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In mathematics, this is insufficient, and the full affine Lie algebra is the vector space [2] ^ = ^ where is the derivation defined above. On this space, the Killing form can be extended to a non-degenerate form, and so allows a root system analysis of the affine Lie algebra.
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 quasigroup with an idempotent element is called a pique ("pointed idempotent quasigroup"); this is a weaker notion than a loop but common nonetheless because, for example, given an abelian group, (A, +), taking its subtraction operation as quasigroup multiplication yields a pique (A, −) with the group identity (zero) turned into a "pointed ...
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.
In mathematics, a Moufang loop is a special kind of algebraic structure. It is similar to a group in many ways but need not be associative. Moufang loops were introduced by Ruth Moufang . Smooth Moufang loops have an associated algebra, the Malcev algebra, similar in some ways to how a Lie group has an associated Lie algebra.
However, n does not equal the dimension of the defining module (a fundamental representation) of the Lie algebra – the index on the Dynkin diagram should not be confused with the index on the Lie algebra. For example, corresponds to + =, which naturally acts on 9-dimensional space, but has rank 4 as a Lie algebra.
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