23 Facts About Lie algebra

1.

In mathematics, a Lie algebra is a vector space together with an operation called the Lie bracket, an alternating bilinear map, that satisfies the Jacobi identity.

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2.

Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to finite coverings .

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3.

The Lie algebra bracket is a measure of the non-commutativity between two rotations: since a rotation commutes with itself, we have the alternating property.

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4.

The name Lie algebra was given by Hermann Weyl in the 1930s; in older texts, the term infinitesimal group is used.

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5.

Lie algebra is a vector space over some field together with a binary operation called the Lie bracket satisfying the following axioms:.

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6.

Elements of a Lie algebra are said to generate it if the smallest subalgebra containing these elements is itself.

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7.

Lie algebra bracket is not required to be associative, meaning that need not equal.

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8.

Lie algebra homomorphism is a linear map compatible with the respective Lie brackets:.

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9.

Since the Lie algebra bracket is a kind of infinitesimal commutator of the corresponding Lie algebra group, we say that two elements commute if their bracket vanishes:.

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10.

Levi's theorem says that a finite-dimensional Lie algebra is a semidirect product of its radical and the complementary subalgebra .

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11.

Derivation on the Lie algebra is a linear map that obeys the Leibniz law, that is,.

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12.

Ado's theorem states that every finite-dimensional Lie algebra has a faithful representation on a finite-dimensional vector space.

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13.

Lie algebra is abelian if the Lie bracket vanishes, i e [x, y] = 0, for all x and y in.

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14.

Every finite-dimensional Lie algebra has a unique maximal solvable ideal, called its radical.

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15.

Lie algebra is "simple" if it has no non-trivial ideals and is not abelian.

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16.

In general, a Lie algebra is called reductive if the adjoint representation is semisimple.

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17.

Conversely, for any finite-dimensional Lie algebra, there exists a corresponding connected Lie group with Lie algebra.

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18.

For instance, the special orthogonal group SO and the special unitary group SU give rise to the same Lie algebra, which is isomorphic to with the cross-product, but SU is a simply-connected twofold cover of SO.

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19.

Therefore, knowing the representations of a Lie algebra settles the question of representations of the group.

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20.

Lie algebra can be equipped with some additional structures that are assumed to be compatible with the bracket.

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21.

For example, a graded Lie algebra is a Lie algebra with a graded vector space structure.

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22.

Conversely to any Lie algebra there is a corresponding ring, called the universal enveloping algebra.

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23.

Lie algebra rings are used in the study of finite p-groups through the Lazard correspondence.

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