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Let $G$ be a simple algebraic group over an algebraically closed field $k$ of characteristic $p$. The classification of the conjugacy classes of unipotent elements of $G(k)$ and nilpotent orbits of $G$ on $\operatorname{Lie}(G)$ is well-established. One knows there are representatives of every unipotent class as a product of root group elements and every nilpotent orbit as a sum of root elements. We give explicit representatives in terms of a Chevalley basis for the eminent classes. A unipotent (resp. nilpotent) element is said to be eminent if it is not contained in any subsystem subgroup (resp. subalgebra), or a natural generalisation if $G$ is of type $D_n$. From these representatives, it is straightforward to generate representatives for any given class. Along the way we also prove recognition theorems for identifying both the unipotent classes and nilpotent orbits of exceptional algebraic groups., 26 pages

This work concerns the representation theory and cohomology of a finite unipotent supergroup scheme $G$ over a perfect field $k$ of positive characteristic $p\ge 3$. It is proved that an element $x$ in the cohomology of $G$ is nilpotent if and only if for every extension field $K$ of $k$ and every elementary sub-supergroup scheme $E\subseteq G_K$, the restriction of $x_K$ to $E$ is nilpotent. It is also shown that a $kG$-module $M$ is projective if and only if for every extension field $K$ of $k$ and every elementary sub-supergroup scheme $E\subseteq G_K$, the restriction of $M_K$ to $E$ is projective. The statements are motivated by, and are analogues of, similar results for finite groups and finite group schemes, but the structure of elementary supergroups schemes necessary for detection is more complicated than in either of these cases. One application is a detection theorem for the nilpotence of cohomology, and projectivity of modules, over finite dimensional Hopf subalgebras of the Steenrod algebra., 46 pages; Sections 12 on Z-graded group schemes and the Steenrod algebra is revised compared to the previous version

We prove a general theorem for constructing integral quantum cluster algebras over Z[q±1/2], namely, that under mild conditions the integral forms of quantum nilpotent algebras always possess integral quantum cluster algebra structures. These algebras are then shown to be isomorphic to the corresponding upper quantum cluster algebras, again defined over Z[q±1/2]. Previously, this was only known for acyclic quantum cluster algebras. The theorem is applied to prove that, for every symmetrizable Kac–Moody algebra g and Weyl group element w, the dual canonical form Aq(n+(w))Z[q±1] of the corresponding quantum unipotent cell has the property that Aq(n+(w))Z[q±1]⊗Z[q±1]Z[q±1/2] is isomorphic to a quantum cluster algebra over Z[q±1/2] and to the corresponding upper quantum cluster algebra over Z[q±1/2].

Let $V$ be a finite-dimensional vector space over a field of characteristic two. As the main result of this paper, for every nilpotent element $e \in \mathfrak{sl}(V)$, we describe the Jordan normal form of $e$ on the $\mathfrak{sl}(V)$-modules $\wedge^2(V)$ and $S^2(V)$. In the case where $e$ is a regular nilpotent element, we are able to give a closed formula. We also consider the closely related problem of describing, for every unipotent element $u \in \operatorname{SL}(V)$, the Jordan normal form of $u$ on $\wedge^2(V)$ and $S^2(V)$. A recursive formula for the Jordan block sizes of $u$ on $\wedge^2(V)$ was given by Gow and Laffey (J. Group Theory 9 (2006), 659-672). We show that their proof can be adapted to give a similar formula for the Jordan block sizes of $u$ on $S^2(V)$., to appear in Linear Algebra and its Applications

Let $G$ be a classical group with natural module $V$ and Lie algebra $\mathfrak{g}$ over an algebraically closed field $K$ of good characteristic. For rational irreducible representations $f: G \rightarrow \operatorname{GL}(W)$ occurring as composition factors of $V \otimes V^*$, $\wedge^2(V)$, and $S^2(V)$, we describe the Jordan normal form of $\mathrm{d} f(e)$ for all nilpotent elements $e \in \mathfrak{g}$. The description is given in terms of the Jordan block sizes of the action of $e$ on $V \otimes V^*$, $\wedge^2(V)$, and $S^2(V)$, for which recursive formulae are known. Our results are in analogue to earlier work (Proc. Amer. Math. Soc., 147 (2019) 4205-4219), where we considered these same representations and described the Jordan normal form of $f(u)$ for every unipotent element $u \in G$., to appear in J. Pure Appl. Algebra

