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Similarly, a set of all smooth maps from S 1 to a Lie group G forms an infinite-dimensional Lie group (Lie group in the sense we can define functional derivatives over it) called the loop group. The Lie algebra of a loop group is the corresponding loop algebra.
Removing a node from a connected diagram may yield a connected diagram (simple Lie algebra), if the node is a leaf, or a disconnected diagram (semisimple but not simple Lie algebra), with either two or three components (the latter for D n and E n). At the level of Lie algebras, these inclusions correspond to sub-Lie algebras.
In mathematics, especially abstract algebra, loop theory and quasigroup theory are active research areas with many open problems.As in other areas of mathematics, such problems are often made public at professional conferences and meetings.
A loop is a quasigroup with an identity element; that is, an element, e, such that x ∗ e = x and e ∗ x = x for all x in Q . It follows that the identity element, e , is unique, and that every element of Q has unique left and right inverses (which need not be the same).
The Weyl group of an affine Lie algebra can be written as a semi-direct product of the Weyl group of the zero-mode algebra (the Lie algebra used to define the loop algebra) and the coroot lattice. The Weyl character formula of the algebraic characters of the affine Lie algebras generalizes to the Weyl-Kac character formula. A number of ...
In its most general form a loop group is a group of continuous mappings from a manifold M to a topological group G.. More specifically, [1] let M = S 1, the circle in the complex plane, and let LG denote the space of continuous maps S 1 → G, i.e.
Starting with a polynomial loop algebra over finite-dimensional simple Lie algebra and performing two extensions, a central extension and an extension by a derivation, one obtains a Lie algebra which is isomorphic with an untwisted affine Kac–Moody algebra. Using the centrally extended loop algebra one may construct a current algebra in two ...
Theorems establishing the uniqueness of the loop representation as defined by Ashtekar et al. (i.e. a certain concrete realization of a Hilbert space and associated operators reproducing the correct loop algebra – the realization that everybody was using) have been given by two groups (Lewandowski, Okolow, Sahlmann and Thiemann) [5] and ...
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