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Algebras associated with posets

100%

Neggers J. , Kim H. S.

2001

tom Vol. 34, nr 1

13-23

In this paper we introduce a class of algebras whose bases over a field K are pogroupoids. We discuss several properties of these algebras as they relate to the structure of their associated pogroupoids and through these to the associated posets also. In particular the Jacobi form is O precisely when the pogroupoid is a semigroup, precisely when the posets is (C2 + 1)-free. Thus, it also follows that a pg-algebra KS over a field K is a Lie algebra with respect to the commutator product iff its associated posets S(<) is (C2 +1)-free. The ideals generated by commutators have some easily identifiable properties m terms of the incomparability graph of the posets associated with the pogroupoid base of the algebra. We conjecture that a fundamental theorem on the relationship between isomorphic algebras and isomorphic pogroupoids holds as well.

Extended lie algebraic stability analysis for switched systems with continuous-time and discrete-time subsystems

100%

Zhai G. , Xu X. , Lin H. , Liu D.

tom 17

nr 4

447-454

We analyze stability for switched systems which are composed of both continuous-time and discrete-time subsystems. By considering a Lie algebra generated by all subsystem matrices, we show that if all subsystems are Hurwitz/Schur stable and this Lie algebra is solvable, then there is a common quadratic Lyapunov function for all subsystems and thus the switched system is exponentially stable under arbitrary switching. When not all subsystems are stable and the same Lie algebra is solvable, we show that there is a common quadratic Lyapunov-like function for all subsystems and the switched system is exponentially stable under a dwell time scheme. Two numerical examples are provided to demonstrate the result.

A note on groups of symplectomorphisms

100%

Rybicki T.

tom [Z] 38

115-126

It is shown that the groups of Hamiltonian diffeomorphisms of a symplectic manifold determine uniquely the smooth and symplectic structures themselves. An analogous result is true for the Lie algebras of Hamiltoniam vector fields.

A note on semidirect sum of Lie algebras

72%

Ostrowski T.

nr 2

233-247

In the paper there are investigated some properties of Lie algebras, the construction which has a wide range of applications like computer sciences (especially to computer visions), geometry or physics, for example. We concentrate on the semidirect sum of algebras and there are extended some theoretic designs as conditions to be a center, a homomorphism or a derivative. The Killing form of the semidirect sum where the second component is an ideal of the first one is considered as well.