The classification of structurable tori with nontrivial involution, which was begun by Allison and Yoshii, is completed. New examples of struct-urable 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 Z 2 and from the geometry of generaliz...

We develop the method of the hamiltonian reduction of affine Lie superal-gebras to obtain explicit and general expressions both for the classical and the quantum extended superconformal algebras. By performing the gauge transformation which connects the diagonal gauge with the Drinfeld-Sokolov gauge and considering the quantum corrections, we get generic expressions for the classical and quantu...

We classify the cohomology spaces H(g,K) for all filiform nilpotent Lie algebras of dimension n ≤ 11 over K and for certain classes of algebras of dimension n ≥ 12. The result is applied to the determination of affine cohomology classes [ω] ∈ H(g,K). We prove the general result that the existence of an affine cohomology class implies an affine structure of canonical type on g, hence a canonical...

Centreless Lie tori have been used by E. Neher to construct all extended affine Lie algebras (EALAs). In this article, we study isotopy for centreless Lie tori, and show that Neher’s construction provides a 1-1 correspondence between centreless Lie tori up to isotopy and families of EALAs up to isomorphism. Also, centreless Lie tori can be coordinatized by unital algebras that are in general no...

We obtain Drinfel ′ d's realization of quantum affine superalgebra U q (gl(1|1)) based on the super version of RS construction method and Gauss decomposition. How to construct quantum algebras (including quantum affine algebras and Yangians) is a important problem in mathematics physics. After Drinfel ′ d [1] and Jimbo [3, 4] independently discovered that the universal enveloping algebra U(g) o...

For any finite graph Γ and any field K of characteristic unequal to 2 we construct an algebraic variety X over K whose K-points parameterise K-Lie algebras generated by extremal elements, corresponding to the vertices of the graph, with prescribed commutation relations, corresponding to the non-edges. After that, we study the case where Γ is a connected, simply laced Dynkin diagram of finite or...