A. S. Dzhumadil'daev
Filipov proved that Jacobian algebra is n-Lie. In our paper we consider algebras defined on associative commutative algebra U with derivation $\der$ by (k+1)-multiplication $V^{0,1,...,k}=\der^0\wedge\der^1\wedge...\wedge \der^k$ (Wronskian). We study whether they have (k+1)-Lie, k-left commutative and homotopical (k+1)-Lie structures.
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Representations of the Schlessinger-Stasheff's associative homotopy Lie algebras in the spaces of higher-order differential operators are analyzed; in particular, a remarkable identity for the Wronskian determinants is obtained. The W-transformations of chiral embeddings, related with the Toda equations, of complex curves into the Kaehler manifolds are shown to be endowed with the homotopy Lie algebra structures. Extensions of the Wronskian determinants that preserve the properties of the Schlessinger-Stasheff's algebras are constructed for the case of $n\geq1$ independent variables.
Explicit class of finite-dimensional polynomial algebras with Wronskians over $\mathbb{R}^d$ as $N$-ary Lie brackets: beyond $\mathfrak{sl}(2)$
本文通过广义完全Wronskian作为$N$元括号，显式构造了$\mathbb{R}^d$上所有有限维多项式强同伦李代数，并得到了结构常数中广义Vandermonde行列式的分解公式。
Wronskians as $N$-ary brackets in finite-dimensional analogues of $sl(2)$
本文研究Wronskian确定在有限维多项式代数中的$N$-ary括号结构，揭示$\mathfrak{sl}(2)$作为二次系数向量场的底层结构。
Irreducible cuspidal modules of simple $n$-Lie algebras
研究简单n-李代数上Wronskians和Jacobian结构的不可约极小模的分类与保持不可约性证明。
The purpose of this paper is to investigate ternary multiplications constructed from a binary multiplication, linear twisting maps and a trace function. We provide a construction of ternary Hom-Nambu and Hom-Nambu-Lie algebras starting from a binary multiplication of a Hom-Lie algebra and a trace function satisfying certain compatibility conditions involving twisting maps. We show that mutual position of kernels of twisting maps and the trace play important role in this context, and provide examples of Hom-Nambu-Lie algebras obtained using this construction.
Frölicher-Nijenhuis bracket and derived bracket associated to a nonsymmetric operad with multiplication
本文通过构造与非对称带乘法算畴相关的两个分次李代数，建立了Maurer-Cartan元素与Nijenhuis元素、Rota-Baxter元素的对应关系，并给出括号显式形式以研究Loday型代数上的相关算子。
We describe the general non-associative version of Lie theory that relates unital formal multiplications (formal loops), Sabinin algebras and non-associative bialgebras. Starting with a formal multiplication we construct a non-associative bialgebra, namely, the bialgebra of distributions with the convolution product. Considering the primitive elements in this bialgebra gives a functor from formal loops to Sabinin algebras. We compare this functor to that of Mikheev and Sabinin and show that although the brackets given by both constructions coincide, the multioperator does not. We also show how…
The notion of $n$-ary algebras, that is vector spaces with a multiplication concerning $n$-arguments, $n \geq 3$, became fundamental since the works of Nambu. Here we first present general notions concerning $n$-ary algebras and associative $n$-ary algebras. Then we will be interested in the notion of $n$-Lie algebras, initiated by Filippov, and which is attached to the Nambu algebras. We study the particular case of nilpotent or filiform $n$-Lie algebras to obtain a beginning of classification. This notion of $n$-Lie algebra admits a natural generalization in Strong Homotopy $n$-Lie algebras …
Left invariant affine structures in a Lie group $G$ are in one-to-one correspondence with left-symmetric algebras over its Lie algebra $\mathfrak g=T_eG$ (``over'' means that the commutator $[x,y]=xy-yx$ coincides with the Lie bracket; left-symmetric algebras can be defined as Lie-admissible algebras such that the multiplication by left defines a representation of the underlying Lie algebra). An affine structure (and the corresponding left symmetric algebra) is complete if $G$ is affinely equivalent to $\mathfrak g$. By the main result of this paper, a complete left symmetric algebra admits a …
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 mult…
We study the algebra of bilinear multiplications of an $n$-dimensional vector space. In particular, we study the Kantor product of some well-known (associative, Lie, alternative, Novikov and some other) multiplications.
