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Group Theory in Statistical Signal Processing and Control
Published in Harish Parthasarathy, Advanced Probability and Statistics: Applications to Physics and Engineering, 2023
Let g be a semisimple Lie algebra with a Cartan subalgebra h and let ∆ denote the set of roots of (g,h). Let ∆+ denote the set of positive roots w.r.t a given Weyl chamber. Let gα denote the root space corresponding to the root α, ie, gα={X∈g:[H,X]=α(H)X∀H∈h} gα = {X ∈ g : [H,X] = α(H)X∀H ∈ h}
Nonassociative Algebras
Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006
Murray R. Bremner, Lucia I. Murakami, Ivan P. Shestakov
Let A be a finite dimensional Malcev algebra over a field F of characteristic 0 and let Solv A be the solvable radical of A. The algebra A is semisimple (it decomposes into a direct sum of simple algebras) if and only if Solv A = {0} (in fact, this is often used as the definition of “semisimple” for Malcev algebras, following the terminology for Lie algebras). If the quotient algebra A/Solv A is separable, then A contains a subalgebra B ≅ A/Solv A and A = B ⊕ Solv A (vector space direct sum).
A review on some classes of algebraic systems
Published in International Journal of Control, 2020
Víctor Ayala, Heriberto Román-Flores
On the other hand, the Lie algebra is semisimple. Thus does not contain ideals of co-dimension one (San Martin, 1999). So, for any positive time Therefore, we already proved the existence of time for Σ such that starting from e it is possible to reach any point of G at exact time Now, we consider the invariant transitive control system on G. Again, such that through , it is possible to reach e from any point of the Lie group G at the exact time . Take . Therefore, any two arbitrary points of G can be connected in exactly time and Σ is controllable at uniform time. In a more general set-up, we obtain