Lie Algebras

In General > s.a. examples and representations.
$ Def: A Lie algebra A is a vector space on some field K, with a Lie bracket [ , ]: A × A → A satisfying:
- Antisymmetry: [x, y] = –[y, x], for all x, y A;
- Linearity: [x, y] = [x, y], [x+y, z] = [x, z] + [y, z], for all x, y, z A, for all K;
- Jacobi identity: [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0, for all x, y, z A.
* Structure constants: Given a basis T_i for the L.a., they are defined in that basis by: [T_i, T_j] = C^k_ij T_k.
@ General references: Jacobson 62; Serre 64; Kaplansky 71; Humphreys 72; Erdmann & Wildon 06 [II/III]; Hall 06; Goze a0805 [rev].
@ Space of Lie algebras: Ritter mp/03 [topology and invariants]; Goze m.RA/06 [classification, deformations and rigidity]; Shirokov a0705 [classification using Clifford algebras].
@ Related topics: Cariñena et al JPA(94) [and Poisson tensors]; Paal CzJP(03)mp-in [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 T_i) = _r=0^m P_m–r(aⁱ) t^r

in powers of t, and substituting T_i to get the polynomials P_m–r (T_i).
@ References: Di Francesco & Zinn-Justin mp/05 [1-parameter family of vector-valued polynomials]; Boyko et al JPA(06)mp [moving frame method].

Subsets, Operations and Other Structure > s.a. Cartan Subalgebra; Center.
* Metric: A metric can be defined on a Lie algebra by

g_ij = C^k_li C^l_kj ;

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.
@ Contractions: Fialowski & de Montigny JPA(05), Sigma(06)m.RT [and contractions]; Nesterenko & Popovych JMP(06)mp; Vitiello IJTP(08) [and quantum field theory].
@ Other operations: de Azcárraga et al NPB(03), Izaurieta et al JMP(06)ht [expansion]; Nurowski JGP(07) [deformations from 2-forms]; > s.a. Central Extension.

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):

* SU(1,1): Generators K_a, a = 1, 2, 3, with [K₁, K₂] = –i K₃, [K₂, K₃] = i K₁, [K₃, K₁] = i K₂.
* SO(2,1), 1+1 de Sitter: Generators P_a, a = 1, 2, and , with [P_a, P_b] = k _ab and [, P_a] = _a^b P_b.
@ References: Su qp/06-in [su(N), Cartan decomposition].

Infinite-Dimensional > s.a. 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 (for even very simple spaces like S¹!).
@ References: Cornwell 89; Kac 90; Wakimoto 01.

Other Types and Related Concepts > s.a. Drinfeld Doubles; Fusion Rules [affine]; knot invariants [and quantum groups].
* Simple: For each simple Lie algebra , one can construct a Hopf algebra U_q(), and a polynomial link invariant.
$ S theorem: Any invariant of a compact semisimple Lie algebra is symmetric wrt the reflections which generate the discrete Weyl group of the algebra.
@ General references: Gruber & O'Raifeartaigh JMP(64) [S theorem]; Cahn 84.
@ 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 algebroids: Iglesias & Marrero; Cattaneo LMP(04)m.SG/03 [integration]; de León et al m.DG/04 [Ham mechanics]; Landsman JGP(06)mp/05 [in physics, rev]; > s.a. 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 ht/05 [representations of sl(2|1)].
@ Other types: Majid JGP(94) [braided, quantum]; Baez et al m.QA/05 [Lie 2-algebras, loop groups and String(n)]; Burde mp/05 [left-symmetric or pre-Lie algebras]; Papadopoulos CQG(08) [structure constants of k-Lie algebras].

In Physics > s.a. algebras [including quantum algebra]; Feynman Diagram; lagrangian dynamics [conserved currents].
@ Texts: Hermann 70; Georgi 82; Azcárraga & Izquierdo 95; Fuchs & Schweigert 97; Prakash 03 [including -dimensional].
@ Specific systems: McLachlan & Ryland mp/02/JMP [classical mechanics].
@ 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-in.

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