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alternate case: complex Lie algebra
Split Lie algebra
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of interest both because they formalize the split real form of a complex Lie algebra, and because split semisimple Lie algebras (more generally, splitCompact Lie algebra (1,192 words) [view diff] exact match in snippet view article find links to article
algebra can be seen as the smallest real form of a corresponding complex Lie algebra, namely the complexification. Formally, one may define a compactAlmost commutative ring (83 words) [view diff] exact match in snippet view article find links to article
differential operators. For example, the enveloping algebra of a complex Lie algebra is almost commutative by the PBW theorem. Similarly, a Weyl algebraSteinberg formula (225 words) [view diff] exact match in snippet view article find links to article
the multiplicity of an irreducible representation of a semisimple complex Lie algebra in a tensor product of two irreducible representations. It is a consequenceVogel plane (246 words) [view diff] exact match in snippet view article find links to article
projective plane (modulo permutations) corresponding to a simple complex Lie algebra is given by three eigenvalues α, β, γ of the Casimir operator actingComplex conjugate representation (256 words) [view diff] exact match in snippet view article find links to article
q). If g {\displaystyle {\mathfrak {g}}} is a *-Lie algebra (a complex Lie algebra with a * operation which is compatible with the Lie bracket), π(X)Chang number (292 words) [view diff] exact match in snippet view article find links to article
mathematics, the Chang number of an irreducible representation of a simple complex Lie algebra is its dimension modulo 1 + h, where h is the Coxeter number. ChangSimple Lie group (2,379 words) [view diff] exact match in snippet view article find links to article
L is a real simple Lie algebra, its complexification is a simple complex Lie algebra, unless L is already the complexification of a Lie algebra, in whichSatake diagram (912 words) [view diff] exact match in snippet view article find links to article
diagrams associated to a Dynkin diagram classify real forms of the complex Lie algebra corresponding to the Dynkin diagram. More generally, the Tits indexChevalley basis (556 words) [view diff] exact match in snippet view article find links to article
In mathematics, a Chevalley basis for a simple complex Lie algebra is a basis constructed by Claude Chevalley with the property that all structure constantsVirasoro algebra (4,140 words) [view diff] exact match in snippet view article find links to article
In mathematics, the Virasoro algebra is a complex Lie algebra and the unique nontrivial central extension of the Witt algebra. It is widely used in two-dimensionalList of humorous names in mathematics (458 words) [view diff] exact match in snippet view article find links to article
everywhere Monogamy Mother Functor Sausage catastrophe Sexy prime Simple complex Lie algebra Taking minors Ten martini problem Tits' Deformation Theorem RetardedIndefinite orthogonal group (1,672 words) [view diff] exact match in snippet view article find links to article
theory, for example. It is the split Lie group corresponding to the complex Lie algebra so2n (the Lie group of the split real form of the Lie algebra); moreCharge (physics) (1,269 words) [view diff] exact match in snippet view article
spinors is a (Lorentz) vector and a (Lorentz) scalar. Note that the complex Lie algebra sl(2,C) has a compact real form su(2) (in fact, all Lie algebrasYozo Matsushima (700 words) [view diff] exact match in snippet view article find links to article
counterexample. He then developed a proof that Cartan subalgebras of a complex Lie algebra are conjugate. However, Japanese researchers were out of touch withCartan's criterion (794 words) [view diff] exact match in snippet view article find links to article
typical counterexample is G = L[t]/tnL[t] where n>1, L is a simple complex Lie algebra with a bilinear form (,), and the bilinear form on G is given byTheorem of the highest weight (1,103 words) [view diff] exact match in snippet view article find links to article
{\displaystyle {\mathfrak {g}}} be a finite-dimensional semisimple complex Lie algebra with Cartan subalgebra h {\displaystyle {\mathfrak {h}}} . Let RLanglands–Shahidi method (1,570 words) [view diff] exact match in snippet view article find links to article
