In this paper, we introduce the cohomology theory of O-operators on Hom-associative algebras. This cohomology can also be viewed as the Hochschild cohomology of a certain Hom-associative algebra with ...coefficients in a suitable bimodule. Next, we study infinitesimal and formal deformations of an O-operator and show that they are governed by the above-defined cohomology. Furthermore, the notion of Nijenhuis elements associated with an O-operator is introduced to characterize trivial infinitesimal deformations. Moreover, we provide relevant constructions that twists objects on associative algebras to objects on Hom-associative algebras along morphisms.
We consider a skew-symmetric n-ary bracket on the polynomial algebra Kx1,…,xn,xn+1 (n≥2) over a field K of characteristic zero defined by {a1,…,an}=Jac(a1,…,an,C), where C is a fixed element of ...Kx1,…,xn,xn+1 and Jac is the Jacobian. If n=2 then this bracket is a Poisson bracket and if n≥3 then it is an n-Lie-Poisson bracket on Kx1,…,xn,xn+1. We describe the center of the corresponding n-Lie-Poisson algebra and show that the quotient algebra Kx1,…,xn,xn+1/(C−λ), where (C−λ) is the ideal generated by C−λ, 0≠λ∈K, is a simple central n-Lie-Poisson algebra if C is a homogeneous polynomial that is not a proper power of any nonzero polynomial. This construction includes the quotients P(sl2(K))/(C−λ) of the Poisson enveloping algebra P(sl2(K)) of the simple Lie algebra sl2(K), where C is the standard Casimir element of sl2(K) in P(sl2(K)). It is also proven that the quotients P(M)/(CM−λ) of the Poisson enveloping algebra P(M) of the exceptional simple seven dimensional Malcev algebra M are central simple, where CM is the standard Casimir element of M in P(M).
A note on $2$-plectic vector spaces Mohammad Shafiee
Journal of Mahani Mathematical Research Center,
11/2023, Letnik:
13, Številka:
1
Journal Article
Odprti dostop
Among the $2$-plectic structures on vector spaces, the canonical ones and the $2$-plectic structures induced by the Killing form on semisimple Lie algebras are more interesting. In this note, we show ...that the group of linear preservers of the canonical $2$-plectic structure is noncompact and disconnected and its dimension is computed. Also, we show that the group of automorphisms of a compact semisimple Lie algebra preserving its $2$-plectic structure, is compact. Furthermore, it is shown that the $2$-plectic structure on a semisimple Lie algebra $\mathfrak{g}$ is canonical if and only if it has an abelian Lie subalgebra whose dimension satisfies in a special condition. As a consequence, we conclude that the $2$-plectic structures induced by the Killing form on some important classical Lie algebras are not canonical.
In this paper, irreducible modules of the diagonal coset vertex operator algebra C(Lg(k+l,0),Lg(k,0)⊗Lg(l,0)) are classified under the assumption that C(Lg(k+l,0),Lg(k,0)⊗Lg(l,0)) is rational, ...C2-cofinite and certain additional assumption. An explicit modular transformation formula of traces functions of C(Lg(k+l,0),Lg(k,0)⊗Lg(l,0)) is obtained. As an application, the fusion rules of C(LE8(k+2,0),LE8(k,0)⊗LE8(2,0)) are determined by using the Verlinde formula.
In this paper, we focus on the structure of the variety of Lie algebras with a finite number of ideals and their graph representations using Hasse diagrams. The large number of necessary conditions ...on the algebraic structure of this type of algebras leads to the explicit description of those algebras in the variety with trivial Frattini subalgebra. To illustrate our results, we have included and discussed lots of examples throughout the paper.
A set grading on the split simple Lie algebra of type D13, that cannot be realized as a group-grading, is constructed by splitting the set of positive roots into a disjoint union of pairs of ...orthogonal roots, following a pattern provided by the lines of the projective plane over GF(3). This answers in the negative 3, Question 1.11.
Similar non-group gradings are obtained for types Dn with n≡1(mod12), by substituting the lines in the projective plane by blocks of suitable Steiner systems.
We consider real 2-step metric nilpotent Lie algebras associated to graphs with possibly repeated edge labels as constructed by Ray in 2016. We determine how the structure of the edge labeling within ...the graph contributes to the abelian factor in these Lie algebras. Furthermore, we explicitly compute the abelian factor of the 2-step nilpotent Lie algebras associated with some families of graphs such as star graphs, cycles, Schreier graphs, and properly edge-colored graphs. We also study the singularity properties of these Lie algebras in certain cases.
The paper is devoted to give a full classification of all finite-dimensional nilpotent Lie algebras L of class 4 with
Moreover, we specify the capable ones.
For q∈(0,1), the q-deformation of the square white noise Lie algebra is introduced using the q-calculus. A representation of this Lie algebra is given, using the q-derivative (or Jackson derivative) ...and the multiplication operator. The free square white noise Lie algebra is defined. Moreover, its representation on the Hardy space is given.
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