We present the subalgebra structure of sl(3,O), a particular real form of E6 chosen for its relevance to particle physics through the connection between its associated Lie group SL(3,O) and generalized Lorentz groups. Given the complications related to the non-associativity of the octonions O and the restriction to working with a real form of E6, we find that the traditional methods used to study Lie algebras must be modified for our purposes. We use an explicit representation of the Lie group SL(3,O) to produce the multiplication table of the corresponding algebra sl(3,O). Both the multiplica…
We investigate Lie algebras whose Lie bracket is also an associative or cubic associative multiplication to characterize the class of nilpotent Lie algebras with a nilindex equal to 2 or 3. In particular we study the class of 2-step nilpotent Lie algebras, their deformations and we compute the cohomology which parametrize the deformations in this class.
Weak central polynomials for algebras of multiplications of simple algebras
本文通过简单证明，确立了有限维单代数乘法代数弱中心多项式的存在性，并给出了三维单李代数与二维若尔当代数的显式例子。
A commutative algebra is exact if its multiplication endomorphisms are trace-free and is Killing metrized if its Killing type trace-form is nondegenerate and invariant. A Killing metrized exact commutative algebra is necessarily neither unital nor associative. Such algebras can be viewed as commutative analogues of semisimple Lie algebras or, alternatively, as nonassociative generalizations of étale (associative) algebras. Some basic examples are described and there are introduced quantitative measures of nonassociativity, formally analogous to curvatures of connections, that serve to facilita…
A Lie algebra $L$ is said to be of breadth $k$ if the maximal dimension of the images of left multiplication by elements of the algebra is $k$. In this paper we give characterization of finite dimensional nilpotent Lie algebras of breadth less than or equal to two. Furthermore, using these characterizations we determined the isomorphism classes of these algebras.
We present the subalgebra structure of sl(3,O), a particular real form of e6 chosen for its relevance to particle physics and its close relation to generalized Lorentz groups. We use an explicit representation of the Lie group SL(3,O) to construct the multiplication table of the corresponding Lie algebra sl(3,O). Both the multiplication table and the group are then utilized to find various nested chains of subalgebras of sl(3,O), in which the corresponding Cartan subalgebras are also nested where possible. Because our construction involves the Lie group, we simultaneously obtain an explicit re…
On the Structure of Low-Dimensional Poisson Algebras over Arbitrary Fields
本文研究了任意域上维数至多三的泊松代数结构，通过结合平方和导李代数等内部结构，给出了维数一、二的完全分类和维数三的结构分类，并揭示了特征2等特殊现象。
This paper deals with the classification of Leibniz central extensions of a naturally graded filiform Lie algebra. We choose a basis with respect to that the table of multiplication has a simple form. In low dimensional cases isomorphism classes of the central extensions are given. In parametric family orbits cases invariant functions (orbit functions) are provided.
The class of matrix algebras on a ladder $\mathcal{L}$ generalizes the class of block upper triangular matrix algebras. It was previously shown that the matrix algebra on a ladder $\mathcal{L}$ is zero product determined under matrix multiplication. In this article, we show that the matrix algebra on a one-step ladder is zero product determined under the Lie bracket.
Depolarization and distributive laws
本文研究向量空间中交换与反交换乘法通过分配律联系时的去极化原理，证明转置泊松代数无法用单一乘法实现去极化，并考察了 Hom-李代数的去极化情形。
Ternary Associativity and Ternary Lie Algebra at Cube Root of Unity
本文基于两类三元结合性提出三元交换子与三元李代数的新构造，得到由 GA(1,5) 置换决定的三元雅可比型恒等式，并给出矩阵构造例子及其与旋转群不可约表示的联系。
Anti-commutative anti-associative algebras. Acaa-algebras
本文通过极化过程研究非结合代数，提出并探讨了满足特定二次恒等式的Acaa-代数（反交换反结合代数）的性质。
Hom-Lie algebras defined on central extensions of a given quadratic Lie algebra that in turn admit an invariant metric, are studied. It is shown how some of these algebras are naturally equipped with other symmetric, bilinear forms that satisfy an invariant condition for their twisted multiplication maps. The twisted invariant bilinear forms so obtained resemble the Cartan-Killing forms defined on ordinary Lie algebras. This fact allows one to reproduce on the Hom-Lie algebras hereby studied, some results that are classically associated to the ordinary Cartan-Killing form.
