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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.
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.
A Bol loop where the aforementioned two-sided inverse satisfies the automorphic inverse property, (ab) −1 = a −1 b −1 for all a,b in L, is known as a (left or right) Bruck loop or K-loop (named for the American mathematician Richard Bruck). The example in the following section is a Bruck loop.
Manin matrices give a new look on this classical subject. This example is related to Lie algebra gl n and serves as a prototype for more complicated applications to loop Lie algebra for gl n, Yangian and integrable systems. Take E ij be matrices with 1 at position (i,j) and zeros everywhere else. Form a matrix E with elements E ij at position ...
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