Let $p$ and $\ell$ be distinct primes, and let $\overline{\unicode[STIX]{x1D70C}}$ be an orthogonal or symplectic representation of the absolute Galois group of an $\ell$-adic field over a finite field of characteristic $p$. We define and study a liftable deformation condition of lifts of $\overline{\unicode[STIX]{x1D70C}}$ ‘ramified no worse than $\overline{\unicode[STIX]{x1D70C}}$’, generalizing the minimally ramified deformation condition for $\operatorname{GL}_{n}$ studied in Clozel et al. [Automorphy for some$l$-adic lifts of automorphic mod$l$Galois representations, Publ. Math. Inst. Hautes Études Sci. 108 (2008), 1–181; MR 2470687 (2010j:11082)]. The key insight is to restrict to deformations where an associated unipotent element does not change type when deforming. This requires an understanding of nilpotent orbits and centralizers of nilpotent elements in the relative situation, not just over fields.

This paper provides a comparison between the K K -structure of unipotent representations and regular sections of bundles on nilpotent orbits for complex groups of type D D . Precisely, let G 0 = Spin ⁡ ( 2 n , C ) G_0 =\operatorname {Spin}(2n,\mathbb {C}) be the Spin complex group as a real group, and let K ≅ G 0 K\cong G_0 be the complexification of the maximal compact subgroup of G 0 G_0 . We compute K K -spectra of the regular functions on some small nilpotent orbits O \mathcal {O} transforming according to characters ψ \psi of C K ( O ) C_{ K}(\mathcal {O}) trivial on the connected component of the identity C K ( O ) 0 C_{ K}(\mathcal {O})^0 . We then match them with the K {K} -types of the genuine (i.e., representations which do not factor to SO ⁡ ( 2 n , C ) \operatorname {SO}(2n,\mathbb {C}) ) unipotent representations attached to O \mathcal {O} .

For an isolated hypersurface singularity \(f:(\mathbb {C}^{n+1},0)\rightarrow (\mathbb {C},0)\) with Milnor number \(\mu \) and good representative \(f:(X,0)\rightarrow (\Delta ,0)\) canonical \(\mu \)-dimensional \(\mathbb {C}\)-bilinear vector spaces are associated: the Jacobian module, \(\Omega ^{f}\), which is isomorphic to the Milnor algebra \(A_f\) up to a choice of coordinates; and the cohomology of the canonical Milnor fiber, H. Indeed, one has defined on \(\Omega ^f\), and hence in \(A_f\), the non-degenerate Grothendieck pairing \(res_{f,0}\) which is a symmetric \(\mathbb {C}\)-bilinear form, and on the vanishing cohomology H it is defined a non-degenerate \(\mathbb {C}\)-bilinear form \(\mathbb {S}\), induced by Poincare duality, which is \((-1)^{n+1}\)-symmetric on the generalized monodromy eigenspace \(H_{1}\) and \((-1)^{n}\)-symmetric on the direct sum of generalized monodromy eigenspaces \(H_{\ne 1}:=\oplus _{\lambda \ne 1}H_{\lambda }\). On the other hand, there are two nilpotent \(\mathbb {C}\)-linear maps defined on \(\Omega ^f\) and H, respectively; the first one is the map \(\{\mathbf {f}\}\) given by multiplication with f, which is \(res_{f,0}\)-symmetric, and the other one is the \(\mathbb {S}\)-antisymmetric endomorphism N given by the logarithm of the unipotent part of the monodromy transformation. New bilinear forms can be constructed by composing on the left (or equivalently on the right) with powers of such nilpotent maps: \(res_{f,0}(\{\mathbf {f}\}^{\ell }\bullet ,\bullet )\) and \(\mathbb {S}(N^{\ell }\bullet ,\bullet )\) for each integer \(\ell \ge 1\). These new bilinear forms are called higher bilinear forms on \(\Omega ^f\) resp. on H. In this paper, we show a formula which relates the powers \(\{\mathbf {f}\}^{\ell }\), \(\ell \ge 1\), to the powers \(N^{j}\), \(j\ge 1\). Our proof, which is inspired by a result of Varchenko obtained in 1981, uses the Laurent series (asymptotic) expansions of elements in the Jacobian module with respect to the Malgrange–Kashiwara’s \(\mathcal {V}\)-filtration. Finally, when the relation between Saito pairing and Grothendieck pairing is considered such a formula provides us with a result that gives an additive expansion for each higher bilinear form on \(\Omega ^f\) expressed in terms of the higher bilinear forms on H and depending on the asymptotic expansions for the top forms on \(\Omega ^f\) where these bilinear forms act.

