Integration of complex polynomial differential equations in higher dimension,a first step: Lie transverse structure

We prove that a one-dimensional holomorphic foliationswith generic singularities on a complex projective space ?P ^m+1, m ≥ 2, exhibiting a Lie group transverse structure in some Zariski open subset, is logarithmic. That is, it is given by a system of m closed rational one-forms with simple poles. The foliation is given by a linear vector field in some affine space ? ^m+1 ? ?P ^m+1 if, and only if, it exhibits only one singularity in this affine space. An application to foliations invariant under Lie group transverse actions is given.     

Quasi-state rigidity for finite-dimensional Lie algebras

We say that a Lie algebra g is quasi-state rigid if every Ad-invariant continuous Lie quasi-state on it is the directional derivative of a homogeneous quasimorphism. Extending work of Entov and Polterovich, we show that every reductive Lie algebra, as well as the algebras C ⁿ ? u(n), n ≥ 1, are rigid. On the other hand, a Lie algebra which surjects onto the three-dimensional Heisenberg algebra is not rigid. For Lie algebras of dimension ≤ 3 and for solvable Lie algebras which split over a codimension one abelian ideal, we show that this is the only obstruction to rigidity.     

Infinite-dimensional super Lie groups

A super Lie group is a group whose operations are G^∞ mappings in the sense of Rogers. Thus the underlying supermanifold possesses an atlas whose transition functions are G^∞ functions. Moreover the images of our charts are open subsets of a graded infinite-dimensional Banach space since our space of supernumbers is a Banach Grassmann algebra with a countably infinite set of generators.In this context, we prove that if h is a closed, split sub-super Lie algebra of the super Lie algebra of a super Lie group G, then h is the super Lie algebra of a sub-super Lie group of G. Additionally, we show that if g is a Banach super Lie algebra satisfying certain natural conditions, then there is a super Lie group G such that the super Lie algebra g is in fact the super Lie algebra of G. We also show that if H is a closed sub-super Lie group of a super Lie group G, then G→G/H is a principal fiber bundle.We emphasize that some of these theorems are known when one works in the super-analytic category and also when the space of supernumbers is finitely generated in which case, one can use finite-dimensional techniques. The issues dealt with here are that our supermanifolds are modeled on graded Banach spaces and that all mappings must be morphisms in the G^∞ category.     

Low dimensional Lie groups admitting left invariant flat projective or affine structures

We prove that any real Lie group of dimension ?5 admits a left invariant flat projective structure. We also prove that a real Lie group L of dimension ?5 admits a left invariant flat affine structure if and only if the Lie algebra of L is not perfect.     

A synopsis of Combinatorial Integral Geometry

A Lie coalgebra is a coalgebra whose comultiplication Δ : M → M ? M satisfies the Lie conditions. Just as any algebra A whose multiplication ? : A ? A → A is associative gives rise to an associated Lie algebra $\text{L}$(A), so any coalgebra C whose comultiplication Δ : C → C ? C is associative gives rise to an associated Lie coalgebra $\text{L}$^c(C). The assignment C ? $\text{L}$^c(C) is functorial. A universal coenveloping coalgebra Uc(M) is defined for any Lie Lie coalgebra M by asking for a right adjoint U^c to $\text{L}$^c. This is analogous to defining a universal enveloping algebra U(L) for any Lie algebra L by asking for a left adjoint U to the functor $\text{L}$. In the case of Lie algebras, the unit (i.e., front adjunction) 1 → $\text{L}$ o U of the adjoint functor pair U ? $\text{L}$ is always injective. This follows from the Poincaré-Birkhoff-Witt theorem, and is equivalent to it in characteristic zero (x = 0). It is, therefore, natural to inquire about the counit (i.e., back adjunction) $\text{L}$^c o U^c → 1 of the adjoint functor pair $\text{L}$^c ? U^c.Theorem. For any Lie coalgebra M, the natural map$\text{L}$^c(U^cM) → M is surjective if and only if M is locally finite, (i.e., each element of M lies in a finite dimensional sub Lie coalgebra of M).An example is given of a non locally finite Lie coalgebra. The existence of such an example is surprising since any coalgebra C whose diagonal Δ is associative is necessarily locally finite by a result of that theory. The present paper concludes with a development of an analog of the Poincaré-Birkhoff-Witt theorem for Lie algebras which we choose to call the Dual Poincaré-Birkhoff-Witt Theorem and abbreviate by “The Dual PBWθ.” The constraints of the present paper, however, allow only a sketch of this theorem. A complete proof will appear in a subsequent paper. The reader may, however, consult [12], in the meantime, for details. The Dual PBWθ shows for any locally finite Lie coalgebra M the existence (in χ = 0) of a natural isomorphism of the graded Hopf algebras ₀E(U^cM) and ₀E(S^cM) associated to U^cM and to S^cM = U^c(TrivM) when U^c(M) and S^c(M) are given the Lie filtrations. [Just as U^c(M) is the analog of the enveloping algebra U(L) of a Lie algebra L, so S^c(V) is the analog of the symmetric algebra S(V) on a vector space V. Triv(M) denotes the trivial Lie coalgebra structure on the underlying vector space of M obtained by taking the comultiplication to be the zero map.]     

