Let $𝔤$ be a complex, semisimple Lie algebra, with an involutive automorphism $\vartheta$ and set $𝔨=\mathrm{Ker}(\vartheta -I)$, $𝔭=\mathrm{Ker}(\vartheta +I)$. We consider the differential operators, $𝒟(𝔭{)}^K$, on $𝔭$ that are invariant under the action of the adjoint group $K$ of $𝔨$. Write $\tau :𝔨\to \mathrm{Der}𝒪\left(𝔭\right)$ for the differential of this action. Then we prove, for the class of symmetric pairs $(𝔤,𝔨)$ considered by Sekiguchi, that $\left\{d\in 𝒟\left(𝔭\right):d\left(𝒪(𝔭{)}^K\right)=0\right\}=𝒟\left(𝔭\right)\tau \left(𝔨\right)$. An immediate consequence of this equality is the following result of Sekiguchi: Let $({𝔤}_0,{𝔨}_0)$ be a real form of one of these symmetric pairs $(𝔤,𝔨)$, and suppose that $T$ is a $K_0$-invariant...

A control system is said to be finite if the Lie algebra generated by its vector fields is finite dimensional. Sufficient conditions for such a system on a compact manifold to be controllable are stated in terms of its Lie algebra. The proofs make use of the equivalence theorem of [Ph. Jouan, ESAIM: COCV 16 (2010) 956–973]. and of the existence of an invariant measure on certain compact homogeneous spaces.

A control system is said to be finite if the Lie algebra generated by its vector fields is finite dimensional. Sufficient conditions for such a system on a compact manifold to be controllable are stated in terms of its Lie algebra. The proofs make use of the equivalence theorem of [Ph. Jouan, ESAIM: COCV 16 (2010) 956–973]. and of the existence of an invariant measure on certain compact homogeneous spaces.

A control system is said to be finite if the Lie algebra generated by its vector fields is finite dimensional. Sufficient conditions for such a system on a compact manifold to be controllable are stated in terms of its Lie algebra. The proofs make use of the equivalence theorem of [Ph. Jouan, ESAIM: COCV 16 (2010) 956–973]. and of the existence of an invariant measure on certain compact homogeneous spaces.

[For the entire collection see Zbl 0742.00067.]The author formulates several theorems about invariant orders in Lie groups (without proofs). The main theorem: a simply connected Lie group $G$ admits a continuous invariant order if and only if its Lie algebra $L\left(G\right)$ contains a pointed invariant cone. V. M. Gichev has proved this theorem for solvable simply connected Lie groups (1989). If $G$ is solvable and simply connected then all pointed invariant cones $W$ in $L\left(G\right)$ are global in $G$ (a Lie wedge $W\subset L\left(G\right)$ is said to...

The vertex algebra ${𝒲}_{1+\infty ,c}$ with central charge $c$ may be defined as a module over the universal central extension of the Lie algebra of differential operators on the circle. For an integer $n\ge 1$, it was conjectured in the physics literature that ${𝒲}_{1+\infty ,-n}$ should have a minimal strong generating set consisting of $n^2+2n$ elements. Using a free field realization of ${𝒲}_{1+\infty ,-n}$ due to Kac–Radul, together with a deformed version of Weyl’s first and second fundamental theorems of invariant theory for the standard representation of ${\text{GL}}_n$,...

We prove a series of "going-up" theorems contrasting the structure of semiprime algebras and their subalgebras of invariants under the actions of Lie color algebras.

The half-liberated orthogonal group $O_n^{*}$ appears as intermediate quantum group between the orthogonal group $O_n$, and its free version $O_n^{+}$. We discuss here its basic algebraic properties, and we classify its irreducible representations. The classification of representations is done by using a certain twisting-type relation between $O_n^{*}$ and $U_n$, a non abelian discrete group playing the role of weight lattice, and a number of methods inspired from the theory of Lie algebras. We use these results for showing that...