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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.
In mathematics and abstract algebra, a Bol loop is an algebraic structure generalizing the notion of group. Bol loops are named for the Dutch mathematician Gerrit Bol who introduced them in . A loop, L, is said to be a left Bol loop if it satisfies the identity (()) = (()), for every a,b,c in L,
Hilbert's basis theorem (commutative algebra,invariant theory) Hilbert's Nullstellensatz (theorem of zeroes) (commutative algebra, algebraic geometry) Hilbert–Schmidt theorem (functional analysis) Hilbert–Speiser theorem (cyclotomic fields) Hilbert–Waring theorem (number theory) Hilbert's irreducibility theorem (number theory)
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