En este artículo se considera un marco general para la precuantización geométrica de una variedad provista de un corchete que no es necesariamente de Jacobi. La existencia de una foliación generalizada permite definir una noción de fibrado de precuantización. Se estudia una aproximación alternativa suponiendo la existencia de un algebroide de Lie sobre la variedad. Se relacionan ambos enfoques y se recuperan los resultados conocidos para variedades de Poisson y Jacobi.

We study the invariants of the universal enveloping algebra of a Lie superalgebra with respect to a certain “twisted” adjoint action.

Given a principal ideal domain $R$ of characteristic zero, containing $1/2$, and a connected differential non-negatively graded free finite type $R$-module $V$, we prove that the natural arrow $𝕃FH\left(V\right)\to FH𝕃\left(V\right)$ is an isomorphism of graded Lie algebras over $R$, and deduce thereby that the natural arrow $UFH𝕃\left(V\right)\to FHU𝕃\left(V\right)$ is an isomorphism of graded cocommutative Hopf algebras over $R$; as usual, $F$ stands for free part, $H$ for homology, $𝕃$ for free Lie algebra, and $U$ for universal enveloping algebra. Related facts and examples are also considered....

For a complex nilpotent finite-dimensional Lie algebra of matrices, and a Jordan-Hölder basis of it, we prove a spectral radius formula which extends a well-known result for commuting matrices.

Let $G$ be a semisimple complex algebraic group and $X$ its flag variety. Let $𝔤=\mathrm{Lie}\left(G\right)$ and let $U$ be its enveloping algebra. Let $𝔥$ be a Cartan subalgebra of $𝔤$. For $\mu \in {𝔥}^{*}$, let $J_{\mu }$ be the corresponding minimal primitive ideal, let $U_{\mu }=U/J_{\mu }$, and let ${𝒯}_{U_{\mu }}:K_0(U_{m}u)\to ℂ$ be the Hattori-Stallings trace. Results of Hodges suggest to study this map as a step towards a classification, up to isomorphism or Morita equivalence, of the $ℂ$-algebras $U_{\mu }$. When $\mu$ is regular, Hodges has shown that $K_0(U_{\mu })\cong K_0\left(X\right)$. In this case $K_0(U_{\mu })$ is generated by the classes corresponding to...

The Banach-Lie algebras ℌκ of all holomorphic infinitesimal isometries of the classical symmetric complex Banach manifolds of compact type (κ = 1) and non compact type (κ = −1) associated with a complex JB*-triple Z are considered and the Lie ideal structure of ℌκ is studied.

Let $𝒩$ denote the class of nilpotent Lie algebras. For any finite-dimensional Lie algebra $L$ over an arbitrary field $𝔽$, there exists a smallest ideal $I$ of $L$ such that $L/I\in 𝒩$. This uniquely determined ideal of $L$ is called the nilpotent residual of $L$ and is denoted by $L^{𝒩}$. In this paper, we define the subalgebra $S\left(L\right)={\bigcap }_{H\le L}{I}_L\left(H^{𝒩}\right)$. Set $S_0\left(L\right)=0$. Define $S_{i+1}\left(L\right)/S_i\left(L\right)=S(L/S_i\left(L\right))$ for $i\ge 1$. By $S_{\infty }\left(L\right)$ denote the terminal term of the ascending series. It is proved that $L=S_{\infty }\left(L\right)$ if and only if $L^{𝒩}$ is nilpotent. In addition, we investigate the basic properties of a Lie algebra...