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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.
One can normalize a Cayley table of a quasigroup in the same manner as a reduced Latin square. Then the quasigroup associated to a reduced Latin square has a (two sided) identity element (namely, the first element among the row headers). A quasigroup with a two sided identity is called a loop. Some, but not all, loops are groups.
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 ...
Talk: List of problems in loop theory and quasigroup theory. Add languages. Page contents not supported in other languages. ... Download as PDF; Printable version
Moufang loops are universal among inverse property loops; that is, a loop Q is a Moufang loop if and only if every loop isotope of Q has the inverse property. It follows that every loop isotope of a Moufang loop is a Moufang loop. One can use inverses to rewrite the left and right Moufang identities in a more useful form:
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Using the language of Lie algebra cohomology, the central extension can be described using a 2-cocycle on the loop algebra. This is the map φ : L g × L g → C {\displaystyle \varphi :L{\mathfrak {g}}\times L{\mathfrak {g}}\rightarrow \mathbb {C} } satisfying φ ( X ⊗ t m , Y ⊗ t n ) = m B ( X , Y ) δ m + n , 0 . {\displaystyle \varphi ...
In mathematics, a quasisimple group (also known as a covering group) is a group that is a perfect central extension E of a simple group S.In other words, there is a short exact sequence