A ring $R$ is a UU ring if every unit is unipotent, or equivalently if every unit is a sum of a nilpotent and an idempotent that commute. These rings have been investigated in C\u{a}lug\u{a}reanu \cite{C} and in Danchev and Lam \cite{DL}. In this paper, two generalizations of UU rings are discussed. We study rings for which every unit is a sum of a nilpotent and an idempotent, and rings for which every unit is a sum of a nilpotent and two idempotents that commute with one another.

In this paper, bifurcation of limit cycles is considered for planar cubic-order systems with an isolated nilpotent critical point. Normal form theory is applied to compute the generalized Lyapunov constants and to prove the existence of at least 9 small-amplitude limit cycles in the neighborhood of the nilpotent critical point. In addition, the method of double bifurcation of nilpotent focus is used to show that such systems can have 10 small-amplitude limit cycles near the nilpotent critical point. These are new lower bounds on the number of limit cycles in planar cubic-order systems near an isolated nilpotent critical point. Moreover, a set of center conditions is obtained for such cubic systems.

An almost nilpotent variety of linear growth is constructed in the paper for any infinite periodic word in an alphabet of two letters. A discrete series of different almost nilpotent varieties is also constructed. Only a few almost nilpotent varieties were studied previously and their existence was proved often under some additional assumptions. The existence of almost nilpotent varieties of arbitrary integer exponential growth with a fractional exponent is proved as well as the existence of a continual family of almost nilpotent varieties with not more than quadratic growth.

Let $G$ be a finitely generated torsion-free nilpotent group. The representation zeta function $\zeta_G(s)$ of $G$ enumerates twist isoclasses of finite-dimensional irreducible complex representations of $G$. We prove that $\zeta_G(s)$ has rational abscissa of convergence $a(G)$ and may be meromorphically continued to the left of $a(G)$ and that, on the line $\{s\in\mathbb{C} \mid \textrm{Re}(s) = a(G)\}$, the continued function is holomorphic except for a pole at $s=a(G)$. A Tauberian theorem yields a precise asymptotic result on the representation growth of $G$ in terms of the position and order of this pole. We obtain these results as a consequence of a more general result establishing uniform analytic properties of representation zeta functions of finitely generated nilpotent groups of the form $\mathbf{G}(\mathcal{O})$, where $\mathbf{G}$ is a unipotent group scheme defined in terms of a nilpotent Lie lattice over the ring $\mathcal{O}$ of integers of a number field. This allows us to show, in particular, that the abscissae of convergence of the representation zeta functions of such groups and their pole orders are invariants of $\mathbf{G}$, independent of $\mathcal{O}$., Comment: 25 pages, final version, to appear in the Transactions of the American Mathematical Society

For a useful class of functions (containing functions whose one finite component is essentially a matrix coefficient of a supercuspidal representation), we establish three results about the unipotent contribution of the Guo-Jacquet relative trace formula for the pair $(GL_n(D),GL_n(E))$. First we get a fine expansion in terms of global nilpotent integrals. Second we express these nilpotent integrals in terms of zeta integrals. Finally we prove that they satisfy certain homogeneity properties. The proof is based on a new kind of truncation introduced in a previous article.

We study the simultaneously nilpotent index of a simultaneously nilpotent set of matrices over an antinegative commutative semiring S . We find an upper bound for this index and give some characterizations of the simultaneously nilpotent sets when this upper bound is met. In the special case of antinegative semirings with all zero divisors nilpotent, we also find a bound on the simultaneously nilpotent index for all nonmaximal simultaneously nilpotent sets of matrices and establish their cardinalities in case of a finite S .

In this paper we prove that any finite nilpotent alternative ring with planar zero-divisor graph is associative.

