Let G be a complex simple algebraic group and let g be its Lie algebra. A nilpotent orbit O in g is an orbit of a nilpotent element of g by the adjoint action of G on g. Then O admits a natural symplectic 2-form ω and the nilpotent orbit closure Ō has symplectic singularities in the sense of [Be] and [Na3] (cf. [Pa], [Hi]). In [Ri], Richardson introduced the notion of so-called the Richadson or...

Let g be a complex simple Lie algebra. Fix a Borel subalgebra b and a Cartan subalgebra t ⊂ b. The nilpotent radical of b is denoted by u. The corresponding set of positive (resp. simple) roots is ∆ (resp. Π). An ideal of b is called ad-nilpotent, if it is contained in [b, b]. The theory of ad-nilpotent ideals has attracted much recent attention in the work of Kostant, Cellini-Papi, Sommers, an...

Let X be a finite set and let k be a commutative ring. We consider the k-algebra of the monoid of all relations on X, modulo the ideal generated by the relations factorizing through a set of cardinality strictly smaller than Card(X), called inessential relations. This quotient is called the essential algebra associated to X. We then define a suitable nilpotent ideal of the essential algebra and...

In this paper, first we introduce the notions of k-nilpotent (solvable) ideals and BCK-algebras. Specially, show that every commutative ideal is 1-nilpotent (solvable). Second, state an equivalent condition to k-nilpotency (solvablity) BCK-algebra. Finally, study n-fold 2-nilpotent BCK-algebras as a generalization BCK-algebras, relation between these two concepts. Basically, compare solvable (B...

An improved version of the well-known Poincaré-Dulac’s normal form theorem is first proposed. It is shown that, for a nonlinear vector field, a normal form near a singular point can always be chosen so that the number of nonlinear components is at most equal to the number of Jordan blocks in the normalized leading matrix, thus leading to the “simplest” form in which a formal vector field can be...

An n × n sign pattern Sn is spectrally arbitrary if, for any given real monic polynomial g(x) of degree n, there is a real matrix having sign pattern Sn and characteristic polynomial g(x). All n × n star sign patterns that are spectrally arbitrary, and all minimal such patterns, are characterized. This subsequently leads to an explicit characterization of all n × n star sign patterns that are p...

The conjugacy classes of the nite general linear and unitary groups are used to de ne probability measures on the set of all partitions of all natural numbers. Probabilistic algorithms for growing random partitions according to these measures are obtained. These algorithms are applied to prove group theoretic results which are typically proved by techniques such as character theory and Moebius ...

A non-negative pluriharmonic polynomial <p(z) on the unit ball of C is used as a weight against the rotationally invariant measure on the unit sphere. The resulting Hardy space carries the canonical n-tuple S of multiplication by the coordinate functions. By means of compressions of S to co-analytically invariant subspaces, and known estimates of the numerical radius of a nilpotent matrix we ob...

The isomorphism problem for centrally nilpotent loops can be tackled by methods of cohomology. We develop tools based on cohomology and linear algebra that either lend themselves to direct count of the isomorphism classes (notably in the case of nilpotent loops of order 2q, q a prime), or lead to efficient classification computer programs. This allows us to enumerate all nilpotent loops of orde...