In mathematics, a free Lie algebra over a field K is a Lie algebra generated by a set X, without any imposed relations other than the defining relations of alternating K-bilinearity and the Jacobi identity.
In mathematics, a freeLiealgebra over a field K is a Liealgebra generated by a set X, without any imposed relations other than the defining relations...
mathematics, a Liealgebra (pronounced /liː/ LEE) is a vector space g {\displaystyle {\mathfrak {g}}} together with an operation called the Lie bracket, an...
enveloping algebra of a Liealgebra is the unital associative algebra whose representations correspond precisely to the representations of that Liealgebra. Universal...
of representation theory, a Liealgebra representation or representation of a Liealgebra is a way of writing a Liealgebra as a set of matrices (or endomorphisms...
and a semisimple algebra over an algebraically closed field. The universal enveloping algebra of a Liealgebra is an associative algebra that can be used...
affine Liealgebra is an infinite-dimensional Liealgebra that is constructed in a canonical fashion out of a finite-dimensional simple Liealgebra. Given...
In mathematics, especially in the area of abstract algebra known as ring theory, a freealgebra is the noncommutative analogue of a polynomial ring since...
matrix Liealgebra, there is a linear group (matrix Lie group) with this algebra as its Liealgebra. On the other hand, Lie groups with isomorphic Lie algebras...
permutation. The shuffle algebra on a finite set is the graded dual of the universal enveloping algebra of the freeLiealgebra on the set. Over the rational...
free object is one of the basic concepts of abstract algebra. Informally, a free object over a set A can be thought of as being a "generic" algebraic...
mathematics, a Lie superalgebra is a generalisation of a Liealgebra to include a Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } ‑grading. Lie superalgebras...
necklace). This counting function also describes the dimensions in a freeLiealgebra and the number of irreducible polynomials over a finite field. The...
algebra of quasisymmetric function. Over the rational numbers it is isomorphic as a Hopf algebra to the universal enveloping algebra of the freeLie algebra...
the free associative algebra on α generators, and the right hand side is the generating function for the universal enveloping algebra of the freeLie algebra...
abstract algebra and topology, a differential graded Liealgebra (or dg Liealgebra, or dgla) is a graded vector space with added Liealgebra and chain...
notion of vertex algebra was introduced by Richard Borcherds in 1986, motivated by a construction of an infinite-dimensional Liealgebra due to Igor Frenkel...
on freeLiealgebras. He proved the Shirshov–Witt theorem, which states that any Lie subalgebra of a freeLiealgebra is itself a freeLiealgebra. Anatoly...
Virasoro algebra (named after the physicist Miguel Ángel Virasoro) is a complex Liealgebra and the unique central extension of the Witt algebra. It is...
mathematical theories of Lie groups and Liealgebras. For the topics in the representation theory of Lie groups and Liealgebras, see Glossary of representation...
series of free groups and freeLiealgebras. The group (Z,+) of integers is free of rank 1; a generating set is S = {1}. The integers are also a free abelian...
established. Free probability is currently undergoing active research. Typically the random variables lie in a unital algebra A such as a C*-algebra or a von...
This (real) Liealgebra has dimension n2 − 1. More information about the structure of this Liealgebra can be found below in § Liealgebra structure. In...
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