Second Homology Groups and Universal Coverings of Steinberg Leibniz Algebras of Small Characteristic

It is known that the second Leibniz homology group HL ₂ (𝔰𝔱𝔩 _n (R)) of the Steinberg Leibniz algebra 𝔰𝔱𝔩 _n (R) is trivial for n ≥ 5. In this article, we determine HL ₂(𝔰𝔱𝔩 _n (R)) explicitly (which are shown to be not necessarily trivial) for n = 3, 4 without any assumption on the base ring.     

On Universal Central Extensions of Leibniz Algebras

We construct the endofunctor 𝔲𝔠𝔢 between the category of Leibniz algebras which assigns to a perfect Leibniz algebra its universal central extension, and we obtain the isomorphism 𝔲𝔠𝔢_Lie(𝔮_Lie) ? (𝔲𝔠𝔢_Leib(𝔮))_Lie, where 𝔮 is a perfect Leibniz algebra satisfying the condition [x, [x, y]] + [[x, y], x] = 0, for all x, y ∈ 𝔮. Moreover, we obtain several results concerning the lifting of automorphisms and derivations in a covering. We also study the relationship between the universal central extension of a semidirect product of perfect Leibniz algebras and the semidirect product of the universal central extension of both of them.     

Lattices of Subalgebras of Leibniz Algebras

I describe the lattice ?(L) of subalgebras of a one-generator Leibniz algebra L. Using this, I show that, apart from one special case, a lattice isomorphism φ: ?(L) → ?(L′) between Leibniz algebras L, L′ maps the Leibniz kernel Leib(L) of L to Leib(L′).     

Leibniz and Hochschild homology

We construct and study the map from Leibniz homology HL_?(𝔥) of an abelian extension 𝔥 of a simple real Lie algebra 𝔤 to the Hochschild homology HH_??1(U(𝔥)) of the universal envelopping algebra U(𝔥). To calculate some homology groups, we use the Hochschild-Serre spectral sequences and Pirashvili spectral sequences. The result shows what part of the non-commutative Leibniz theory is detected by classical Hochschild homology, which is of interest today in string theory.     

Formal Expansion of Harish-Chandra Homomorphism

Abstract

Let 𝔤 be a complex semisimple Lie algebra with adjoint group G and let 𝔥 be a Cartan subalgebra of 𝔤. Let Â(𝔤) and Â(𝔥) denote the algebra of differential operators with formal power series coefficients on 𝔤 and 𝔥 respectively. We construct a subalgebra A _𝔤 of Â(𝔤) containing all the pull-backs of the differential operators in G attached to any element x in 𝔤. We also consider the projection P: A _𝔤 → Â _𝔥. Then, we calculate explicity the pull-back of the differential operator in G attached to an element h in 𝔥 modulo Ker P.     

Automorphisms of Free Left Nilpotent Leibniz Algebras and Fixed Points

ABSTRACT

Let F _m (N) be the free left nilpotent (of class two) Leibniz algebra of finite rank m, with m ≥ 2. We show that F _m (N) has non-tame automorphisms and, for m ≥ 3, the automorphism group of F _m (N) is generated by the tame automorphisms and one more non-tame IA-automorphism. Let F(N) be the free left nilpotent Leibniz algebra of rank greater than 1 and let G be an arbitrary non-trivial finite subgroup of the automorphism group of F(N). We prove that the fixed point subalgebra F(N) ^G is not finitely generated.     