In this work, we consider Lie algebras L containing a subalgebra isomorphic to sl3 and such that L decomposes as a module for that sl3 subalgebra into copies of the adjoint module, the natural 3-dimensional module and its dual, and the trivial one-dimensional module. We determine the multiplication in L and establish connections with structurable algebras by exploiting symmetry relative to the symmetric group S4.
Transposed Poisson structures on quasi-filiform Lie algebras of maximum length
该研究聚焦最大长度拟丝状李代数的$\frac{1}{2}$-导子，通过建立交换结合乘法，构造出以这类李代数为伴随代数的非平凡转置泊松代数，为相关代数结构研究提供新构造。
Transposed Poisson structures on solvable Lie algebras with filiform nilradical
该研究利用具有线状幂零根基的可解李代数的 1/2-导子，建立交换结合乘法，从而构造出关联给定李代数的非平凡转置 Poisson 代数。
We show that for a given nilpotent Lie algebra $\mathfrak{g}$ with $Z(\mathfrak{g})\subseteq [\mathfrak{g},\mathfrak{g}]$ all commutative post-Lie algebra structures, or CPA-structures, on $\mathfrak{g}$ are complete. This means that all left and all right multiplication operators in the algebra are nilpotent. Then we study CPA-structures on free-nilpotent Lie algebras $F_{g,c}$ and discover a strong relationship to solving systems of linear equations of type $[x,u]+[y,v]=0$ for generator pairs $x,y\in F_{g,c}$. We use results of Remeslennikov and Stöhr concerning these equations to prove that…
Let $\mathfrak{A}$ and $\mathfrak{A}'$ be two $C^*$-algebras with identities $I_{\mathfrak{A}}$ and $I_{\mathfrak{A}'}$, respectively, and $P_1$ and $P_2 = I_{\mathfrak{A}} - P_1$ nontrivial symmetric projections in $\mathfrak{A}$. In this paper we study the characterization of multiplicative $*$-Lie-type maps. In particular, if $\mathcal{M}$ is a factor von Neumann algebra then every complex scalar multiplication bijective unital multiplicative $*$-Lie-type map is $*$-isomorphism.
We compute all complex structures on indecomposable 6-dimensional real Lie algebras and their equivalence classes. We also give for each of them a global holomorphic chart on the connected simply connected Lie group associated to the real Lie algebra and write down the multiplication in that chart.
A Hom-algebra structure is a multiplication on a vector space where the structure is twisted by a homomorphism. The structure of Hom-Lie algebra was introduced by Hartwig, Larsson and Silvestrov and extended by Larsson and Silvestrov to quasi-hom Lie and quasi-Lie algebras. In this paper we introduce and study Hom-associative, Hom-Leibniz, and Hom-Lie admissible algebraic structures which generalize the well known associative, Leibniz and Lie admissible algebras. Also, we characterize the flexible Hom-algebras in this case. We also explain some connections between Hom-Lie algebras and Santilli…
A pseudo-Euclidean non-associative algebra $(\mathfrak{g}, \bullet)$ is a real algebra of finite dimension that has a metric, i.e., a bilinear, symmetric, and non-degenerate form $\langle\;\rangle$. The metric is considered $\mathrm{L}$-invariant (resp. $\mathrm{R}$-invariant) if all left multiplications (resp. right multiplications) are skew-symmetric. The metric is called associative if $\langle u\bullet v,w\rangle= \langle u,v\bullet w\rangle$ for all $u, v, w \in \mathfrak{g}$. These three notions coincide when $\mathfrak{g}$ is a Lie algebra and in this case $\mathfrak{g}$ endowed with th…
We determine normal forms of the multiplication of four-dimensional anti-commutative algebras over a field $\mathbb K$ of characteristic zero having an analogous family of flags of subalgebras as the four-dimensional non-Lie binary Lie algebras, and hence can be considered as the closest relatives of binary Lie algebras. These algebras are extensions of $\mathbb K$ by the 3-dimensional nilpotent Lie algebra and at the same time extensions of a two-dimensional Lie algebra by a two-dimensional abelian algebra. We describe their groups of automorphisms as extensions of a subgroup of the group of …
