arXiv subjects
Publications and source records attributed to Ruipu Bai.
After endowing with a 3-Lie-Rinehart structure on Hom 3-Lie algebras, we obtain a class of special Hom 3-Lie algebras, which have close relationships with representations of commutative associative algebras. We provide a special class of Hom 3-Lie-Rinehart algebras, called split regular Hom 3-Lie-Rinehart algebras, and we then characterize their structures by means of root systems and weight systems associated to a splitting Cartan subalgebra.
In this paper, we study the structure of 3-Lie algebras with involutive derivations. We prove that if $A$ is an $m$-dimensional 3-Lie algebra with an involutive derivation $D$, then there exists a compatible 3-pre-Lie algebra $(A, \{ , , , \}_D)$ such that $A$ is the sub-adjacent 3-Lie algebra, and there is a local cocycle $3$-Lie bialgebraic structure on the $2m$-dimensional semi-direct product 3-Lie algebra $A\ltimes_{ad^*} A^*$, which is associated to the adjoint representation $(A, ad)$. By means of involutive derivations, the skew-symmetric solution of the 3-Lie classical Yang-Baxter equation in the 3-Lie algebra $A\ltimes_{ad^*}A^*$, a class of 3-pre-Lie algebras, and eight and ten dimensional local cocycle 3-Lie bialgebras are constructed.
A new 3-ary non-associative algebra, which is called a semi-associative $3$-algebra, is introduced, and the double modules and double extensions by cocycles are provided. Every semi-associative $3$-algebra $(A, \{ , , \})$ has an adjacent 3-Lie algebra $(A, [ , , ]_c)$. From a semi-associative $3$-algebra $(A, \{, , \})$, a double module $(ϕ, ψ, M)$ and a cocycle $θ$, a semi-direct product semi-associative $3$-algebra $A\ltimes_{ϕψ} M $ and a double extension $(A\dot+A^*, \{ , , \}_θ)$ are constructed, and structures are studied.
In this paper, we define the induced modules of Lie algebra ad$(B)$ associated with a 3-Lie algebra $B$-module, and study the relation between 3-Lie algebra $A_ω^δ$-modules and induced modules of inner derivation algebra ad$(A_ω^δ)$. We construct two infinite dimensional intermediate series modules of 3-Lie algebra $A_ω^δ$, and two infinite dimensional modules $(V, ψ_{λμ})$ and $(V, ϕ_μ)$ of the Lie algebra ad$(A_ω^δ)$, and prove that only $(V, ψ_{\lambda0})$ and $(V, ψ_{\lambda1})$ are induced modules.
For any $n$-dimensional 3-Lie algebra $A$ over a field of characteristic zero with an involutive derivation $D$, we investigate the structure of the 3-Lie algebra $B_1=A\ltimes_{ad^*} A^* $ associated with the coadjoint representation $(A^*, ad^*)$. We then discuss the structure of the dual 3-Lie algebra $B_2$ of the local cocycle 3-Lie bialgebra $(A\ltimes_{ad^*} A^*, Δ)$. By means of the involutive derivation $D$, we construct the $4n$-dimensional Manin triple $(B_1\oplus B_2,$ $ [ \cdot, \cdot, \cdot]_1,$ $ [ \cdot, \cdot, \cdot]_2,$ $ B_1, B_2)$ of 3-Lie algebras, and provide concrete multiplication in a special basis $Π_1\cupΠ_2$. We also construct a sixteen dimensional Manin triple $(B, [ \cdot, \cdot, \cdot])$ with $\dim B^1=12$ using an involutive derivation on a four dimensional 3-Lie algebra $A$ with $\dim A^1=2$.
In this paper, we define a class of 3-algebras which are called 3-Lie-Rinehart algebras. A 3-Lie-Rinehart algebra is a triple $(L, A, ρ)$, where $A$ is a commutative associative algebra, $L$ is an $A$-module, $(A, ρ)$ is a 3-Lie algebra $L$-module and $ρ(L, L)\subseteq Der(A)$. We discuss the basic structures, actions and crossed modules of 3-Lie-Rinehart algebras and construct 3-Lie-Rinehart algebras from given algebras, we also study the derivations from 3-Lie-Rinehart algebras to 3-Lie $A$-algebras. From the study, we see that there is much difference between 3-Lie algebras and 3-Lie-Rinehart algebras.