Lie properties of symmetric elements in group rings

Let ¹ be an involution of a group G extended linearly to the group algebra KG. We prove that if G contains no 2-elements and K is a field of characteristic $p\ne 2$, then the 1-symmetric elements of KG are Lie nilpotent (Lie n-Engel) if and only if KG is Lie nilpotent (Lie n-Engel).     

On Bounded Positive (m,p)-Circle Domains

Let D be a bounded positive (m, p)-circle domain in ?². The authors prove that if dim(Iso(D)⁰) = 2, then D is holomorphically equivalent to a Reinhardt domain; if dim(Iso(D)⁰) = 4, then D is holomorphically equivalent to the unit ball in ?². Moreover, the authors prove the Thullen’s classification on bounded Reinhardt domains in ?² by the Lie group technique.     

Lie properties of symmetric elements in group rings II

Let F be a field of characteristic different from 2, and G a group with involution ∗. Write (FG)⁺ for the set of elements in the group ring FG that are symmetric with respect to the induced involution. Recently, Giambruno, Polcino Milies and Sehgal showed that if G has no 2-elements, and (FG)⁺ is Lie nilpotent (resp. Lie n-Engel), then FG is Lie nilpotent (resp. Lie m-Engel, for some m). Here, we classify the groups containing 2-elements such that (FG)⁺ is Lie nilpotent or Lie n-Engel.     

Lie bialgebra structures on derivation Lie algebra over quantum tori

We investigate Lie bialgebra structures on the derivation Lie algebra over the quantum torus. It is proved that, for the derivation Lie algebra W over a rank 2 quantum torus, all Lie bialgebra structures on W are the coboundary triangular Lie bialgebras. As a by-product, it is also proved that the first cohomology group H ¹(W, W ? W) is trivial.     

Nilpotent Lie algebras with 2-dimensional commutator ideals

We classify all (finitely dimensional) nilpotent Lie k-algebras h with 2-dimensional commutator ideals h^′, extending a known result to the case where h^′ is non-central and k is an arbitrary field. It turns out that, while the structure of h depends on the field k if h^′ is central, it is independent of k if h^′ is non-central and is uniquely determined by the dimension of h. In the case where k is algebraically or real closed, we also list all nilpotent Lie k-algebras h with 2-dimensional central commutator ideals h^′ and dim_kh?11.     

A note on strongly Lie nilpotency

In this note the authors studies strongly Lie nilpotent rings and proves that if a ringR is strongly Lie nilpotent thenR ⁽²⁾, the ideal generated by all commutators, is nilpotent.     

A class of Sasakian 5-manifolds

We obtain some general results on Sasakian Lie algebras and prove as a consequence that a (2n + 1)-dimensional nilpotent Lie group admitting left-invariant Sasakian structures is isomorphic to the real Heisenberg group H _{2n + 1}. Furthermore, we classify Sasakian Lie algebras of dimension five and determine which of them carries a Sasakian α-Einstein structure. We show that a five-dimensional solvable Lie group with a left-invariant Sasakian structure and which admits a compact quotient by a discrete subgroup is isomorphic to either H ₅ or a semidirect product ? ? (H ₃ × ?). In particular, the compact quotient is an S ¹-bundle over a four-dimensional Kähler solvmanifold.     

