LIE RING Meaning and
Definition
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A lie ring, in abstract algebra, refers to an algebraic structure that exhibits properties similar to a ring but with slight modifications. It is a set equipped with two binary operations: addition and the Lie bracket operation, usually denoted as [a, b].
Formally, a lie ring is defined as an additive abelian group paired with a binary operation, the Lie bracket, which satisfies three fundamental axioms. The first axiom, called the alternating property, states that [a, a] = 0 for any element a in the lie ring. The second axiom, known as the Jacobi identity, expresses that [a, [b, c]] + [c, [a, b]] + [b, [c, a]] = 0 holds for any a, b, and c in the lie ring. Lastly, the Lie bracket operation is required to be bilinear, meaning it must distribute over addition and satisfy [a + b, c] = [a, c] + [b, c] and [a, b + c] = [a, b] + [a, c] for all elements a, b, and c.
Lie rings are closely related to Lie algebras, which are vector spaces equipped with a Lie bracket satisfying the same axioms as the lie ring. The structure of lie rings originates from the work of Norwegian mathematician Sophus Lie, known for his contributions to the theory of continuous transformation groups, and they find applications in various areas of mathematics and physics, particularly in the study of symmetries and transformation groups.
Common Misspellings for LIE RING
Etymology of LIE RING
The term "lie ring" actually does not have a standard etymology as a specific word. However, it can be broken down into its constituent parts to understand its approximate meaning.
The word "ring" generally refers to a mathematical structure known as a ring. In mathematics, a ring is an algebraic structure consisting of a set along with two binary operations: addition and multiplication. These operations follow certain rules like associativity, distributivity, and closure.
On the other hand, "lie" could be a reference to the mathematician Sophus Lie, who was influential in the field of Lie algebras. A Lie algebra is a vector space equipped with a binary operation called the Lie bracket, which satisfies certain properties. Lie algebras are often used to study symmetries and transformations in mathematics, physics, and other sciences.