Let F be an algebraically closed field of positive characteristic p > 3 , and A the divided power algebra in one indeterminate, which, as a vector space, coincides with the truncated polynomial ring of F [ T ] by T p n . Let g be the special derivation algebra over A which is a simple Lie algebra, and additionally non-restricted as long as n > 1 . Let N be the nilpotent cone of g , and G = Aut ( g ) , the automorphism group of g . In contrast with only finitely many nilpotent orbits in a classical simple Lie algebra, there are infinitely many nilpotent orbits in g . In this paper, we parameterize all nilpotent orbits, and obtain their dimensions. Furthermore, the nilpotent cone N is proven to be reducible and not normal. There are two irreducible components in N . The dimension of N is determined.

In this note we report on recent differential geometric constructions aimed at devising representations of braid groups in various contexts, together with some applications in different domains of mathematical physics. First, the classical Kohno construction for the 3- and 4-strand pure braid groups \(P_3\) and \(P_4\) is explicitly implemented by resorting to the Chen-Hain-Tavares nilpotent connections and to hyperlogarithmic calculus, yielding unipotent representations able to detect Brunnian and nested Brunnian phenomena. Physically motivated unitary representations of Riemann surface braid groups are then described, relying on Bellingeri’s presentation and on the geometry of Hermitian–Einstein holomorphic vector bundles on Jacobians, via representations of Weyl-Heisenberg groups.

The nilpotent graph of a group G is a simple graph whose vertex set is G∖nil(G), where nil(G) = {y ∈ G | ⟨ x, y ⟩ is nilpotent ∀ x ∈ G}, and two distinct vertices x and y are adjacent if ⟨ x, y ⟩ is nilpotent. In this article, we show that the collection of finite non-nilpotent groups whose nilpotent graphs have the same genus is finite, derive explicit formulas for the genus of the nilpotent graphs of some well-known classes of finite non-nilpotent groups, and determine all finite non-nilpotent groups whose nilpotent graphs are planar or toroidal.

Let H be a quadratic homogeneous polynomial map of dimension n over an infinite field in which 2 is invertible such that its Jacobian JH is nilpotent. Meisters and Olech have shown that JH is strongly nilpotent if n ≤ 4 . They also proved that it is not true when n = 5 . We show that if rank J H ≤ 2 and n arbitrary, then JH is strongly nilpotent. We also give examples to show that this is no longer true for any rank and dimension as long as the rank is greater than 2.

In studying nilpotent groups, the lower central series and other variations can be used to construct an associated $\mathbb{Z}^+$-graded Lie ring, which is a powerful method to inspect a group. Indeed, the process can be generalized substantially by introducing $\mathbb{N}^d$-graded Lie rings. We compute the adjoint refinements of the lower central series of the unipotent subgroups of the classical Chevalley groups over the field $\mathbb{Z}/p\mathbb{Z}$ of rank $d$. We prove that, for all the classical types, this characteristic filter is a series of length $\Theta(d^2)$ with nearly all factors having $p$-bounded order., Comment: 13 pages, 3 figures

Our goal is to classify all generically transitive actions of commutative unipotent groups on flag varieties up to conjugation. We establish relationship between this problem and classification of multiplications with certain properties on Lie algebra representations. Then we classify multiplications with the desired properties and solve the initial classification problem., Comment: 25 pages, master thesis (improved afterwards)

Fixing an arithmetic lattice $\Gamma$ in an algebraic group $G$, the commensurability growth function assigns to each $n$ the cardinality of the set of subgroups $\Delta$ with $[\Gamma : \Gamma \cap \Delta] [\Delta: \Gamma \cap \Delta] = n$. This growth function gives a new setting where methods of F. Grunewald, D. Segal, and G. C. Smith's "Subgroups of finite index in nilpotent groups" apply to study arithmetic lattices in an algebraic group. In particular, we show that for any unipotent algebraic $\mathbb{Z}$-group with arithmetic lattice $\Gamma$, the Dirichlet function associated to the commensurability growth function satisfies an Euler decomposition. Moreover, the local parts are rational functions in $p^{-s}$, where the degrees of the numerator and denominator are independent of $p$. This gives regularity results for the set of arithmetic lattices in $G$., Comment: 10 pages: revised introduction, made minor corrections, and added references

Let U be the unipotent radical of a Borel subgroup of a connected reductive algebraic group G, which is defined over an algebraically closed field k. In this paper, we extend work by Goodwin-R\"ohrle concerning the commuting variety of Lie(U) for char(k)=0 to fields, whose characteristic is good for G, Comment: Final version