Lie algebras generated by bounded linear operators on Hilbert spaces

It is proved that the operator Lie algebra ε(T,T^∗) generated by a bounded linear operator T on Hilbert space H is finite-dimensional if and only if T=N+Q, N is a normal operator, [N,Q]=0, and dimA(Q,Q^∗)<+∞, where ε(T,T^∗) denotes the smallest Lie algebra containing T,T^∗, and A(Q,Q^∗) denotes the associative subalgebra of B(H) generated by Q,Q^∗. Moreover, we also give a sufficient and necessary condition for operators to generate finite-dimensional semi-simple Lie algebras. Finally, we prove that if ε(T,T^∗) is an ad-compact E-solvable Lie algebra, then T is a normal operator.     

TWISTED CONJUGACY CLASSES IN LATTICES IN SEMISIMPLE LIE GROUPS

Given a group automorphism ?: Γ?→?Γ, one has an action of Γ on itself by ?-twisted conjugacy, namely, g.x?=?gx?(g ^?1). The orbits of this action are called ?-conjugacy classes. One says that Γ has the R∞-property if there are infinitely many ?-conjugacy classes for every automorphism ? of Γ. In this paper we show that any irreducible lattice in a connected semisimple Lie group having finite centre and rank at least 2 has the R∞-property.     

Flat orbits and kernels of irreducible representations of the group algebra of a completely solvable Lie group

We show that the kernel of an irreducible unitary representation π of the group algebra L¹(G) of a completely solvable Lie group G is given by the functions, whose abelian Fourier transform vanish on the Kirillov orbit Oπ of π if and only if this orbit Oπ is flat. This is a generalization of a result obtained before for nilpotent Lie groups.     

On twin and anti-twin words in the support of the free Lie algebra

Let ${\mathcal{L}}_{K}(A)$ be the free Lie algebra on a finite alphabet A over a commutative ring K with unity. For a word u in the free monoid A ^? let $\tilde{u}$ denote its reversal. Two words in A ^? are called twin (resp. anti-twin) if they appear with equal (resp. opposite) coefficients in each Lie polynomial. Let l denote the left-normed Lie bracketing and ?? be its adjoint map with respect to the canonical scalar product on the free associative algebra K??A??. Studying the kernel of ?? and using several techniques from combinatorics on words and the shuffle algebra , we show that, when K is of characteristic zero, two words u and v of common length n that lie in the support of ${\mathcal{L}}_{K}(A)$ ??i.e., they are neither powers a ⁿ of letters a??A with exponent n>1 nor palindromes of even length??are twin (resp. anti-twin) if and only if u=v or $u = \tilde{v}$ and n is odd (resp. $u =\tilde{v}$ and n is even).     

Graded Lie algebras with few nontrivial components

We prove that if a (?/n?)-graded Lie algebra L = ? _i=0 ^n?1 L _i has d nontrivial components L _i and the null component L ₀ has finite dimension m, then L has a homogeneous solvable ideal of derived length bounded by a function of d and of codimension bounded by a function of m and d. An analogous result holds also for the (?/n?)-graded Lie rings L = ? _i=0 ^n?1 with few nontrivial components L _i if the null component L ₀ has finite order m. These results generalize Kreknin’s theorem on the solvability of the (?/n?)-graded Lie rings L = ? _i=0 ^n?1 L _i with trivial component L ₀ and Shalev’s theorem on the solvability of such Lie rings with few nontrivial components L _i . The proof is based on the method of generalized centralizers which was created by E. I. Khukhro for Lie rings and nilpotent groups with almost regular automorphisms of prime order [1], as well as on the technique developed in the work of N. Yu. Makarenko and E. I. Khukhro on the almost solvability of Lie algebras with an almost regular automorphism of finite order [2].