Restrictions of Symmetric Invariants of a Complex Semisimple Lie Algebra and Some Applications

Let 𝔤 be a finite-dimensional complex semisimple Lie algebra and σ an arbitrary semisimple automorphism of 𝔤. Let 𝔱 be a Cartan subalgebra of 𝔨 = 𝔤^σ and 𝔥 =Z _𝔤(𝔱) be the centralizer of 𝔱 in 𝔤. Then 𝔥 is a σ-invariant Cartan subalgebra of 𝔤 and 𝔱 = 𝔥^σ. Let W(𝔤, 𝔥) be the Weyl group. One knows that Δ(𝔤, 𝔱), the set of roots of 𝔤 in 𝔱, is also a root system. It is proved that the corresponding Weyl group W(𝔤, 𝔱) is isomorphic to W(𝔤, 𝔥)^σ, which is the subgroup of W(𝔤, 𝔥) consisting of those elements commuting with σ. It is also shown that the image of the restriction map S(𝔥*) ^{W(𝔤, 𝔥)} → S(𝔱*) ^{W(𝔨, 𝔱)}, where S(𝔥*) and S(𝔱*) are the polynomial algebras on 𝔥 and 𝔱, respectively, is exactly S(𝔱*) ^{W(𝔤, 𝔱)}. Based on the above result, we also get a complete classification of the pairs (𝔤, σ) such that 𝔤^σ is noncohomologous to zero in 𝔤.     

Algèbres de Poisson et Algèbres de Lie Résolubles

Let 𝔤 be a solvable Lie algebra and Q an (ad 𝔤)-stable prime ideal of the symmetric algebra S(𝔤) of 𝔤. If E denotes the set of nonzero elements of S(𝔤)/Q which are eigenvectors for the adjoint action of 𝔤 on S(𝔤)/Q, then the localization (S(𝔤)/Q) _E has a natural structure of Poisson algebra. We study this algebra here.

Soient 𝔤 une algèbre de Lie résoluble et Q un idéal premier (ad 𝔤)-stable de l'algèbre symétrique S(𝔤) de 𝔤. Si E est l'ensemble des éléments non nuls de S(𝔤)/Q qui sont vecteurs propres pour l'action adjointe de 𝔤 dans S(𝔤)/Q, l'algèbre localisée (S(𝔤)/Q) _E a une structure naturelle d'algèbre de Poisson. On étudie ici cette algèbre.     

Representations of Nonstandard Poincaré Hopf Algebras

Let 𝔭 _q (1 + 1) be a nonstandard Poincaré Hopf algebra, we characterize all finite dimensional completely E-semisimple modules of 𝔭 _q (1 + 1). We also classify all finite dimensional E-semisimple modules of 𝔭 _q for a special quotient algebra of 𝔭 _q (1 + 1). Moreover, the decomposition of tensor product of two finite dimensional E-semisimple indecomposable modules is obtained.     

From Leibniz Homology to Cyclic Homology

For an algebra R over a commutative ring k, a natural homomorphism _*: HL_*+1(R) HH_* (R) from Leibniz to Hochschild homology is constructed that is induced by an antisymmetrization map on the chain level. The map _* is surjective when R = gl(A), A an algebra over a characteristic zero field. If f: A B is an algebra homomorophism, the relative groups HL_* (gl(f)) are studied, where gl(f): gl(A) gl(B) is the induced map on matrices. Letting HC_* denote cyclic homology, if f is surjective with nilpotent kernel, there is a natural surjection HL_*+1(gl(f)) HC_* (f) in the characteristic zero setting.     

Global Well-Posedness of mKdV in L 2 (𝕋, ℝ)

ABSTRACT

We show that the Miura map L ²(𝕋) → H ^?1(𝕋), r?r _x  + r ² is a global fold and then apply our results on global well-posedness of KdV in H ^?1(𝕋) to show that mKdV is globally well-posed in L ²(𝕋).     

An Example of Constructing Versal Deformation for Leibniz Algebras

In this work we compute a versal deformation of the 3-dimensional nilpotent Leibniz algebra over ?, defined by the nontrivial brackets [e ₁, e ₃] = e ₂ and [e ₃, e ₃] = e ₁.     

