, we introduce and study $3$-NS-Lie-algebras as the underlying structure of twisted $\mathcal{O}$-operators on $3$-Lie algebras. Finally, we investigate twisted $\mathcal{O}$-o…
The main purpose of this paper is to introduce the notion of $n$-L-dendriform algebra which can be seen as a dendrification of $n$-pre-Lie algebras by means of $\mathcal{O}$-operators. We investigate the representation theory of $n$-pre-Lie algebras and provide some related constructions. Furthermore, we introduce the notion of phase space of a $n$-Lie algebra and show that a $n$-Lie algebra has a phase space if and only if it is sub-adjacent to a $n$-pre-Lie algebra. Moreover, we present a procedure to construct $(n + 1)$-pre-Lie algebras from $n$-pre-Lie algebras equipped with a generalized …
A morphism Lie algebra is a triple $(\mathfrak{g}, \mathfrak{h}, ϕ)$ consisting of two Lie algebras $\mathfrak{g}, \mathfrak{h}$ and a Lie algebra homomorphism $ϕ: \mathfrak{g} \rightarrow \mathfrak{h}$. We define representations and cohomology of morphism Lie algebras. As applications of our cohomology, we study some aspects of deformations, abelian extensions of morphism Lie algebras and classify skeletal morphism sh Lie algebras. Finally, we consider the cohomology of morphism Lie groups and find a relation with the cohomology of morphism Lie algebras.
In this paper, first we introduce the notion of a twilled 3-Lie algebra, and construct an $L_\infty$-algebra, whose Maurer-Cartan elements give rise to new twilled 3-Lie algebras by twisting. In particular, we recover the Lie $3$-algebra whose Maurer-Cartan elements are O-operators (also called relative Rota-Baxter operators) on 3-Lie algebras. Then we introduce the notion of generalized matched pairs of 3-Lie algebras using generalized representations of 3-Lie algebras, which will give rise to twilled 3-Lie algebras. The usual matched pairs of 3-Lie algebras correspond to a special class of t…
