Related papers: Separating invariants of three nilpotent 3x3 matri…

We study the ring R(n,m) of invariants for the left-right action of SL_n \times SL_n on m-tuples of n by n complex matrices. We show that R(3,m) is generated by invariants of degree less equal 309 for all m. Then, we use a combinatorial…

We give an algebraic classification of complex $5$-dimensional one-generated nilpotent terminal algebras.

The paper is devoted to give a full classification of all finite dimensional nilpotent Lie algebras $ L $ of class $4$ such that $ \dim L^2=3. $ Moreover, we classify the capable ones.

We consider the conjugation-action of an arbitrary upper-block parabolic subgroup of ${\rm GL}_n(\mathbf{C})$ on the variety of $x$-nilpotent complex matrices and translate it to a representation-theoretic context. We obtain a criterion as…

The set of matrix tuples with invariant subspaces whose dimensions sum up to the dimension of the space, but which do not span the whole space form an algebraic hypersurface. We found the equation of this hypersurface. This generalizes…

We present some results concerning the generalized homologies associated with nilpotent endomorphisms $d$ such that $d^N=0$ for some integer $N\geq 2$. We then introduce the notion of graded $q$-differential algebra and describe some…

Given an nxn nilpotent matrix over an algebraically closed field K, we prove some properties of the set of all the nxn nilpotent matrices over K which commute with it. Then we give a proof of the irreducibility of the variety of all the…

In this paper and its sequel we continue our study of nilpotent symplectic alternating algebras. In particular we give a full classification of such algebras of dimension $10$ over any field. It is known that symplectic alternating algebras…

Let $k$ be an algebraically closed field of any characteristic except 2, and let $G = \GL_n(k)$ be the general linear group, regarded as an algebraic group over $k$. Using an algebro-geometric argument and Dynkin-Kostant theory for $G$ we…

The notion of defining relations is well-defined for any nilpotent Lie algebra. Therefore a conventional way to present a simple Lie algebra G is by splitting it into the direct sum of a commutative Cartan subalgebra and two maximal…

For modular indecomposable representations of a cyclic group $G$ of prime order $p$ we propose a list of polynomial invariants of degree $\leq 3$ that, together with a simple invariant of degree $p$, separate generic orbits and generate the…

For a nilpotent group $G$, let $\Xi(G)$ be the difference between the complement of the generating graph of $G$ and the commuting graph of $G$, with vertices corresponding to central elements of $G$ removed. That is, $\Xi(G)$ has vertex set…

In this paper we classify filiform associative algebras of degree $k$ over a field of characteristic zero. Moreover, we also classify naturally graded complex filiform and quasi-filiform nilpotent associative algebras which are described by…

We develop a structure theory for nilpotent symplectic alternating algebras.

We construct all solvable Lie algebras with a specific n-dimensional nilradical n_{n,3} which contains the previously studied filiform nilpotent algebra n_{n-2,1} as a subalgebra but not as an ideal. Rather surprisingly it turns out that…

Let $k$ be an algebraically closed field of characteristic different from 2. Up to isomorphism, the algebra $\operatorname{Mat}_{n \times n}(k)$ can be endowed with a $k$-linear involution in one way if $n$ is odd and in two ways if $n$ is…

The conjugation action of the complex orthogonal group on the polynomial functions on $n \times n$ matrices gives rise to a graded algebra of invariant polynomials. A spanning set of this algebra is in bijective correspondence to a set of…

We study the subspace of the exterior algebra of a simple complex Lie algebra linearly spanned by the copies of the little adjoint representation or, in the case of the Lie algebra of traceless matrices, by the copies of the n-th symmetric…

The index of a Lie algebra is an important invariant which arises in several areas, e.g. in the study of coadjoint orbits for a Lie group, in invariant theory and in representation theory. We study the index for several classes of nilpotent…

Let N be a nilpotent Lie group and let S be an invariant geometric structure on N (cf. symplectic, complex or hypercomplex). We define a left invariant Riemannian metric on N compatible with S to be "minimal", if it minimizes the norm of…