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Algebras extend a traditionally finite-dimensional universe

Many questions about the structure of Kac-Moody algebras, generalisations of Lie algebras to infinite dimensions remain unanswered. EU research helped relate their topological properties with analytical ones.
Algebras extend a traditionally finite-dimensional universe
Kac-Moody algebras were independently introduced in the 1960s by Victor Kac and Robert Moody to generalise the finite-dimensional Lie algebras. In recent years, the theory has undergone such tremendous developments in various directions that it has become a standard tool in mathematics.

The EU-funded project KMLIEGROUPS (Infinite-dimensional Lie theory and Kac-Moody groups) was centred on Kac-Moody groups. Since the structure theory of Kac-Moody algebras was developed in complete analogy with the classical theory, these groups are infinite-dimensional analogues of Lie groups.

The KMLIEGROUPS team looked at Kac-Moody groups from an analytic perspective. They carried out a detailed analysis of positive energy representations of Lie algebras and groups that generalise Kac-Moody algebras and groups to infinite rank.

Mathematicians isolated irreducible positive energy representations of the semi-direct product of any continuous action of additive groups of real numbers on Lie groups. These were considered as a subset of the equivalence classes of irreducible unitary representations of every Lie group.

The team carried out representations of Hilbert-Lie groups with the highest weight and affinisations of such groups. Such Lie algebras are natural generalisations of affine Kac-Moody groups to infinite rank. Next, the KMLIEGROUPS study focused on maximal Kac-Moody groups over finite fields.

The action of a rank two maximal Kac-Moody group on the boundary of its building was investigated. Mathematicians showed that it always acts transitively on the boundary while subgroups are primitive at. Importantly, they contributed to a better understanding of locally normal subgroups.

KMLIEGROUPS shed new light on the correspondence between a Kac-Moody group and its algebra with theorems on morphisms between algebras that can be transferred to corresponding groups. In the past, unlike the correspondence between a Lie group and its algebra, the understanding of the infinite dimensional theory was poorly understood.

Related information

Keywords

Kac-Moody algebras, Lie algebras, mathematics, KMLIEGROUPS, Lie groups
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