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Free Lie algebra information


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.

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Free Lie algebra

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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...

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Lie algebra

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mathematics, a Lie algebra (pronounced /liː/ LEE) is a vector space g {\displaystyle {\mathfrak {g}}} together with an operation called the Lie bracket, an...

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Universal enveloping algebra

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enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal...

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Graded Lie algebra

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Lie algebra is a Lie algebra endowed with a gradation which is compatible with the Lie bracket. In other words, a graded Lie algebra is a Lie algebra...

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Lie algebra representation

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of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms...

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Associative algebra

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and a semisimple algebra over an algebraically closed field. The universal enveloping algebra of a Lie algebra is an associative algebra that can be used...

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Affine Lie algebra

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affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. Given...

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Free algebra

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In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring since...

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Lie group

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matrix Lie algebra, there is a linear group (matrix Lie group) with this algebra as its Lie algebra. On the other hand, Lie groups with isomorphic Lie algebras...

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Shuffle algebra

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permutation. The shuffle algebra on a finite set is the graded dual of the universal enveloping algebra of the free Lie algebra on the set. Over the rational...

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Lie algebra extension

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Lie groups, Lie algebras and their representation theory, a Lie algebra extension e is an enlargement of a given Lie algebra g by another Lie algebra...

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Free object

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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...

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Lie superalgebra

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mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } ‑grading. Lie superalgebras...

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Necklace polynomial

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necklace). This counting function also describes the dimensions in a free Lie algebra and the number of irreducible polynomials over a finite field. The...

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Noncommutative symmetric function

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algebra of quasisymmetric function. Over the rational numbers it is isomorphic as a Hopf algebra to the universal enveloping algebra of the free Lie algebra...

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Cyclotomic identity

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the free associative algebra on α generators, and the right hand side is the generating function for the universal enveloping algebra of the free Lie algebra...

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Differential graded Lie algebra

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abstract algebra and topology, a differential graded Lie algebra (or dg Lie algebra, or dgla) is a graded vector space with added Lie algebra and chain...

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Vertex operator algebra

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notion of vertex algebra was introduced by Richard Borcherds in 1986, motivated by a construction of an infinite-dimensional Lie algebra due to Igor Frenkel...

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Anatoly Shirshov

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on free Lie algebras. He proved the Shirshov–Witt theorem, which states that any Lie subalgebra of a free Lie algebra is itself a free Lie algebra. Anatoly...

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Virasoro algebra

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Virasoro algebra (named after the physicist Miguel Ángel Virasoro) is a complex Lie algebra and the unique central extension of the Witt algebra. It is...

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List of things named after Sophus Lie

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infinite-dimensional Lie algebras Free Lie algebra Graded Lie algebra Differential graded Lie algebra Homotopy Lie algebra Malcev Lie algebra Modular Lie algebra Monster...

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Glossary of Lie groups and Lie algebras

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mathematical theories of Lie groups and Lie algebras. For the topics in the representation theory of Lie groups and Lie algebras, see Glossary of representation...

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Free group

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series of free groups and free Lie algebras. The group (Z,+) of integers is free of rank 1; a generating set is S = {1}. The integers are also a free abelian...

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Free probability

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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...

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Special unitary group

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This (real) Lie algebra has dimension n2 − 1. More information about the structure of this Lie algebra can be found below in § Lie algebra structure. In...

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