Invariants of the coadjoint representation of two classes of Lie algebras are calculated. The first class consists of the nilpotent Lie algebras T (M), isomorphic to the algebras of upper triangular M × M matrices. The Lie algebra T (M) is shown to have [M/2] functionally independent invariants. They can all be chosen to be polynomials and they are presented explicitly. The second class consist...

Given a sequence d→=(d1,…,dk) of natural numbers, we consider the Lie subalgebra h gl(d,F), where d=d1+⋯+dk and F is field characteristic 0, generated by two block upper triangular matrices D E partitioned according to d→, study problem computing nilpotency degree m nilradical n h. We obtain complete answer when belong certain family that arises naturally attempting classify indecomposable modu...

Parabolic subalgebras p of semisimple Lie algebras define a Z-grading of the Lie algebra. If there exists a nilpotent element in the first graded part of g on which the adjoint group of p acts with a dense orbit, the parabolic subalgebra is said to be nice. The corresponding nilpotent element is also called admissible. Nice parabolic subalgebras of simple Lie algebras have been classified. In t...

An algebraic algorithm is developed for computation of invariants (‘generalized Casimir operators’) of general Lie algebras over the real or complex number field. Its main tools are the Cartan’s method of moving frames and the knowledge of the group of inner automorphisms of each Lie algebra. Unlike the first application of the algorithm in [J.Phys. A: Math. Gen., 2006, V.39, 5749; math-ph/0602...

Let G be a simple algebraic group over the complex numbers containing a Borel subgroup B. Given a B-stable ideal I in the nilradical of the Lie algebra of B, we define natural numbers m1,m2, . . . ,mk which we call ideal exponents. We then propose two conjectures where these exponents arise, proving these conjectures in types An, Bn, Cn and some other types. When I = 0, we recover the usual exp...

We study the ∆-filtered modules for the Auslander algebra of k[T ]/Tn ⋊C2 where C2 is the cyclic group of order two. The motivation for this is the bijection between parabolic orbits in the nilradical of a parabolic subgroup of SLn and certain ∆-filtered modules for the Auslander algebra of k[T ]/Tn as found by Hille and Röhrle and Brüstle et al., cf. [HR99] [BHRR99]. Under this bijection, the ...

Let $R$ be a commutative ring with identity and $N(R)$ and
 $J\left(R\right)$ denote the nilradical Jacobson radical
 of $R$, respectively. A proper ideal $I$ is called an
 n-ideal if for every $a,b\in R$, whenever $ab\in I$\ $a\notin
 N(R)$, then $b\in I$. In this paper, we introduce study
 J-ideals as new generalization n-ideals in rings.
 $I$\ $R$\ J-ideal $ab\i...