* Nilpotent cone, the set of elements that act nilpotently in all representations of a finite-dimensional semisimple Lie algebra

The geometric interpretation of curl as rotation corresponds to identifying bivectors ( 2-vectors ) in 3 dimensions with the special orthogonal Lie algebra so ( 3 ) of infinitesimal rotations ( in coordinates, skew-symmetric 3 × 3 matrices ), while representing rotations by vectors corresponds to identifying 1-vectors ( equivalently, 2-vectors ) and so ( 3 ), these all being 3-dimensional spaces.

By using the commutator as a Lie bracket, every associative algebra can be turned into a Lie algebra.

Use of the same expansion expresses the above Lie group commutator in terms of a series of nested Lie bracket ( algebra ) commutators,

Then is a derivation and is linear, i. e., and, and a Lie algebra homomorphism, i. e.,, but it is not always an algebra homomorphism, i. e. the identity does not hold in general.

Lie algebras: Assigning to every real ( complex ) Lie group its real ( complex ) Lie algebra defines a functor.

The Georgi – Glashow model was preceded by the Semisimple Lie algebra Pati – Salam model by Abdus Salam and Jogesh Pati, who pioneered the idea to unify gauge interactions.

If the 3 generations are then put in a 3x3 hermitian matrix with certain additions for the diagonal elements then these matrices form an exceptional ( grassman -) Jordan algebra, which has the symmetry group of one of the exceptional Lie groups ( F < sub > 4 </ sub >, E < sub > 6 </ sub >, E < sub > 7 </ sub > or E < sub > 8 </ sub >) depending on the details.

A GUT model basically consists of a gauge group which is a compact Lie group, a connection form for that Lie group, a Yang-Mills action for that connection given by an invariant symmetric bilinear form over its Lie algebra ( which is specified by a coupling constant for each factor ), a Higgs sector consisting of a number of scalar fields taking on values within real / complex representations of the Lie group and chiral Weyl fermions taking on values within a complex rep of the Lie group.

# REDIRECT Lie algebra

In mathematics, a Lie algebra (, not ) is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds.

The term " Lie algebra " ( after Sophus Lie ) was introduced by Hermann Weyl in the 1930s.

In some categories — notably groups, rings, and Lie algebras — it is possible to separate automorphisms into two types, called " inner " and " outer " automorphisms.

For Lie algebras the definition is slightly different.

The Langlands program is a far-reaching web of these ideas of ' unifying conjectures ' that link different subfields of mathematics, e. g. number theory and representation theory of Lie groups ; some of these conjectures have since been proved.

That is is the Lie derivative of f in the direction X, and one has df ( X )= X ( f ).

Their classification is divided into the small and large rank cases, where the rank is the largest rank of an odd abelian subgroup normalizing a nontrivial 2-subgroup, which is often ( but not always ) the same as the rank of a Cartan subalgebra when the group is a group of Lie type in characteristic 2.

If a group has an involution with a 2-component that is a group of Lie type of odd characteristic, the goal is to show that it has a centralizer of involution in " standard form " meaning that a centralizer of involution has a component that is of Lie type in odd characteristic and also has a centralizer of 2-rank 1.

The problem is to show that if a group has a centralizer of involution in " standard form " then it is a group of Lie type of odd characteristic.

* 1946 – Trygve Lie of Norway is picked to be the first United Nations Secretary General.

The smallest simple Lie group which contains the standard model, and upon which the first Grand Unified Theory was based, is

The next simple Lie group which contains the standard model is

This is a significant result, as other Lie groups lead to different normalizations.

It is known that every Lie algebra can be embedded into one that arises from an associative algebra in this fashion.

According to Cartan's theorem, a closed subgroup of G admits a unique smooth structure which makes it an embedded Lie subgroup of G -- i. e. a Lie subgroup such that the inclusion map is a smooth embedding.

A subspace that is closed under the Lie bracket is called a Lie subalgebra.

* The subspace of the general linear Lie algebra consisting of matrices of trace zero is a subalgebra, the special linear Lie algebra, denoted

Let g be a Lie algebra, h a maximal commutative Lie subalgebra consisting of semi-simple elements ( sometimes called Cartan subalgebra ) and let V be a finite dimensional representation of g. If g is semisimple, then g = g and so all weights on g are trivial.

Note that the even subalgebra of a Lie superalgebra forms a ( normal ) Lie algebra as all the signs disappear, and the superbracket becomes a normal Lie bracket.

* If G is a matrix Lie group ( i. e. a closed subgroup of GL ( n, C )), then its Lie algebra is an algebra of n × n matrices with the commutator for a Lie bracket ( i. e. a subalgebra of ).

Any smooth function on a symplectic manifold gives rise, by definition, to a Hamiltonian vector field and the set of all such form a subalgebra of the Lie Algebra of symplectic vector fields.

The Lie algebra of real polynomial vector fields on the circle is a dense subalgebra of the Lie algebra of diffeomorphisms of the circle.

If is a Lie group, and is a subgroup obtained by exponentiating a closed subalgebra of the Lie algebra of, then is foliated by cosets of.

* The Levi decomposition writes a finite dimensional Lie algebra as a semidirect product of a normal solvable subalgebra by a semisimple subalgebra.

Thus, one has the important result that the space of vector fields over M, equipped with the Lie bracket, forms a Lie algebra.

When applying to the state, it should be understood as the Lie derivative if the vector bundle is equipped with non-vanishing connection.

A < sup >*</ sup > Lie superalgebra is a complex Lie superalgebra equipped with an involutive antilinear map from itself to itself which respects the Z < sub > 2 </ sub > grading and satisfies

In mathematics, an anyonic Lie algebra is a U ( 1 ) graded vector space L over C equipped with a bilinear operator and linear maps ε: L -> C and Δ: L -> L ⊗ L satisfying

Cartan reformulated the differential geometry of ( pseudo ) Riemannian geometry, as well as the differential geometry of manifolds equipped with some non-metric structure, including Lie groups and homogeneous spaces.

A < sup >*</ sup > Lie superalgebra is a complex Lie superalgebra equipped with an involutive antilinear map < sup >*</ sup > such that * respects the grading and

It is equipped to handle certain non-commutative algebras which are extensively used in theoretical high energy physics: Clifford algebras, SU ( 3 ) Lie algebras, and Lorentz tensors.

If G is a Lie group and H is a Lie subgroup, then the quotient space G / H is a manifold ( subject to certain technical restrictions like H being a closed subset ) and is also a homogeneous space of G or in other words, a nonlinear realization of G. In many cases, G / H can be equipped with a Riemannian metric which is G-invariant.

More examples are provided by compact, semi-simple Lie groups equipped with a bi-invariant Riemannian metric.