Lie Algebras |
In General
> s.a. examples / types and representations.
$ Def: A Lie algebra A
is a vector space on a field K, with a Lie bracket [ , ]: A
× A → A satisfying:
- Antisymmetry: [x, y]
= −[y, x], for all x, y ∈ A;
- Linearity: [λx, y]
= λ [x, y],
[x+y, z] = [x, z] + [y, z],
for all x, y, z ∈ A,
for all λ ∈ K;
- Jacobi identity: [x,
[y, z]] + [y, [z, x]]
+ [z, [x, y]] =
0, for all x, y, z ∈ A.
* Structure constants:
Given a basis Ti for the Lie algebra,
they are defined in that basis by [Ti,
Tj]
= Ckij
Tk.
@ General references: Jacobson 62;
Serre 64;
Kaplansky 71;
Humphreys 72;
Bourbaki 75;
Erdmann & Wildon 06 [II/III];
Hall 06;
Abbaspour & Moskowitz 07;
Goze a0805 [rev];
Steeb et al 12 [problems and solutions];
Belhaj a1205-ln;
Feger & Kephart a1206
[LieART, Mathematica application for Lie algebras and representation theory].
@ Space of Lie algebras: Ritter mp/03 [topology and invariants];
Goze m.RA/06-ln [classification, deformations and rigidity];
Shirokov AACA(10)-a0705 [classification using Clifford algebras].
@ Related topics:
Cariñena et al JPA(94) [and Poisson tensors];
Paal CzJP(03)mp-conf [and Moufang loops];
Sabinin 04 [mirror symmetry].
Invariants > s.a. Casimir
Operator; Weil Homomorphism.
* Result: All invariants of
a Lie algebra are obtained by expanding the m × m
determinant
det(t I + ai Ti) = ∑r=0m Pm−r(ai) tr
in powers of t, and substituting Ti
to get the polynomials Pm−r
(Ti).
@ References: Di Francesco & Zinn-Justin mp/05 [1-parameter family of vector-valued polynomials];
Boyko et al JPA(06)mp [moving frame method];
Hrivnák a1506-PhD [and Jordan algebras].
Subsets, Operations and Other Structure > s.a. Cartan
Subalgebra; Center.
* Metric: A metric can be defined on a Lie algebra by
gij = Ckli Clkj ;
If the Lie algebra is semisimple, this metric is non-degenerate.
* Deformations and contractions: They
are mutually opposite procedures, but whereas for every contraction there exists
a reverse deformation, the converse is not true in general; For examples, global
deformations of the Witt, Virasoro, and affine Kac-Moody algebras allow one to retrieve
Lie algebras of Krichever-Novikov type; > s.a. deformation quantization.
@ Extensions: Landi & Marmo in(89) [and gauge theories];
Forte & Sciarrino JMP(06)ht/05;
> s.a. Central Extension.
@ Contractions:
Fialowski & de Montigny JPA(05),
Sigma(06)m.RT [and contractions];
Nesterenko & Popovych JMP(06)mp;
Vitiello IJTP(08) [and quantum field theory];
Doikou & Sfetsos JPA(09)-a0904 [and central extensions].
@ Other operations:
de Azcárraga et al NPB(03),
Izaurieta et al JMP(06)ht [expansion];
Nurowski JGP(07) [deformations from 2-forms].
Of a Lie Group G
> s.a. conformal group [SO(3,2)]; lie group;
lorentz; poincaré;
SU(2).
$ Def: The vector space
of left-invariant vector fields on the Lie group G, together
with Lie bracket multiplication.
* Examples: (special group)
implies (traceless algebra):
GL(n, \(\mathbb R\)), real matrices; GL(n, \(\mathbb C\)), complex matrices; SL(n, \(\mathbb R\)), real traceless matrices; SL(n, \(\mathbb C\)), complex traceless matrices; |
O(n), real antisymmetric matrices; SO(n), real antisymmetric matrices; U(n), complex antihermitian matrices; SU(n), complex antihermitian matrices. |
* SU(1,1): Generators
Ka, a
= 1, 2, 3, with [K1,
K2]
= −i\(\hbar\) K3,
[K2, K3]
= i\(\hbar\) K1,
[K3, K1]
= i\(\hbar\) K2.
* SO(2,1), 1+1 de Sitter:
Generators Pa, a =
1, 2, and Λ, with [Pa,
Pb]
= k εab
Λ and [Λ, Pa]
= εab
Pb.