From a commutative associative algebra $A$, the infinite dimensional unital 3-Lie Poisson algebra~$\mathfrak{L}$~is constructed, which is also a canonical Nambu 3-Lie algebra, and the structure of $\mathfrak{L}$ is discussed. It is proved that: (1) there is a minimal set of generators $S$ consisting of six vectors; (2) the quotient algebra $\mathfrak{L}/\mathbb{F}L_{0, 0}^0$ is a simple 3-Lie Poisson algebra; (3) four important infinite dimensional 3-Lie algebras: 3-Virasoro-Witt algebra $\mathcal{W}_3$, $A_ω^δ$, $A_ω$ and the 3-$W_{\infty}$ algebra can be embedded in $\mathfrak{L}$.
In this paper we study $k$-order homogeneous Rota-Baxter operators with weight $1$ on the simple $3$-Lie algebra $A_ω$ (over a field of characteristic zero), which is realized by an associative commutative algebra $A$ and a derivation $Δ$ and an involution $ω$ (Lemma \mref{lem:rbd3}). A $k$-order homogeneous Rota-Baxter operator on $A_ω$ is a linear map $R$ satisfying $R(L_m)=f(m+k)L_{m+k}$ for all generators $\{ L_m~ |~ m\in \mathbb Z \}$ of $A_ω$ and a map $f : \mathbb Z \rightarrow\mathbb F$, where $k\in \mathbb Z$. We prove that $R$ is a $k$-order homogeneous Rota-Baxter operator on $A_ω$ of weight $1$ with $k\neq 0$ if and only if $R=0$ (see Theorems 3.2, and $R$ is a $0$-order homogeneous Rota-Baxter operator on $A_ω$ of weight $1$ if and only if $R$ is one of the forty possibilities which are described in Theorems3.5, 3.7, 3.9, 3.10, 3.18, 3.21 and 3.22.
We study an extension algebra $A$ from two given $3$-Lie algebras $M$ and $H$, and discuss the extensibility of a pair of derivations, one from the derivation algebra of $M$ and the other from that of $H$, to a derivation of $A$. In particular, we give conditions for such an extension to be a $3$-Lie algebra, and provide necessary and sufficient conditions of the pair of derivations to be extendable.
The $n$-Lie bialgebras are studied. In Section 2, the $n$-Lie coalgebra with rank $r$ is defined, and the structure of it is discussed. In Section 3, the $n$-Lie bialgebra is introduced. A triple $(L, μ, Δ)$ is an $n$-Lie bialgebra if and only if $Δ$ is a conformal $1$-cocycle on the $n$-Lie algebra $L$ associated to $L$-modules $(L^{\otimes n}, ρ_s^μ)$, $1\leq s\leq n$, and the structure of $n$-Lie bialgebras is investigated by the structural constants. In Section 4, two-dimensional extension of finite dimensional $n$-Lie bialgebras are studied. For an $m$ dimensional $n$-Lie bialgebra $(L, μ, Δ)$, and an $ad_μ$-invariant symmetric bilinear form on $L$, the $m+2$ dimensional $(n+1)$-Lie bialgebra is constructed. In the last section, the bialgebra structure on the finite dimensional simple $n$-Lie algebra $A_n$ is discussed. It is proved that only bialgebra structures on the simple $n$-Lie algebra $A_n$ are rank zero, and rank two.
Generalized derivations, quasiderivations and quasicentroid of $3$-algebras are introduced, and basic relations between them are studied. Structures of quasiderivations and quasicentroid of $3$-Lie algebras, which contains a maximal diagonalized tours, are systematically investigated. The main results are: for all $3$-Lie algebra $A$, 1) the generalized derivation algebra $GDer(A)$ is the sum of quasiderivation algebra $QDer(A)$ and quasicentroid $QΓ(A)$; 2) quasiderivations of $A$ can be embedded as derivations in a larger algebra; 3) quasiderivation algebra $QDer(A)$ normalizer quasicentroid, that is, $[QDer(A), QΓ(A)]\subseteq QΓ(A)$; 4) if $A$ contains a maximal diagonalized tours $T$, then $QDer(A)$ and $QΓ(A)$ are diagonalized $T$-modules, that is, as $T$-modules, $(T, T)$ semi-simplely acts on $QDer(A)$ and $QΓ(A)$, respectively.
