Structurable Tori

The classification of structurable tori with nontrivial involution, which was begun by Allison and Yoshii, is completed. New examples of structurable tori are obtained using a construction of structurable algebras from a semilinear version of cubic forms satisfying the adjoint identity. The classification uses techniques borrowed from quadratic forms over ? ₂ and from the geometry of generalized quadrangles. Since structurable tori are the coordinate algebras for the centerless cores of extended affine Lie algebras of type BC₁, the results of this article provide a classification and new examples for this class of Lie algebras.     

On the McCool group M3 and its associated Lie algebra

We prove that the Lie algebra of the McCool group M₃ is torsion free. As a result, we are able to give a presentation for the Lie algebra of M₃. Furthermore, M₃ is a Magnus group.     

Some Forms of Exceptional Lie Algebras

Some forms of Lie algebras of types E ₆, E ₇, and E ₈ are constructed using the exterior cube of a rank 9 finitely generated projective module.     

Nijenhuis forms on Lie-infinity algebras associated to Lie algebroids

Introducing Nijenhuis forms on L_∞-algebras gives a general frame to understand deformations of the latter. We give here a Nijenhuis interpretation of a deformation of an arbitrary Lie algebroid into an L_∞-algebra. Then we show that Nijenhuis forms on L_∞-algebras also give a short and e?cient manner to understand Poisson-Nijenhuis structures and, more generally, the so-called exact Poisson quasi-Nijenhuis structures with background.     

Modular <Emphasis Type="Italic">W</Emphasis> and <Emphasis Type="Italic">H</Emphasis> Type Lie Algebras

The modular Witt algebra W(p, n) and H(p, 2n) are defined on the polynomial rings Z_p[x₁,...,x_n] and Z_p[X₁,...,x_n, y₁,...,y_n] respectively. We generalize Z_p[x₁,...,xn] and Zp[x₁,...,xn, y₁,...,y_n], so we get the generalized W-type and H-type modular Lie algebras. We find all the derivations of W(p, 1).AMS Subject Classification: Primary 17B40, 17B56.     

Group Gradings on G 2

In this article, we describe all group gradings by a finite abelian group Γ of a simple Lie algebra of type G ₂ over an algebraically closed field F of characteristic zero.     

Modular Lie powers and the Solomon descent algebra

Let V be an r-dimensional vector space over an infinite field F of prime characteristic p, and let L_n(V) denote the nth homogeneous component of the free Lie algebra on V. We study the structure of L_n(V) as a module for the general linear group GL_r(F) when n=pk and k is not divisible by p and where r≥n. Our main result is an explicit 1-1 correspondence, multiplicity-preserving, between the indecomposable direct summands of L_k(V) and the indecomposable direct summands of L_n(V) which are not isomorphic to direct summands of V^⊗n. Our approach uses idempotents of the Solomon descent algebras, and in addition a correspondence theorem for permutation modules of symmetric groups. Second author supported by Deutsche Forschungsgemeinschaft (DFG-Scho 799).     

Graded automorphism group of TKK algebra

The classification of extended affine Lie algebras of type A_1 depends on the Tits-Kantor- Koecher (TKK) algebras constructed from semilattices of Euclidean spaces.One can define a unitary Jordan algebra J(S) from a semilattice S of R~v (v≥1),and then construct an extended affine Lie algebra of type A_1 from the TKK algebra T(J(S)) which is obtained from the Jordan algebra J(S) by the so-called Tits-Kantor-Koecher construction.In R~2 there are only two non-similar semilattices S and S′,where S is a lattice and S′is a non-lattice semilattice.In this paper we study the Z~2-graded automorphisms of the TKK algebra T(J(S)).     

Integrable cocycles and global deformations of Lie algebra of type G2 in characteristic 2

All classes of integrable cocycles in H²(L,L) are obtained for Lie algebra of type G₂ over an algebraically closed field of characteristic 2. It is proved that there exist only two orbits of classes of integrable cocycles with respect to automorphism group. The global deformation is shown to exist for any nontrivial class of integrable cocycles. These deformations are isomorphic to one of the two algebras of Cartan type, one of which being S(3:1,ω) while the other H(4:1,ω).     

On the Simplicity of Lie Algebras Associated to Leavitt Algebras

For any field 𝕂 and integer n ≥ 2, we consider the Leavitt algebra L _𝕂(n); for any integer d ≥ 1, we form the matrix ring S = M _d (L _𝕂(n)). S is an associative algebra, but we view S as a Lie algebra using the bracket [a, b] = ab ? ba for a, b ∈ S. We denote this Lie algebra as S ^?, and consider its Lie subalgebra [S ^?, S ^?]. In our main result, we show that [S ^?, S ^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1 and char(𝕂) does not divide d. In particular, when d = 1, we get that [L _𝕂(n)^?, L _𝕂(n)^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1.     

Automorphism group and representation of a twisted multi-loop algebra

Let G be the complexification of the real Lie algebra so（3） and A = C[t1^±1, t2^±1] be the Lau-ent polynomial algebra with commuting variables. Let L：（t1, t2, 1） = G c .A be the twisted multi-loop Lie algebra. Recently we have studied the universal central extension, derivations and its vertex operator representations. In the present paper we study the automorphism group and bosonic representations ofL（t1, t2, 1）.     

A Generalized Global Cartan Decomposition: A Basic Example

Suppose G ₁ ?  G are complex linear simple Lie groups. Let  ₁ ?  be the corresponding pair of Lie algebras. For the Killing-orthogonal  of  ₁ in  we have a vector space direct sum  =  ₁⊕ , which generalizes the classical Cartan decomposition on the Lie algebras level. In this article we study the corresponding problem of a ‘generalized global Cartan decomposition’ on the Lie groups level for the pair of groups ( G , G ₁) = (SL (4,?),Sp (2,?)); here  =  (4,?),  ₁ =  (2,?), and  = {X ?  | X ^? = X}, where X? X ^? is the symplectic involution. We prove that G  =  G ₁exp  ∪ i G ₁exp . The key point of the proof is to study in detail the set exp ; and for that purpose we introduce the J-twisted Pfaffian of size 2n defined on the set of all 2n × 2n matrices X satisfying X ^? = X, which is here a natural counterpart of the standard Pfaffian.     

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.     

Deformations of the infinite-dimensional Lie algebra L 3

For the nilpotent infinite-dimensional Lie algebra L ₃, we compute the second cohomology group H ²(L ₃, L ₃) with coefficients in the adjoint module. Nontrivial cocycles are found in closed form, and Massey powers are computed for them.     

Ideals and simplicity of unitary Lie algebras

Double graded ideals and simplicity of elementary unitary Lie algebra eu _n (R,⁻, γ) and Steinberg unitary Lie algebra stu _n (R,⁻, γ) are characterized, where R is a unital involutory associative algebra over a field F of characteristic zero, n ⩾ 5.     

Solvable Indecomposable Extensions of Two Nilpotent Lie Algebras

A pair of sequences of nilpotent Lie algebras denoted by N_{n, 7} and N_{n, 16} are introduced. Here, n denotes the dimension of the algebras that are defined for n ≥ 6; the first terms in the sequences are denoted by 6.7 and 6.16, respectively, in the standard list of six-dimensional Lie algebras. For each of them, all possible solvable extensions are constructed so that N_{n, 7} and N_{n, 16} serve as the nilradical of the corresponding solvable algebras. The construction continues Winternitz’ and colleagues’ program of investigating solvable Lie algebras using special properties rather than trying to extend one dimension at a time.