Lie algebras admitting non-singular prederivations

The aim of this paper is to prove that any real or complex Lie algebra admitting a non-singular prederivation is necessarily a nilpotent Lie algebra. As to the reciprocal statement, an example is given of a nilpotent Lie algebra with only singular prederivations.     

Erratum to “Separation of representations with quadratic overgroups” [Bull. Sci. Math. 135 (2) (2011) 141–165]

In the paper entitled “Separation of representations with quadratic overgroups”, we defined the notion of quadratic overgroups, and announced that the 6-dimensional nilpotent Lie algebra g6,20 admits such a quadratic overgroup. There is a mistake in the proof. The present Erratum explains that the proposed overgroup is only weakly quadratic, and g6,20 does not admit any natural quadratic overgroup.     

Triple Lie Systems Associated with (−1, 1) Algebras

We introduce a Lie triple system associated with the central isotope of (− 1, 1)-algebra. The associator ideal of (−1, 1)-algebra is nilpotent if and only if the Lie triple system is nilpotent. The relationship of the constructed Lie triple system with other known Lie triple systems is discussed.     

GROUPS WITH ALL PROPER SUBGROUPS (FINITE RANK)-BY-NILPOTENT. II

The authors obtain results concerning the structure of locally soluble-by-finite groups with all proper subgroups (finite rank)-by- nilpotent.     

On <Emphasis Type="Italic">p</Emphasis>-nilpotence and solubility of groups

Recall a result due to O. J. Schmidt that a finite group whose proper subgroups are nilpotent is soluble. The present note extends this result and shows that if all non-normal maximal subgroups of a finite group are nilpotent, then (i) it is soluble; (ii) it is p-nilpotent for some prime p; (iii) if it is not nilpotent, then the number of prime divisors contained in its order is between 2 and k + 2, where k is the number of normal maximal subgroups which are not nilpotent.     

Roe–Strichartz theorem on two-step nilpotent Lie groupsa

Strichartz characterized eigenfunctions of the Laplacian on Euclidean spaces by boundedness conditions which generalized a result of Roe for the one-dimensional case. He also proved an analogous statement for the sub-Laplacian on the Heisenberg groups. In this paper, we extend this result to connected, simply connected two-step nilpotent Lie groups.     

Corrigendum to “The Dunkl intertwining operator” [J. Funct. Anal. 256 (8) (2009) 2697-2709]

In this note we correct the statement related to the regularity characterization of parameter functions and give some related new results.     

On the number of <Emphasis Type="Italic">B</Emphasis><Subscript>π′</Subscript>-characters in a nilpotent π-block

Broué and Puig set the definition of nilpotent p-blocks, stated the existence of such blocks, and then proved that there is a unique Brauer character in a nilpotent p-block. The present paper, based on the works of Slattery and Robinson, generalizes the above idea to the π-block theory of a π-separable group, defines the nilpotency of a π-block, and proves that there is a unique B _π′-character in a nilpotent π-block. This work was partially supported by the National Natural Science Foundation of China (Grant No. 10771132) and Beijing Educational Committee (Grant No. Km200510028002)     

Lie rings admitting an automorphism of order 4 with few fixed points. II

We consider a Lie ring (algebra) L that admits an automorphism φ of order 4 with a finite number m of fixed points (with a fixed-point subalgebra of finite dimension m). It is proved that L contains a subring S of m-bounded index in the additive group L (a subalgebra S of m-bounded codimension), which possesses a nilpotent ideal I of class bounded by some constant, such that the factor-ring S/I is nilpotent of class ≤2. As a consequence, it is proved that, under the same conditions, L has a subring G of m-bounded index in the additive group of L (a subalgebra G of m-bounded codimension), in which an ideal generated by the Lie subring [G, ?²]=«ng?g+g^{? 2} | g∈G»ng (the subalgebra [G, ?²]=«ng?g+g^{? 2} | g∈G»ng is an ideal in G which) is nilpotent of class bounded by some constant (and its factor-algebra G/[G, ?²] is nilpotent of class ≤2 with a derived algebra (square) of m-bounded dimension). In proofs, we use the results of [1] and develop further the version of the method of generalized centralizers employed therein.     

Conjugacy of Cartan subalgebras inn-Lie algebras

One of the most profound results in the theory of Lie algebras states that any two Cartan subalgebras of a finite-dimensional Lie algebra over an algebraically closed field of characteristic 0 are conjugate relative to the group of special automorphisms generated by the exponents of nilpotent inner derivations. Using some new ideas, we prove an analog of this statement for n-ary n-Lie algebras. Other interesting properties of Cartan algebras, which are known to be shared by Lie algebras, are carried over to n-Lie algebras.Translated fromAlgebra i Logika, Vol. 34, No. 4, pp. 405–419, July-August, 1995.     

