Lie algebra extension

Extensions may arise naturally, for instance, when forming a Lie algebra from projective group representations.

[4][5] A large portion towards the end is devoted to background material for applications of Lie algebra extensions, both in mathematics and in physics, in areas where they are actually useful.

A systematic study of group extensions was performed by the Austrian mathematician Otto Schreier in 1923 in his PhD thesis and later published.

Let ν:g∗ →g be the vector space isomorphism associated to the nondegenerate Killing form K, and define a linear map d:g → g by

Since, for semisimple Lie algebras, all derivations are inner, one has d = adGd for some Gd ∈ g. Then Let f be the 1-cochain defined by Then showing that φ is a coboundary.

One such set of derivations is In order to manufacture a non-degenerate bilinear associative antisymmetric form L on g, attention is focused first on restrictions on the arguments, with m, n fixed.

Nonetheless, if, in a physical application, the eigenvalues of g0 or its representative are interpreted as (ordinary) quantum numbers, the additional superscript on the generators is referred to as the level.

To construct this algebra mathematically, let g be the centrally extended polynomial loop algebra of the previous section with as one of the commutation relations, or, with a switch of notation (l→m, m→n, i→a, j→b, λm⊗Ga→Tma) with a factor of i under the physics convention,[nb 3] Define using elements of g, One notes that so that it is defined on a circle.

Define a space of formal Laurent polynomials with coefficients in the finite-dimensional simple Lie algebra.

With the support of a symmetric non-degenerate alternating bilinear form and a derivation, a 2-cocycle is defined, subsequently used in the standard prescription for a central extension by a 2-cocycle.

and using antisymmetry of η one obtains In the extension, the commutation relations for the element d0 are It is desirable to get rid of the central charge on the right hand side.

With β ≠ 0 the following change of basis, the commutation relations take the form showing that the part linear in m is trivial.

These are calculated to satisfy They are interpreted as creation and annihilation operators acting on Hilbert space, increasing or decreasing the quantum of their respective modes.

In quantum mechanics, Wigner's theorem asserts that if G is a symmetry group, then it will be represented projectively on Hilbert space by unitary or antiunitary operators.

[21] In order to adequately discuss extensions, structure that goes beyond the defining properties of a Lie algebra is needed.

Thus one can produce an infinite list of possible gauge theories using the Cartan catalog of simple Lie algebras on their compact form (i.e., sl(n,

By defining gαβ = K[Gα,Gβ] and expanding the inner brackets in terms of structure constants, one finds that the Killing form satisfies the invariance condition of above.

Attention here is focused on polynomial loop algebras of the form To see this, consider elements H(λ) near the identity in G for H in the loop group, expressed in a basis {G_k} for g where the hk(λ) are real and small and the implicit sum is over the dimension K of g. Now write to obtain Thus the functions constitute the Lie algebra.

coefficients in g. In terms of a basis and structure constants, It is also common to have a different notation, where the omission of λ should be kept in mind to avoid confusion; the elements really are functions S1 → g. The Lie bracket is then which is recognizable as one of the commutation relations in an untwisted affine Kac–Moody algebra, to be introduced later, without the central term.

It generates (as seen by tracing backwards in the definitions) the set of constant maps from S1 into G, which is obviously isomorphic with G when exp is onto (which is the case when G is compact.

Current algebras arise in quantum field theories as a consequence of global gauge symmetry.

Let the Lagrangian density be This Lagrangian is invariant under the transformation[nb 10] where {F1, F1, ..., Fr} are generators of either U(N) or a closed subgroup thereof, satisfying Noether's theorem asserts the existence of r conserved currents, where πk0 ≡ πk is the momentum canonically conjugate to Φk.

Consider a conserved current with a generic Schwinger term By taking the vacuum expectation value (VEV), one finds where S10 and Heisenberg's equation of motion have been used as well as H|0⟩ = 0 and its conjugate.

(regarding the copies as distinct), setting as a vector space and assigning the commutation relations If C = D = 0, then the subalgebra spanned by the Gmi is obviously identical to the polynomial loop algebra of above.

A basis for su(1, 1) is given, see classical group, by Now compute The map preserves brackets and there are thus Lie algebra isomorphisms between the subalgebra of W spanned by {d0, d−1, d1} with real coefficients, sl(2,

The same holds for any subalgebra spanned by {d0, d−n, dn}, n ≠ 0, this follows from a simple rescaling of the elements (on either side of the isomorphisms).

The commutation relations in m for a basis, become in u so in order for u to be closed under the bracket (and hence have a chance of actually being a Lie algebra) a central charge I must be included.

These rather formidable equations simplify considerably with a clever choice of parametrization called the light cone gauge.

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.

Some are zero (hence missing in the equations above), and the "minus coefficients" satisfy The quantity on the left is given a name, the transverse Virasoro mode.