THE STRUCTURE OF THE TANGENT CONE: AN INTERESTING BIJECTION#

ABSTRACT

Let X = Spec(R) be a reduced equidimensional algebraic variety over an algebraically closed field k. Let Y = Spec(R/𝔮) be a codimension one ordinary multiple subvariety, where 𝔮 is a prime ideal of height 1 of R. If U is a nonempty open subset of Y and 𝔪 a closed point of U, we denote by A ? R _𝔪 its local ring in X, by 𝔭 the extension of 𝔮 in A, and by K the algebraic closure of the residue field k(𝔭).

Then there exists a bijection γ^𝔪:Proj(G _𝔭(A) ?  _A/𝔭 k) → Proj(G(A _𝔭) ?  _k(𝔭)K) such that for every subset Σ of Proj(G _𝔭(A) ?  _A/𝔭 k), the Hilbert function of Σ coincides with the Hilbert function of γ^𝔪(Σ). We examine some applications. We study the structure of the tangent cone at a closed point of a codimension one ordinary multiple subvariety.     

Some notes on the second homology of Leibniz algebras

Some properties of the second homology and cover of Leibniz algebras are established. By constructing a stem cover, the second Leibniz homology and cover of abelian, Heisenberg Lie algebras and cyclic Leibniz algebras are described. Also, for the dimension of a non-cyclic nilpotent Leibniz algebra L, we obtain dim(HL₂(L))≥2.     

On Exact Courant Algebras

In this note, we will show that exact Courant algebras over a Lie algebra 𝔤 can be characterized via Leibniz 2-cocycles, and the automorphism group of a given exact Courant algebra is in a one-to-one correspondence with first Leibniz cohomology space of 𝔤.     

TORSION THEORIES FOR FINITE VON NEUMANN ALGEBRAS

ABSTRACT

The study of modules over a finite von Neumann algebra 𝒜 can be advanced by the use of torsion theories. In this work, some torsion theories for 𝒜 are presented, compared, and studied. In particular, we prove that the torsion theory (T, P) (in which a module is torsion if it is zero-dimensional) is equal to both Lambek and Goldie torsion theories for 𝒜.

Using torsion theories, we describe the injective envelope of a finitely generated projective 𝒜-module and the inverse of the isomorphism K ₀(𝒜) → K ₀ (𝒰), where 𝒰 is the algebra of affiliated operators of 𝒜. Then the formula for computing the capacity of a finitely generated module is obtained. Lastly, we study the behavior of the torsion and torsion-free classes when passing from a subalgebra ? of a finite von Neumann algebra 𝒜 to 𝒜. With these results, we prove that the capacity is invariant under the induction of a ?-module.     

Generalized skew derivations on triangular algebras determined by action on zero products

For a triangular algebra 𝒜 and an automorphism σ of 𝒜, we describe linear maps F,G:𝒜→𝒜 satisfying F(x)y+σ(x)G(y) = 0 whenever x,y∈𝒜 are such that xy = 0. In particular, when 𝒜 is a zero product determined triangular algebra, maps F and G satisfying the above condition are generalized skew derivations of the form F(x) = F(1)x+D(x) and G(x) = σ(x)G(1)+D(x) for all x∈𝒜, where D:𝒜→𝒜 is a skew derivation. When 𝒜 is not zero product determined, we show that there are also nonstandard solutions for maps F and G.     

Weak Cayley table groups III: PSL(2,q)

A weak Cayley table isomorphism is a bijection φ:G→H of groups such that φ(xy)～φ(x)φ(y) for all x,y∈G. Here ～ denotes conjugacy. When G = H the set of all weak Cayley table isomorphisms φ:G→G forms a group 𝒲(G) that contains the automorphism group Aut(G) and the inverse map I:G→G,x?x^?1. Let 𝒲₀(G) = ?Aut(G),I?≤𝒲(G) and say that G has trivial weak Cayley table group if 𝒲(G) = 𝒲₀(G). We show that PSL(2,pⁿ) has trivial weak Cayley table group, where p≥5 is a prime and n≥1.