Let $n$ be a maximal nilpotent subalgebra of a complex symmetric Kac-Moody Lie algebra. Lusztig has introduced a basis of U(n) called the semicanonical basis, whose elements can be seen as certain constructible functions on varieties of nilpotent modules over a preprojective algebra of the same type as $n$. We prove a formula for the product of two elements of the dual of this semicanonical basis, and more generally for the product of two evaluation forms associated to arbitrary modules over the preprojective algebra. This formula plays an important role in our work on the relationship between…
BiHom-$Ω$-associative algebras and related structures
该研究提出四类BiHom-$\Omega$代数概念，借助权$\lambda$的Rota-Baxter族构造新的BiHom-$\Omega$结合与李代数，并将经典代数间关系推广到BiHom-$\Omega$情形，丰富了同态代数结构理论。
Transposed Poisson structures on solvable and perfect Lie algebras
本文分类描述了振荡李代数等多种李代数上的所有转置Poisson代数结构，并构造性地回答了关于转置Poisson代数的一个遗留问题，找到了一个具有非平凡$\ rac{1}{2}$-导数但无非平凡转置Poisson结构的有限维李代数。
Braided m-Lie algebras induced by multiplication are introduced, which generalize Lie algebras, Lie color algebras and quantum Lie algebras. The necessary and sufficient conditions for the braided m-Lie algebras to be strict Jacobi braided Lie algebras are given. Two classes of braided m-Lie algebras are given, which are generalized matrix braided m-Lie algebras and braided m-Lie subalgebras of $End_F M$, where $M$ is a Yetter-Drinfeld module over $B$ with dim $B< \infty $ . In particular, generalized classical braided m-Lie algebras $sl_{q, f}(GM_G(A), F)$ and $osp_{q, t} (GM_G(A), M, F)$ of …
On $q$-pre-Lie algebras
本文从李代数表示角度引入q-预李代数概念，统一预李代数和反预李代数，介绍相关概念及关系并给出构造，对Witt代数等进行分类，证明特定条件下某些代数上q-预李代数结构的存在性与不存在性。
Every Lie algebra over a field $E$ gives rise to new Lie algebras over any subfield $F \subseteq E$ by restricting the scalar multiplication. This paper studies the structure of these underlying Lie algebra in relation to the structure of the original Lie algebra, in particular the question how much of the original Lie algebra can be recovered from its underlying Lie algebra over subfields $F$. By introducing the conjugate of a Lie algebra we show that in some specific cases the Lie algebra is completely determined by its underlying Lie algebra. Furthermore we construct examples showing that t…
An algebra is called skew-symmetric if its multiplication operation is a skew-symmetric bilinear application. We determine all these algebras in dimension $3$ over a field of characteristic different from $2$. As an application, we determine the subvariety of $3$-dimensional Hom-Lie algebras. For this type of algebras, we study also the dimension $4$.
Let F_m=F_m(var(sl(2,K))) be the relatively free algebra of rank m in the variety of Lie algebras generated by the algebra sl(2,K) over a field K of characteristic 0. Translating an old result of Baker from 1901 we present a multiplication rule for the inner automorphisms of the completion of F_m with respect to the formal power series topology. Our results are more precise for m=2 when F_2 is isomorphic to the Lie algebra L generated by two generic traceless 2 x 2 matrices. We give a complete description of the group of inner automorphisms of the completion of L. As a consequence we obtain si…
The algebraic and geometric classification of commutative post-Lie algebras
本文从代数和几何角度研究交换后李代数的分类，发现其代数结构特性并给出3、4维nilpotent CPA的分类结果。
Associative ternary algebras and ternary Lie algebras at cube roots of unity
本文提出基于$\omega$-对称性（$\omega$为本原立方根）将李代数推广到三元结构，定义三元$\omega$-李代数，并证明结合三元代数上的$\omega$-换位子构成三元$\omega$-李代数，且研究了二阶立方矩阵的8维三元$\omega$-李代数及其低维子代数。
Sharp commutator bounds on complex simple Lie algebras: the Böttcher--Wenzel inequality and the comass of the Cartan $3$-form
本文研究复单李代数的交换子界，证明Cartan 3-形式的谱范数复化后不增大，得到Dynkin指数为ℓ的表示像上Frobenius范数的最优常数2/ℓ，改进了相关不等式并揭示了紧致单李群的曲率性质。
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