Non-linear maps preserving ideals on a parabolic subalgebra of a simple algebra

Let P be an arbitrary parabolic subalgebra of a simple associative F-algebra. The ideals of P are determined completely; Each ideal of P is shown to be generated by one element; Every non-linear invertible map on P that preserves ideals is described in an explicit formula.     

Automorphisms of the standard Borel subalgebra of Lie algebra of Cm type over a commutative ring

All automorphisms of the standard Borel subalgebra of the symplectic algebra sp(2m, R) are determined, provided that R is a commutative ring with identity, 2 is invertible in R.     

Automorphisms of the standard Borel subalgebra of Lie algebra of Cm type over a commutative ring

All automorphisms of the standard Borel subalgebra of the symplectic algebra sp(2m,?R) are determined, provided that R is a commutative ring with identity, 2 is invertible in R.     

Abelian ideals of Borel subalgebras and affine Weyl groups

This paper is devoted to a detailed study of certain remarkable posets which form a natural partition of all abelian ideals of a Borel subalgebra. Our main result is a nice uniform formula for the dimension of maximal ideals in these posets. We also obtain results on ad-nilpotent ideals which complete the analysis started in (J. Algebra 225 (2000) 130, 258 (2002) 112).     

The isotropy subalgebra of the canonical 3-form of a semisimple Lie algebra

The isotropy subalgebra of the canonical 3-form of a semisimple Lie algebra over a field of characteristic zero is computed. Its isotropy subgroup is also studied.     

Ad-nilpotent ideals of a parabolic subalgebra

We extend the results of Cellini and Papi [P. Cellini, P. Papi, Ad-nilpotent ideals of a Borel subalgebra, J. Algebra 225 (2000) 130–140; P. Cellini, P. Papi, Ad-nilpotent ideals of a Borel subalgebra II, J. Algebra 258 (2002) 112–121] on the characterizations of ad-nilpotent and abelian ideals of a Borel subalgebra to parabolic subalgebras of a simple Lie algebra. These characterizations are given in terms of elements of the affine Weyl group and faces of alcoves. In the case of a parabolic subalgebra of a classical simple Lie algebra, we give formulas for the number of these ideals.     

Skew-symmetric derivations of a complex Lie algebra

Let be a complex Lie algebra, its underlying real Lie algebra, a real form of and ·, · the euclidean product induced by the real part of an hermitian inner product on . Let aut be the Lie algebra of skew-symmetric derivations of . We give necessary and sufficient conditions to ensure that aut is composed of skew-hermitian derivations. As an application, we study holomorphy in large subgroups of isometries of Lie groups.     

Powers of the Euler product and commutative subalgebras of a complex simple Lie algebra

If is a complex simple Lie algebra, and k does not exceed the dual Coxeter number of , then the absolute value of the k^th coefficient of the power of the Euler product may be given by the dimension of a subspace of defined by all abelian subalgebras of of dimension k. This has implications for all the coefficients of all the powers of the Euler product. Involved in the main results are Dale Petersons 2^rank theorem on the number of abelian ideals in a Borel subalgebra of , an element of type and my heat kernel formulation of Macdonalds -function theorem, a set D_alcove of special highest weights parameterized by all the alcoves in a Weyl chamber (generalizing Young diagrams of null m-core when ), and the homology and cohomology of the nil radical of the standard maximal parabolic subalgebra of the affine Kac–Moody Lie algebra.     

The adjoint representation inside the exterior algebra of a simple Lie algebra

For a simple complex Lie algebra g$\mathfrak{g}$ we study the space of invariants A=(?g^?⊗g^?)^g$A={(?{\mathfrak{g}}^{?}\otimes {\mathfrak{g}}^{?})}^{\mathfrak{g}}$, which describes the isotypic component of type g$\mathfrak{g}$ in ?g^?$?{\mathfrak{g}}^{?}$, as a module over the algebra of invariants (?g^?)^g${(?{\mathfrak{g}}^{?})}^{\mathfrak{g}}$. As main result we prove that A   is a free module, of rank twice the rank of g$\mathfrak{g}$, over the exterior algebra generated by all primitive invariants in (?g^?)^g${(?{\mathfrak{g}}^{?})}^{\mathfrak{g}}$, with the exception of the one of highest degree.     

Graded Lie algebras whose Cartan subalgebra is the algebra of polynomials in one variable

We define a class of infinite-dimensional Lie algebras that generalize the universal enveloping algebra of the algebra sl(2, ℂ) regarded as a Lie algebra. These algebras are a special case of ℤ-graded Lie algebras with a continuous root system, namely, their Cartan subalgebra is the algebra of polynomials in one variable. The continuous limit of these algebras defines new Poisson brackets on algebraic surfaces. In memory of M. V. Saveliev Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 2, pp. 345–352, May, 2000.     

The subalgebra lattice of a finite algebra

The aim of this paper is to characterize pairs (L, A), where L is a finite lattice and A a finite algebra, such that the subalgebra lattice of A is isomorphic to L. Next, necessary and sufficient conditions are found for pairs of finite algebras (of possibly distinct types) to have isomorphic subalgebra lattices. Both of these characterizations are particularly simple in the case of distributive subalgebra lattices. We do not restrict our attention to total algebras only, but we consider the more general case of partial algebras. Moreover, we use connections between algebras and hypergraphs to solve these problems.     

Cofinalities of Borel ideals

We study the possible values of the cofinality invariant for various Borel ideals on the natural numbers. We introduce the notions of a fragmented and gradually fragmented ideal and prove a dichotomy for fragmented ideals. We show that every gradually fragmented ideal has cofinality consistently strictly smaller than the cardinal invariant and produce a model where there are uncountably many pairwise distinct cofinalities of gradually fragmented ideals.     

The solubility of a Lie algebra which decomposes into a direct sum of an abelian and a nilpotent subalgebra

We prove that a Lie algebra over a field of characteristic 2 is soluble if it is a sum of an abelian and a nilpotent subalgebra.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 986–991, July–August, 1991.