@ References: Su qp/06-conf [su(N), Cartan decomposition];
Isaev & Provorov a2012 [projectors onto invariant subspaces].
Infinite-Dimensional
> s.a. 2D manifolds; Surfaces [deformations].
* Examples: > see Kac-Moody,
Virasoro, and Witt Algebra.
* History: They first appeared
in 1909 in a paper by Cartan.
* Applications: The main ones in
physics are in gauge theories, where the locality of the gauge transformations
causes the infinite-dimensionality, and in the study of diffeomorphism groups
(even for very simple spaces like S1!).
@ References: Cornwell 89;
Kac 90;
Wakimoto 01.
Other Types and Related Concepts > s.a. Drinfel'd
Doubles; Dynkin Diagrams; Fusion
Rules [affine]; knot invariants [and quantum groups].
* Simple: For each
simple Lie algebra \(\cal G\), one can construct a Hopf algebra
Uq(\(\cal G\)),
and a polynomial link invariant.
* Semisimple: A Lie algebra is
semisimple if it is a direct sum of simple Lie algebras; > s.a. Wikipedia
page.
$ S theorem: Any invariant of
a compact semisimple Lie algebra is symmetric with respect to the reflections
which generate the discrete Weyl group of the algebra.
@ General references:
Gruber & O'Raifeartaigh JMP(64) [S theorem];
Cahn 84;
Yamatsu a1511 [finite-dimensional, and representations].
@ Bialgebras: Golubchik & Sokolov TMP(06) [and solutions of Yang-Baxter equation].
Generalizations > s.a. poisson
structure; Quantum Algebra.
* Soft Lie algebra:
One with structure functions rather than structure constants, e.g.,
the 7-sphere.
* Lie 2-algebra:
A "categorified" version of a Lie algebra, a category equipped
with structures analogous those of a Lie algebra, for which the usual laws
hold up to isomorphism.
@ Lie algebroids: Iglesias & Marrero;
Cattaneo LMP(04)m.SG/03 [integration];
de León et al JPA(05)m.DG/04 [Hamiltonian mechanics];
Landsman JGP(06)mp/05 [in physics, rev];
Kotov & Strobl a1004-en [and sigma models];
Ezuck a2105 [for classical and quantum systems];
> s.a. unified theories; yang-mills theories.
@ Superalgebras: Zhang & Gould JMP(05)m.QA/04 [representations of gl(2|2)];
Faux & Gates PRD(05) [Adinkras graphical technique];
Gotz et al JA(07)ht/05 [representations of sl(2|1)].
@ Lie n-algebras: Baez et al m.QA/05 [2-algebras, loop groups and String(n)];
Papadopoulos CQG(08) [structure constants];
Figueroa-O'Farrill JMP(08) [with Lorentzian inner product],
JMP(09)-a0904 [3-algebras, deformations];
Baez et al CMP(09) [2-algebras and classical strings];
Chen et al SChM(12)-a1203 [non-abelian extensions of Lie 2-algebras].
@ Other types:
Majid JGP(94) [braided, quantum];
Burde CEJM(06)mp/05 [left-symmetric or pre-Lie algebras];
Ovsienko a0705,
AIP(08)-a0810 [Lie antialgebras];
Goze et al JAPM-a0909,
Bai et al a1006 [n-Lie algebras];
Azcárraga & Izquierdo JPA(10)-a1005 [n-ary algebras, rev];
Dubois-Violette & Landi a1005 [Lie pre-algebras, and quantum groups];
Leidwanger & Morier-Genoud a1210-conf [Lie antialgebras].
In Physics > s.a. algebras [including
quantum algebra]; Feynman Diagram; lagrangian
dynamics [conserved currents].
@ Texts: Hermann 70;
Azcárraga & Izquierdo 95;
Fuchs & Schweigert 97;
Georgi 99;
Prakash 03 [including ∞-dimensional];
Neumaier & Westra a0810-book [classical and quantum mechanics, statistical mechanics].
@ Specific systems: McLachlan & Ryland JMP(03)mp/02 [classical mechanics];
Reiterer & Trubowitz a1412
[graded Lie algebra of general relativity].
@ Deformations: Chryssomalakos & Okon IJMPD(04)ht [quantum relativistic kinematic algebra];
> s.a. deformation quantization.
@ Lie-algebra cohomology and applications:
de Azcárraga et al RRAC(01)phy/98-proc.
main page
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