If a geometry $\Gamma$ is isomorphic to the residue of a point $A$ of a shadow geometry of a spherical building $\Delta$, a representation $\varepsilon_\Delta^A$ of $\Gamma$ can be given in the unipotent radical $U_{A^*}$ of the stabilizer in $\mathrm{Aut}(\Delta)$ of a flag $A^*$ of $\Delta$ opposite to $A$, every element of $\Gamma$ being mapped onto a suitable subgroup of $U_{A^*}$. We call such a representation a unipotent representation. We develope some theory for unipotent representations and we examine a number of interesting cases, where a projective embedding of a Lie incidence geometry $\Gamma$ can be obtained as a quotient of a suitable unipotent representation $\varepsilon_\Delta^A$ by factorizing over the derived subgroup of $U_{A^*}$, while $\varepsilon^A_\Delta$ itself is not a proper quotient of any other representation of $\Gamma$., Comment: 35 pages

Встановлено, що в напівпервинному кільці всі диференціювання (відповідно внутрішні диференціювання) нільпотентні тоді і тільки тоді, коли воно диференційно тривіальне (відповідно комутативне). Радикал Джекобсона кільця з нільпотентними диференціюваннями індексів $\leq 2$ містить всі нільпотентні елементи кільця.

We prove a Freiman-Ruzsa-type theorem valid in an arbitrary nilpotent group. Specifically, we show that a K-approximate subgroup A of an s-step nilpotent group G is contained in a coset nilprogression of rank at most f(K) and cardinality at most exp(g(K))|A|, with f and g polynomials depending only on the step s of G. To motivate this, we give a direct proof of Breuillard and Green's analogous result for torsion-free nilpotent groups, avoiding the use of Mal'cev's embedding theorem., Comment: 39 pages. Version 2 contains a corrected proof of Theorem 1.7; improved exposition in Section 7, incorporating suggestions from an anonymous referee; and various minor presentational and organisational changes. Version 3 reflects changes made by a copy editor. Appears in Proc. London Math. Soc. at http://plms.oxfordjournals.org/cgi/content/abstract/pdu005? ijkey=AVizKoIO9JYHczu&keytype=ref

We show that a conjecture of Groves for modules over nilpotent groups of class 2 holds for the codimension 2 case with certain assumptions.

Let $G$ be a solvable subgroup of the group $\diff{}{n}$ of local complex analytic diffeomorphisms. Analogously as for groups of matrices we bound the solvable length of $G$ by a function of $n$. Moreover we provide the best possible bounds for connected, unipotent and nilpotent groups., 27 pages

The concept of the derived π-length for finite π-soluble groups is introduced and its elementary properties are described. The dependence between the π-length, nilpotent π-length, and derived π-length, and also between the derived and nilpotent lengths of a π-Hall subgroup, is determined.

In this paper, we introduce the smallest equivalence relation ν⁎ on a hypergroup H such that the quotient H/ν⁎, the set of all equivalence classes, is a nilpotent group. The characterization of nilpotent groups via strongly regular relations is investigated and several results on the topic are presented.

For the p-adic group G = SL(2), we present results of the computations of the sums of the Bernstein projectors of a given depth. Motivation for the computations is based on a conversation with Roger Howe in August 2013. The computations are elementary, but they provide an expansion of the delta distribution \(\delta _{1_{G}}\) into an infinite sum of G-invariant locally integrable essentially compact distributions supported on the set of topologically unipotent elements. When these distributions are transferred, by the exponential map, to the Lie algebra, they give G-invariant distributions supported on the set of topologically nilpotent elements, whose Fourier transforms turn out to be characteristic functions of very natural G-domains. The computations in particular rely on the SL(2) discrete series character tables computed by Sally–Shalika in 1968. This new phenomenon for general rank has also been independently noticed in recent work of Bezrukavnikov, Kazhdan, and Varshavsky.

Let G be a solvable group and H a solvable subgroup of Aut ⁡ ( G ) $\operatorname{Aut}(G)$ whose elements are n-unipotent. When H is finitely generated, we show that it stabilizes a finite series in G and conclude that H is nilpotent. If G furthermore has a characteristic series with torsion-free factors. the same conclusion as above holds without the extra assumption that H is finitely generated.

