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 T_i for the Lie algebra, 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; 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 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]; 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

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; > 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):

* SU(1,1): Generators K_a, 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 P_a, a = 1, 2, and Λ, with [P_a, P_b] = k ε_ab Λ and [Λ, P_a] = ε_a^b P_b.
@ 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 U_q(\(\cal G\)), 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.

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