Structure of the semigroup <Emphasis Type="Italic">OT</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

We study the structure of the semigroup OT _n , which is a unique (up to an isomorphism) R-section of the semigroup T _n . For this semigroup, we describe Green relations, determine regular and nilpotent elements, describe maximal nilpotent subsemigroups, and determine the unique irreducible system of generatrices and maximal subsemigroups.     

Lie symmetries and the center problem

In this short survey we study the narrow relation between the center problem and the Lie symmetries. It is well known that an analytic vector eld X having a non-degenerate center has a non-trivial analytic Lie symmetry in a neighborhood of it, i.e. there exists an analytic vector eld Y such that [X;Y] = \(\mu\)X. The same happens for a nilpotent center with an analytic rst integral as can be seen from the last results about nilpotent centers. From the last results for nilpotent and degenerate centers it also can be proved that any nilpotent or degenerate center has a trivial smooth (of class \(C^{\infty} \) ) ) Lie symmetry. Remains open if always exists also a non-trivial Lie symmetry for any nilpotent and degenerate center.     

An equational theory for a nilpotent <Emphasis Type="Italic">A</Emphasis>-loop

It is shown that a variety generated by a nilpotent A-loop has a decidable equational (quasiequational ) theory. Thereby the question posed by A. I. Mal’tsev in [6] is answered in the negative, and moreover, a finitely presented nilpotent A-loop has a decidable word problem.     

Endotrivial modules for nilpotent restricted Lie algebras

Let $$\mathfrak {g}$$ be a finite dimensional nilpotent p-restricted Lie algebra over a field k of characteristic p. For $$p\geqslant 5$$, we show that every endotrivial $$\mathfrak {g}$$-module is a direct sum of a syzygy of the trivial module and a projective module. The proof includes a theorem that the intersection of the maximal linear subspaces of the null cone of a nilpotent restricted p-Lie algebra for $$p \geqslant 5$$ has dimension at least two. We give an example to show that the statement about endotrivial modules is false in characteristic two. In characteristic three, another example shows that our proof fails, and we do not know a characterization of the endotrivial modules in this case.     

On Locally Nilpotent Subgroups of GL 1(D)

In this article we study locally nilpotent subgroups of D*: = GL ₁(D), where D is a division ring. It is proved that every locally nilpotent subnormal subgroup of D* is central. If D is algebraic over its centre then every locally solvable subnormal subgroup of D* is central. Also, in this case, it is shown that every locally nilpotent maximal subgroup of D* can occur as the multiplicative group of some maximal subfield of D.     

Topological multiple recurrence for polynomial configurations in nilpotent groups

We establish a general multiple recurrence theorem for an action of a nilpotent group by homeomorphisms of a compact space. This theorem can be viewed as a nilpotent version of our recent polynomial Hales-Jewett theorem (Ann. Math. 150 (1999) 33) and contains nilpotent extensions of many known “abelian” results as special cases.     

Formality in generalized Kähler geometry

We prove that no nilpotent Lie algebra admits an invariant generalized Kähler structure. This is done by showing that a certain differential graded algebra associated to a generalized complex manifold is formal in the generalized Kähler case, while it is never formal for a generalized complex structure on a nilpotent Lie algebra.     

Nilpotent groups admitting an almost regular automorphism of order four

We consider locally nilpotent periodic groups admitting an almost regular automorphism of order 4. The following are results are proved: (1) If a locally nilpotent periodic group G admits an automorphism ϕ of order 4 having exactly m<∞ fixed points, then (a) the subgroup {ie176-1} contains a subgroup of m-bounded index in {ie176-2} which is nilpotent of m-bounded class, and (b) the group G contains a subgroup V of m-bounded index such that the subgroup {ie176-3} is nilpotent of m-bounded class (Theorem 1); (2) If a locally nilpotent periodic group G admits an automorphism ϕ of order 4 having exactly m<∞ fixed points, then it contains a subgroup V of m-bounded index such that, for some m-bounded number f(m), the subgroup {ie176-4}, generated by all f(m) th powers of elements in {ie176-5} is nilpotent of class ≤3 (Theorem 2). Supported by RFFR grant No. 94-01-00048 and by ISF grant NQ7000. Translated fromAlgebra i Logika, Vol. 35, No. 3, pp. 314–333, May–June, 1996.     

ON 2-ABELIAN (n – 5)-FILIFORM LIE ALGEBRAS

We classify the (n ? 5)-filiform Lie algebras which have the additional property of a non-abelian derived subalgebra. Moreover we show that if a (n ? 5)-filiform Lie algebra is characteristically nilpotent, then it must be 2-abelian.