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Every Lie algebra over a field E gives rise to new Lie algebras over any subfield F ⊆ E by restricting the scalar multiplication. This paper studies the structure of these underlying Lie algebra in relation to the structure of the original Lie algebra, in particular the question how much of the original Lie algebra can be recovered from its underlying Lie algebra over subfields F. By introducing the conjugate of a Lie algebra we show that in some specific cases the Lie algebra is completely determined by its underlying Lie algebra. Furthermore we construct examples showing that these assumptions are necessary. As an application, we give for every positive n an example of a real 2-step nilpotent Lie algebra which has exactly n different bi-invariant complex structures. This answers an open question by Di Scala, Lauret and Vezzoni motivated by their work on quasi-Kähler Chern-flat manifolds in differential geometry. The methods we develop work for general Lie algebras and for general Galois extensions F ⊆ E , in contrast to the original question which only considered nilpotent Lie algebras of nilpotency class 2 and the field extension R ⊆ C. We demonstrate this increased generality by characterizing the complex Lie algebras of dimension ≤4 which are defined over R and over Q. [ABSTRACT FROM AUTHOR]

For any two complex numbers a and b , V i r ( a , b ) is a central extension of W ( a , b ) which is universal in the case ( a , b ) ≠ ( 0 , 1 ) , where W ( a , b ) is the Lie algebra with basis { L n , W n | n ∈ Z } and relations [ L m , L n ] = ( n − m ) L m + n , [ L m , W n ] = ( a + n + b m ) W m + n , [ W m , W n ] = 0 . In this paper, we construct and classify a class of non-weight modules over the algebra V i r ( a , b ) which are free U ( C L 0 ⊕ C W 0 ) -modules of rank 1. It is proved that such modules can only exist for a ∈ Z . [ABSTRACT FROM AUTHOR]

In this paper, we study the incidence algebra T of the attenuated space poset A q ( N , M ) . We consider the following topics. We consider some generators of T : the raising matrix R , the lowering matrix L , and a certain diagonal matrix K . We describe some relations among R , L , K . We put these relations in an attractive form using a certain matrix S in T . We characterize the center Z ( T ) . Using Z ( T ) , we relate T to the quantum group U τ ( sl 2 ) with τ 2 = q . We consider two elements A , A ⁎ in T of a certain form. We find necessary and sufficient conditions for A , A ⁎ to satisfy the tridiagonal relations. Let W denote an irreducible T -module. We find necessary and sufficient conditions for the above A , A ⁎ to act on W as a Leonard pair. [ABSTRACT FROM AUTHOR]

Motivated by results of Aujla and Silva [3], we give several more precise weak majorization and eigenvalue inequalities for some matrix versions of the famous Jensen inequality with regard to a convexity. Our main results are then applied to log convex functions. As an application, we obtain refinements of some well-known matrix inequalities. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, some properties of algebras satisfying local symmetric triality relations have been studied, including both generalized symmetric composition algebras and the conjugate algebras of any structurable algebras. We also discuss a general method of constructing Lie algebras from such a system, which can permit a construction of the magic square. [Copyright &y& Elsevier]

As is known, the notion of homotopy equivalence is a fundamental notion of mathematics and was introduced in order to formalize a relation that is weaker than isomorphism. In this note we define a homotopy equivalence of Rosenbrock systems and show that it coincides with the classical equivalences of Rosenbrock and Fuhrmann. Next, we show that the homotopy equivalence does preserve the important properties of a system (including the properties at infinity when these are properly understood). Finally, we define in a simple manner the states and motions of a system and claim that they are homotopy invariants. [Copyright &y& Elsevier]