This chapter is based on a seminal paper entitled “The Group Ring of a Class of Infinite Nilpotent Groups” by S. A. Jennings. In Sect. 6.1, we consider the group ring of a finitely generated torsion-free nilpotent group over a field of characteristic zero. We prove that its augmentation ideal is residually nilpotent. We introduce the dimension subgroups of a group in Sect. 6.2. These subgroups are defined in terms of the augmentation ideal of the corresponding group ring. We prove that the nth dimension subgroup coincides with the isolator of the nth lower central subgroup. This is a major result involving a succession of clever reductions where nilpotent groups play a prominent role. Section 6.3 deals with nilpotent Lie algebras. We show that there exists a nilpotent Lie algebra over a field of characteristic zero which is associated to a finitely generated torsion-free nilpotent group. As it turns out, the underlying vector space of this Lie algebra has dimension equal to the Hirsch length of the given group.

A non-nilpotent variety of algebras is almost nilpotent if any proper subvariety is nilpotent. Let the base field be of characteristic zero. It has been shown that for associative or Lie algebras only one such variety exists. Here we present infinite families of such varieties. More precisely we shall prove the existence of 1) a countable family of almost nilpotent varieties of at most linear growth and 2) an uncountable family of almost nilpotent varieties of at most quadratic growth., 7 pages

Let G be a simple, simply-connected algebraic group over the complex numbers with Lie algebra $\mathfrak g$. The main result of this article is a proof that each irreducible representation of the fundamental group of the orbit O through a nilpotent element $e \in \mathfrak g$ lifts to a representation of a Jacobson-Morozov parabolic subgroup of G associated to e. This result was shown in some cases by Barbasch and Vogan in their study of unipotent representations for complex groups and, in general, in an unpublished part of the author's doctoral thesis. In the last section of the article, we state two applications of this result, whose details will appear elsewhere: to answering a question of Lusztig regarding special pieces in the exceptional groups (joint work with Fu, Juteau, and Levy); and to computing the G-module structure of the sections of an irreducible local system on O. A key aspect of the latter application is some new cohomological statements that generalize those in earlier work of the author., minor corrections. 16 pages. Accepted in Bulletin of the Institute of Mathematics Academia Sinica

We prove that the unipotent horocyclic group of a Moufang twin tree of prime order is nilpotent of class at most 2.

In this paper, we show that quantum twist maps, introduced by Lenagan-Yakimov, induce bijections between dual canonical bases of quantum nilpotent subalgebras. As a corollary, we show the unitriangular property between dual canonical bases and Poincar\'e-Birkhoff-Witt type bases under the "reverse" lexicographic order. We also show that quantum twist maps induce bijections between certain unipotent quantum minors., Comment: 13 pages, minor revision, to appear in Algebras and Representation Theory

We study the structure of the nilpotent commutator $\nb$ of a nilpotent matrix $B$. We show that $\nb$ intersects all nilpotent orbits for conjugation if and only if $B$ is a square--zero matrix. We describe nonempty intersections of $\nb$ with nilpotent orbits in the case the $n \times n$ matrix $B$ has rank $n-2$. Moreover, we give some results on the maximal nilpotent orbit that $\nb$ intersects nontrivially., Comment: 14 pages

This is the second article by the authors on extraction of roots in exponential A-groups. We prove results on ω-torsion and ω-isolated subgroups, 𝒰ω-groups, ℰω-groups, and 𝒟ω-groups in the category of exponential A-groups, where A is a unique factorization domain (UFD) and ω is a set of primes in A. In particular, we prove that every ω-torsion-free -group is a 𝒰ω-group. We also prove that if R is a principal ideal domain (PID) and G is a finitely R-generated nilpotent R-powered group, then the R-subgroup of ω'-torsion elements of G equals its maximal ℰω-subgroup (ω' denotes the set of primes in R not in ω).

In this paper, nilpotent subsemigroups in the matrix semigroup over a commutative antiring are discussed. Some basic properties and characterizations for the nilpotent subsemigroups are given, and some equivalent conditions for the matrix semigroup over a commutative antiring to have a maximal nilpotent subsemigroup are obtained. Also, the maximal nilpotent subsemigroups in the matrix semigroup are described.

The notions of the nilpotent and the strong-nilpotent Leibniz 3-algebras are defined. And the three dimensional two-step nilpotent, strong-nilpotent Leibniz 3-algebras are classified.

In this article, the nilpotent orbits of the Witt algebra W 1 are determined under the automorphism group over an algebraically closed field F of characteristic p > 3. In contrast with a finite num...