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Register a new userOn Weakly Complete Universal Enveloping Algebras of pro-Lie algebras
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We study universal enveloping Hopf algebras of Lie algebras in the category of weakly complete vector spaces over the real and complex field.
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A weakly complete vector space over $mathbb{K}=mathbb{R}$ or $mathbb{K}=mathbb{C}$ is isomorphic to $mathbb{K}^X$ for some set $X$ algebraically and topologically. The significance of this type of topological vector spaces is illustrated by the fact that the underlying vector space of the Lie algebra of any pro-Lie group is weakly complete. In this study, weakly complete real or complex associative algebras are studied because they are necessarily projective limits of finite dimensional algebras. The group of units $A^{-1}$ of a weakly complete algebra $A$ is a pro-Lie group with the associated topological Lie algebra $A_{rm Lie}$ of $A$ as Lie algebra and the globally defined exponential function $expcolon Ato A^{-1}$ as the exponential function of $A^{-1}$. With each topological group, a weakly complete group algebra $mathbb{K}[G]$ is associated functorially so that the functor $Gmapsto mathbb{K}[G]$ is left adjoint to $Amapsto A^{-1}$. The group algebra $mathbb{K}[G]$ is a weakly complete Hopf algebra. If $G$ is compact, the $mathbb{R}[G]$ contains $G$ as the set of grouplike elements. The category of all real Hopf algebras $A$ with a compact group of grouplike elements whose linear span is dense in $A$ is shown to be equivalent to the category of compact groups. The group algebra $A=mathbb{R}[G]$ of a compact group $G$ contains a copy of the Lie algebra $mathcal{L}(G)$ in $A_{rm Lie}$; it also contains a copy of the Radon measure algebra $M(G,mathbb{R})$. The dual of the group algebra $mathbb{R}[G]$ is the Hopf algebra ${mathcal R}(G,mathbb{R})$ of representative functions of $G$. The rather straightforward duality between vector spaces and weakly complete vector spaces thus becomes the basis of a duality ${mathcal R}(G,mathbb{R})leftrightarrow mathbb{R}[G]$ and thus yields a new aspect of Tannaka duality.
Universal enveloping algebras of braided m-Lie algebras and PBW theorem are obtained by means of combinatorics on words.
We prove how the universal enveloping algebra constructions for Lie-Rinehart algebras and anchored Lie algebras are naturally left adjoint functors. This provides a conceptual motivation for the universal properties these constructions satisfy. As a supplement, the categorical approach offers new insights into the definitions of Lie-Rinehart algebra morphisms, of modules over Lie-Rinehart algebras and of the infinitesimal gauge algebra of a module.
In this paper, we define and study the universal enveloping algebra of Poisson superalgebras. In particular, a new PBW theorem for Lie-Rinehart superalgebras is proved, leading to a PBW theorem for Poisson superalgebras; we show the universal enveloping algebra of a Poisson Hopf superalgebra (resp. Poisson-Ore extension) is a Hopf superalgebra (resp. iterated Ore extension); and we determine the universal enveloping algebra for examples such as quadratic polynomial Poisson superalgebras and Poisson symmetric superalgebras.
Let L be the space of spinors on the 3-sphere that are the restrictions of the Laurent polynomial type harmonic spinors on C^2. L becomes an associative algebra. For a simple Lie algebra g, the real Lie algebra Lg generated by the tensor product of L and g is called the g-current algebra. The real part K of L becomes a commutative subalgebra of L. For a Cartan subalgebra h of g, h tensored by K becomes a Cartan subalgebra Kh of Lg. The set of non-zero weights of the adjoint representation of Kh corresponds bijectively to the root space of g. Let g=h+e+ f be the standard triangular decomposition of g, and let Lh, Le and Lf respectively be the Lie subalgebras of Lg generated by the tensor products of L with h, e and f respectively . Then we have the triangular decomposition: Lg=Lh+Le+Lf, that is also associated with the weight space decomposition of Lg. With the aid of the basic vector fields on the 3-shpere that arise from the infinitesimal representation of SO(3) we introduce a triple of 2-cocycles {c_k; k=0,1,2} on Lg. Then we have the central extension: Lg+ sum Ca_k associated to the 2-cocycles {c_k; k=0,1,2}. Adjoining a derivation coming from the radial vector field on S^3 we obtain the second central extension g^=Lg+ sum Ca_k + Cn. The root space decomposition of g^ as welll as the Chevalley generators of g^ will be given.
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