リー対応（英語版）のため，理論は，したがってリー環の拡大の歴史は，群の拡大の理論と歴史と密接に関係している．群の拡大の系統的な研究はオーストリアの数学者オットー・シュライアー（英語版） (Otto Schreier) によって1923年の彼の PhD 論文（後に出版）においてなされた[nb 1][6][7]．オットー・ヘルダー (Otto Hölder) によってシュライアーの論文のために出された問題は次のものであった：「2つの群 G と H が与えられたとき，群 E であって G と同型な正規部分群N を持ち剰余群E/N が H と同型であるものをすべて求めよ．」
A classical relativistic string traces out a world sheet in spacetime, just like a point particle traces out a world line. This world sheet can locally be parametrized using two parameters σ and τ. Points xμ in spacetime can, in the range of the parametrization, be written xμ = xμ(σ, τ). One uses a capital X to denote points in spacetime actually being on the world sheet of the string. Thus the string parametrization is given by (σ, τ) ↦(X0(σ, τ), X1(σ, τ), X2(σ, τ), X3(σ, τ)). The inverse of the parametrization provides a local coordinate system on the world sheet in the sense of manifolds.
The equations of motion of a classical relativistic string derived in the Lagrangian formalism from the Nambu–Goto action are[29]
A dot over a quantity denotes differentiation with respect to τ and a prime differentiation with respect to σ. A dot between quantities denotes the relativistic inner product.
These rather formidable equations simplify considerably with a clever choice of parametrization called the light cone gauge. In this gauge, the equations of motion become
the ordinary wave equation. The price to be paid is that the light cone gauge imposes constraints,
so that one cannot simply take arbitrary solutions of the wave equation to represent the strings. The strings considered here are open strings, i.e. they don't close up on themselves. This means that the Neumann boundary conditions have to be imposed on the endpoints. With this, the general solution of the wave equation (excluding constraints) is given by
where α' is the slope parameter of the string (related to the string tension). The quantities x0 and p0 are (roughly) string position from the initial condition and string momentum. If all the αμ n are zero, the solution represents the motion of a classical point particle.
This is rewritten, first defining
and then writing
In order to satisfy the constraints, one passes to light cone coordinates. For I = 2, 3, ...d, where d is the number of space dimensions, set
Not all αnμ, n ∈ ℤ, μ ∈ {+, −, 2, 3, ..., d} are independent. Some are zero (hence missing in the equations above), and the "minus coefficients" satisfy
The quantitity on the left is given a name,
the transverse Virasoro mode.
When the theory is quantized, the alphas, and hence the Ln become operators.
^オットー・シュライアー (1901– 1929) は群の拡大の理論の開拓者である．彼の豊富な研究論文とともにレクチャーノートは死後 Einführung in die analytische Geometrie und Algebra (Vol I 1931, Vol II 1935) の名で（Emanuel Sperner（英語版）により編集され）出版された．後に1951年に英語に Introduction to Modern Algebra and Matrix Theory において翻訳された．さらなる文献は MacTutor 2015 を参照．
^ヤコビ恒等式が成り立つことを示すには，one writes everything out, uses the fact that the underlying Lie algebras have a Lie product satisfying the Jacobi identity, and that δ[X, Y] = [δ(X), Y] + [X, δ(Y)].
^ abRoughly, the whole Lie algebra is multiplied by i, there is an i occurring in the definition of the structure constants and the exponent in the exponential map (Lie theory) acquires a factor of (minus) i. the main reason for this convention is that physicists like their Lie algebra elements to be Hermitian (as opposed to skew-Hermitian) in order for them to have real eigenvalues and hence be candidates for observables.
^Aut h) のリー環が Der h, h のすべての導分の集合（それ自身明らかなブラケットによりリー環である）であるという事実は Rossmann 2002, p. 51 において見つけられる．
^Since U = −i∑αaTa and U† are constant, they may be pulled out of partial derivatives. The U and U† then combine in U†U = I by unitarity.
^This follows from Gauss law is based on the assumption of a sufficiently rapid fall-off of the fields at infinity.
^There are alternative routes to quantization, e.g. one postulates the existence of creation and annihilation operators for all particle types with certain exchange symmetries based on which statistics, Bose–Einstein or Fermi–Dirac, the particles obey, in which case the above are derived for scalar bosonic fields using mostly Lorentz invariance and the demand for the unitarity of the S-matrix. In fact, all operators on Hilbert space can be built out of creation and annihilation operators. See e.g. Weinberg (2002), chapters 2–5.
^This step is ambiguous, since the classical fields commute whereas the operators don't. Here it is pretended that this problem doesn't exist. In reality, it is never serious as long as one is consistent.
Bäuerle, G.G.A; de Kerf, E.A. (1990). Finite and infinite dimensional Lie algebras and their application in physics. Studies in mathematical physics. 1. North-Holland. ISBN0-444-88776-8.
Greiner, W.; Reinhardt, J. (1996). Field Quantization. Springer Publishing. ISBN3-540-59179-6.
Humphreys, J. E. (1972). Introduction to Lie Algebras and Representation Theory (3rd ed.). Berlin·Heidelberg·New York: Springer-Verlag. ISBN3-540-90053-5.
Knapp, A. (2002). Lie groups beyond an introduction. Progress in mathematics. 140 (2nd ed.). Boston·Basel·Berlin: Birkhäuser. ISBN0-8176-4259-5.
Rossmann, Wulf (2002). Lie Groups - An Introduction Through Linear Groups. Oxford Graduate Texts in Mathematics. Oxford Science Publications. ISBN0 19 859683 9
Schottenloher, M. (2008). A Mathematical Introduction to Conformal Field Theory (2nd ed.). Berlin, Heidelberg: Springer-Verlag. ISBN978-3-540-68625-5.
Schreier, O. (1926). “Uber die Erweiterung von Gruppen I” (German). Monatshefte für Mathematik34 (1): 165–180. doi:10.1007/BF01694897.
Schreier, O. (1925). “Uber die Erweiterung von Gruppen II” (German). Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg4 (1): 321–346. doi:10.1007/BF02950735.
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Murray Gell-Mannは，1969年のノーベル物理学賞受賞者で，1960年代にカレント代数の分野を創始した．それは，Adler–Weisberger sum rule のような，予測を引き出す台となる力学の知識さえなしに，知られている局所的な対称性を開発する．
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