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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.
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 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.
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 ...
Two loops a, b in a torus. In mathematics, a loop in a topological space X is a continuous function f from the unit interval I = [0,1] to X such that f(0) = f(1). In other words, it is a path whose initial point is equal to its terminal point. [1]
Moufang's theorem (loop theory) Mountain pass theorem (calculus of variations) Moving equilibrium theorem ; Multinomial theorem (algebra, combinatorics) Multiplication theorem (special functions) Multiplicity-one theorem (group representations) Mumford vanishing theorem (algebraic geometry) Mutual fund separation theorem (financial mathematics)
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