{\displaystyle r=\oplus r_{i}} is the adjoint action of M on the complex Lie algebra of a specific subgroup of the Langlands dual group of G. When G isG2 (mathematics) (2,056 words) [view diff] exact match in snippet view article
associated with this root system: The underlying real Lie algebra of the complex Lie algebra G2 has dimension 28. It has complex conjugation as an outer automorphismE8 (mathematics) (6,100 words) [view diff] exact match in snippet view article
is, Ek is infinite-dimensional for any k > 8. There is a unique complex Lie algebra of type E8, corresponding to a complex group of complex dimensionFunctor (3,550 words) [view diff] no match in snippet view article find links to article
Lie algebras Assigning to every real (complex) Lie group its real (complex) Lie algebra defines a functor. Tensor products If C denotes the category of vectorE7 (mathematics) (2,831 words) [view diff] exact match in snippet view article
dimension of its fundamental representation is 56. There is a unique complex Lie algebra of type E7, corresponding to a complex group of complex dimensionHermann Nicolai (1,434 words) [view diff] exact match in snippet view article find links to article
contains the largest finite-dimensional exceptional semi-simple complex Lie algebra E 8 {\displaystyle E_{8}} , which has been studied as a candidateE6 (mathematics) (3,820 words) [view diff] exact match in snippet view article
E6 plays a role in some grand unified theories. There is a unique complex Lie algebra of type E6, corresponding to a complex group of complex dimensionAffine Lie algebra (2,549 words) [view diff] exact match in snippet view article find links to article
structure of affine Lie algebras. Fix a finite-dimensional, simple, complex Lie algebra g {\displaystyle {\mathfrak {g}}} with Cartan subalgebra h {\displaystyleSpecial linear Lie algebra (1,947 words) [view diff] exact match in snippet view article find links to article
{\displaystyle {\mathfrak {sl}}_{2}\mathbb {C} } is a three-dimensional complex Lie algebra. Its defining feature is that it contains a basis e , h , f {\displaystyleKnizhnik–Zamolodchikov equations (3,059 words) [view diff] exact match in snippet view article find links to article
ranges over the elements of the underlying finite-dimensional simple complex Lie algebra g {\displaystyle {\mathfrak {g}}} . There is an energy 2 eigenvectorMutation (Jordan algebra) (15,817 words) [view diff] exact match in snippet view article
e2L(a), the corresponding complex Lie algebra contains the operators L(a). The commutators [L(a),L(b)] span the complex Lie algebra of derivations of A. TheLie algebra representation (4,312 words) [view diff] no match in snippet view article find links to article
says that every representation of a finite-dimensional (real or complex) Lie algebra lifts to a unique representation of the associated simply connectedSemisimple Lie algebra (5,743 words) [view diff] exact match in snippet view article find links to article
with a given complexification, which are known as real forms of the complex Lie algebra; this can be done by Satake diagrams, which are Dynkin diagrams withSymmetric space (4,599 words) [view diff] exact match in snippet view article find links to article
classification of commuting pairs of antilinear involutions of a complex Lie algebra. The composite σ∘τ determines a complex symmetric space, while τLorentz group (9,875 words) [view diff] exact match in snippet view article find links to article
2 , C ) {\displaystyle {\mathfrak {sl}}(2,\mathbf {C} )} . As a complex Lie algebra s l ( 2 , C ) {\displaystyle {\mathfrak {sl}}(2,\mathbf {C} )} isHermitian symmetric space (7,418 words) [view diff] exact match in snippet view article find links to article
eigenspaces for the diagonal matrices in SU(2). It is a three-graded complex Lie algebra, with the Weyl group element of SU(2) providing the involution. EachGlossary of representation theory (5,011 words) [view diff] exact match in snippet view article find links to article
h → C {\displaystyle \chi :{\mathfrak {h}}\to \mathbb {C} } of a complex Lie algebra h {\displaystyle {\mathfrak {h}}} , χ {\displaystyle \chi } is aRepresentation theory of the Lorentz group (19,763 words) [view diff] exact match in snippet view article find links to article
{\displaystyle {\mathfrak {g}}\oplus {\mathfrak {g}}} . When complexifying a complex Lie algebra, it should be thought of as a real Lie algebra of real dimension