Let L be a Lie pseudoalgebra, a ∈ L. We show that, if a generates a (finite) solvable subalgebra S = 〈a〉 ⊂ L, then one may find a lifting ā ∈ S of [a] ∈ S/S such that 〈ā〉 is nilpotent. We then apply this result towards vertex algebras: we show that every finite vertex algebra V admits a decomposition into a semi-direct product V = U⋉N , where U is a subalgebra of V whose underlying Lie conforma...

We study the classification problem for left-symmetric algebras with commutation Lie algebra gl(n) in characteristic 0. The problem is equivalent to the classification of étale affine representations of gl(n). Algebraic invariant theory is used to characterize those modules for the algebraic group SL(n) which belong to affine étale representations of gl(n). From the classification of these modu...

Let R be a discrete valuation ring with maximal ideal πR and residue field k. We study obstructions to lifting a Lie algebra L over R/πR to one over R/πR. If L̄ is the Lie algebra over k obtained from reducing L then we show there exists a well-defined class in H(L̄, ad) which vanishes if and only if L lifts. Furthermore, if L lifts, the lifts are shown to be in one to one correspondence with H(L̄...

We consider a skew-symmetric n -ary bracket on the polynomial algebra K [ x 1 , … + ] ( ≥ 2 ) over field of characteristic zero defined by { } = Jac C where is fixed element and Jacobian. If then this Poisson if 3 it an -Lie-Poisson . describe center corresponding show that quotient / − λ ideal generated 0 ≠ ∈ simple central homogeneous not proper power any nonzero polynomial. This construction...

Let φ : Rm → Rd be a map of free modules over a commutative ring R. Fitting’s Lemma shows that the “Fitting ideal,” the ideal of d × d minors of φ, annihilates the cokernel of φ and is a good approximation to the whole annihilator in a certain sense. In characteristic 0 we define a Fitting ideal in the more general case of a map of graded free modules over a Z/2graded skew-commutative algebra a...

We call (i?) 1 the Lie ring associated with R, and denote it by 9Î. The question of how far the properties of SR determine those of R is of considerable interest, and has been studied extensively for the case when R is an algebra, but little is known of the situation in general. In an earlier paper the author investigated the effect of the nilpotency of 9î upon the structure of R if R contains ...

The Royden ideal boundary of a domain is the set of all points in the maximal ideal space of the domain's Royden algebra that do not lie in the domain. Elements of the Royden ideal boundary can be characterized as nets convergent in both the weak and Euclidean topologies that have no subnet which is a sequence. As with other function algebras, boundary bers can be deened as subsets of all point...

a unital $c^*$ -- algebra $mathcal a,$ endowed withthe lie product $[x,y]=xy- yx$ on $mathcal a,$ is called a lie$c^*$ -- algebra. let $mathcal a$ be a lie $c^*$ -- algebra and$g,h:mathcal a to mathcal a$ be $bbb c$ -- linear mappings. a$bbb c$ -- linear mapping $f:mathcal a to mathcal a$ is calleda lie $(g,h)$ -- double derivation if$f([a,b])=[f(a),b]+[a,f(b)]+[g(a),h(b)]+[h(a),g(b)]$ for all ...

We study symplectic structures on characteristically nilpotent Lie algebras (CNLAs) by computing the cohomology space H(g, k) for certain Lie algebras g. Among these Lie algebras are filiform CNLAs of dimension n ≤ 14. It turns out that there are many examples of CNLAs which admit a symplectic structure. A generalization of a sympletic structure is an affine structure on a Lie algebra.

A subalgebra $B$ of a Lie algebra $L$ is called weak c-ideal if there subideal $C$ such that $L=B+C$ and $B\cap C\leq B_{L} $ where $B_{L}$ the largest ideal contained in $B.$ This analogous to concept weakly c-normal subgroups, which has been studied by number authors. We obtain some properties c-ideals use them give characterisations solvable supersolvable algebras. also note one-dimensional ...