Anyonic Lie algebra
Encyclopedia
In mathematics
, an anyonic Lie algebra
is a U(1) graded vector space
L over C equipped with a bilinear operator [.,.] and linear
map
s ε:L->C and Δ:L -> L⊗L satisfying
for pure graded elements X, Y, and Z.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an anyonic Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
is a U(1) graded vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
L over C equipped with a bilinear operator [.,.] and linear
Linear
In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...
map
Map (mathematics)
In most of mathematics and in some related technical fields, the term mapping, usually shortened to map, is either a synonym for function, or denotes a particular kind of function which is important in that branch, or denotes something conceptually similar to a function.In graph theory, a map is a...
s ε:L->C and Δ:L -> L⊗L satisfying
- ε([X,Y]) = ε(X)ε(Y)
for pure graded elements X, Y, and Z.