In this paper we construct two kinds of infinite-dimensional 3-Lie algebras from a given commutative associative algebra, and show that they are all canonical Nambu 3-Lie algebras. We relate their inner derivation algebras to Witt algebras, and then study the regular representations of these 3-Lie algebras and the natural representations of the inner derivation algebras. In particular, for the second kind of 3-Lie algebras, we find that their regular representations are Harish-Chandra modules, and the inner derivation algebras give rise to intermediate series modules of the Witt algebras and contain the smallest full toroidal Lie algebras without center.
In the paper we study homogeneous Rota-Baxter operators with weight zero on the infinite dimensional simple $3$-Lie algebra $A_ω$ over a field $F$ ( $ch F=0$ ) which is realized by an associative commutative algebra $A$ and a derivation $Δ$ and an involution $ω$ ( Lemma \mref{lem:rbd3} ). A homogeneous Rota-Baxter operator on $A_ω$ is a linear map $R$ of $A_ω$ satisfying $R(L_m)=f(m)L_m$ for all generators of $A_ω$, where $f : A_ω \rightarrow F$. We proved that $R$ is a homogeneous Rota-Baxter operator on $A_ω$ if and only if $R$ is the one of the five possibilities $R_{0_1}$, $R_{0_2}$,$R_{0_3}$,$R_{0_4}$ and $R_{0_5}$, which are described in Theorem \mref{thm:thm1}, \mref{thm:thm4}, \mref{thm:thm01}, \mref{thm:thm03} and \mref{thm:thm04}. By the five homogeneous Rota-Baxter operators $R_{0_i}$, we construct new $3$-Lie algebras $(A, [ , , ]_i)$ for $1\leq i\leq 5$, such that $R_{0_i}$ is the homogeneous Rota-Baxter operator on $3$-Lie algebra $(A, [ , , ]_i)$, respectively.
The symplectic structures on $3$-Lie algebras and metric symplectic $3$-Lie algebras are studied. For arbitrary $3$-Lie algebra $L$, infinite many metric symplectic $3$-Lie algebras are constructed. It is proved that a metric $3$-Lie algebra $(A, B)$ is a metric symplectic $3$-Lie algebra if and only if there exists an invertible derivation $D$ such that $D\in Der_B(A)$, and is also proved that every metric symplectic $3$-Lie algebra $(\tilde{A}, \tilde{B}, \tildeω)$ is a $T^*_θ$-extension of a metric symplectic $3$-Lie algebra $(A, B, ω)$. Finally, we construct a metric symplectic double extension of a metric symplectic $3$-Lie algebra by means of a special derivation.
3-Lie algebras are constructed by Lie algebras, derivations and linear functions, associative commutative algebras, whose involutions and derivations. Then the 3-Lie algebras are obtained from group algebras $F[G]$. An infinite dimensional simple 3-Lie algebra $(A, [,,]_{ω, δ_0})$ and a non-simple 3-Lie algebra $(A, [,,]_{ω_1, δ})$ are constructed by Laurent polynomials $A=F[t, t^{-1}]$ and its involutions $ω$ and $ω_1$ and derivations $δ$ and $δ_0$. At last of the paper, we summarize the methods of constructing $n$-Lie algebras for $n\geq 3$ and provide a problem.
In this paper we introduce the concepts of a Rota-Baxter operator and a differential operator with weights on an $n$-algebra. We then focus on Rota-Baxter 3-Lie algebras and show that they can be derived from Rota-Baxter Lie algebras and pre-Lie algebras and from Rota-Baxter commutative associative algebras with derivations. We also establish the inheritance property of Rota-Baxter 3-Lie algebras.
3-Lie algebras have close relationships with many important fields in mathematics and mathematical physics. The paper concerns 3-Lie algebras. The concepts of 3-Lie coalgebras and 3-Lie bialgebras are given. The structures of such categories of algebras, and the relationships with 3-Lie algebras are studied. And the classification of 3-dimensional 3-Lie coalgebras and 3-Lie bialgebras over an algebraically closed field of characteristic zero are provided.
In this paper, we give a systematic study on isotropic ideals of metric n-Lie algebras. As an application, we show that the center of a non-abelian (n+k)-dimensional metric n-Lie algebra (1< k< n+2), whose center is isotropic, is of dimension k-1. Furthermore, we classify (n+k)-dimensional metric n-Lie algebras for 1< k < n+2.