In the theory of
Lie groups
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additi ...
,
Lie algebras and their
representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
, a Lie algebra extension is an enlargement of a given Lie algebra by another Lie algebra . Extensions arise in several ways. There is the trivial extension obtained by taking a direct sum of two Lie algebras. Other types are the split extension and the central extension. Extensions may arise naturally, for instance, when forming a Lie algebra from
projective group representations. Such a Lie algebra will contain
central charge
In theoretical physics, a central charge is an operator ''Z'' that commutes with all the other symmetry operators. The adjective "central" refers to the center of the symmetry group—the subgroup of elements that commute with all other elemen ...
s.
Starting with a
polynomial loop algebra over finite-dimensional
simple
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnn ...
Lie algebra and performing two extensions, a central extension and an extension by a derivation, one obtains a Lie algebra which is isomorphic with an untwisted affine
Kac–Moody algebra
In mathematics, a Kac–Moody algebra (named for Victor Kac and Robert Moody, who independently and simultaneously discovered them in 1968) is a Lie algebra, usually infinite-dimensional, that can be defined by generators and relations through a g ...
. Using the centrally extended loop algebra one may construct a
current algebra
Certain commutation relations among the current density operators in quantum field theories define an infinite-dimensional Lie algebra called a current algebra. Mathematically these are Lie algebras consisting of smooth maps from a manifold into a ...
in two spacetime dimensions. The
Virasoro algebra
In mathematics, the Virasoro algebra (named after the physicist Miguel Ángel Virasoro) is a complex Lie algebra and the unique central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string the ...
is the universal central extension of the
Witt algebra.
Central extensions are needed in physics, because the symmetry group of a quantized system usually is a central extension of the classical symmetry group, and in the same way the corresponding symmetry Lie algebra of the quantum system is, in general, a central extension of the classical symmetry algebra. Kac–Moody algebras have been conjectured to be symmetry groups of a unified superstring theory. The centrally extended Lie algebras play a dominant role in
quantum field theory, particularly in
conformal field theory
A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometime ...
,
string theory and in
M-theory
M-theory is a theory in physics that unifies all consistent versions of superstring theory. Edward Witten first conjectured the existence of such a theory at a string theory conference at the University of Southern California in 1995. Witten's ...
.
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 parenthetical link, (
background material), is provided where it might be beneficial.
History
Due to the
Lie correspondence
A lie is an assertion that is believed to be false, typically used with the purpose of deceiving or misleading someone. The practice of communicating lies is called lying. A person who communicates a lie may be termed a liar. Lies can be int ...
, the theory, and consequently the history of Lie algebra extensions, is tightly linked to the theory and history of group extensions. A systematic study of group extensions was performed by the Austrian mathematician
Otto Schreier
Otto Schreier (3 March 1901 in Vienna, Austria – 2 June 1929 in Hamburg, Germany) was a Jewish-Austrian mathematician who made major contributions in combinatorial group theory and in the topology of Lie groups.
Life
His parents were the arch ...
in 1923 in his PhD thesis and later published.
[
]Otto Schreier
Otto Schreier (3 March 1901 in Vienna, Austria – 2 June 1929 in Hamburg, Germany) was a Jewish-Austrian mathematician who made major contributions in combinatorial group theory and in the topology of Lie groups.
Life
His parents were the arch ...
(1901 - 1929) was a pioneer in the theory of extension of groups. Along with his rich research papers, his lecture notes were posthumously published (edited by Emanuel Sperner
Emanuel Sperner (9 December 1905 – 31 January 1980) was a German mathematician, best known for two theorems. He was born in Waltdorf (near Neiße, Upper Silesia, now Nysa, Poland), and died in Sulzburg-Laufen, West Germany. He was a student at ...
) under the name ''Einführung in die analytische Geometrie und Algebra'' (Vol I 1931, Vol II 1935), later in 1951 translated to English i
Introduction to Modern Algebra and Matrix Theory
See for further reference. The problem posed for his thesis by
Otto Hölder
Ludwig Otto Hölder (December 22, 1859 – August 29, 1937) was a German mathematician born in Stuttgart.
Early life and education
Hölder was the youngest of three sons of professor Otto Hölder (1811–1890), and a grandson of professor Chris ...
was "given two groups and , find all groups having a normal subgroup isomorphic to such that the factor group is isomorphic to ".
Lie algebra extensions are most interesting and useful for infinite-dimensional Lie algebras. In 1967,
Victor Kac
Victor Gershevich (Grigorievich) Kac (russian: link=no, Виктор Гершевич (Григорьевич) Кац; born 19 December 1943) is a Soviet and American mathematician at MIT, known for his work in representation theory. He co-disco ...
and
Robert Moody independently generalized the notion of classical Lie algebras, resulting in a new theory of infinite-dimensional Lie algebras, now called
Kac–Moody algebra
In mathematics, a Kac–Moody algebra (named for Victor Kac and Robert Moody, who independently and simultaneously discovered them in 1968) is a Lie algebra, usually infinite-dimensional, that can be defined by generators and relations through a g ...
s. They generalize the finite-dimensional simple Lie algebras and can often concretely be constructed as extensions.
Notation and proofs
Notational abuse to be found below includes for the
exponential map given an argument, writing for the element in a direct product ( is the identity in ), and analogously for Lie algebra direct sums (where also and are used interchangeably). Likewise for semidirect products and semidirect sums. Canonical injections (both for groups and Lie algebras) are used for implicit identifications. Furthermore, if , , ..., are groups, then the default names for elements of , , ..., are , , ..., and their Lie algebras are , , ... . The default names for elements of , , ..., are , , ... (just like for the groups!), partly to save scarce alphabetical resources but mostly to have a uniform notation.
Lie algebras that are ingredients in an extension will, without comment, be taken to be over the same
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
.
The
summation convention
In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of i ...
applies, including sometimes when the indices involved are both upstairs or both downstairs.
Caveat: Not all proofs and proof outlines below have universal validity. The main reason is that the Lie algebras are often infinite-dimensional, and then there may or may not be a Lie group corresponding to the Lie algebra. Moreover, even if such a group exists, it may not have the "usual" properties, e.g. the
exponential map might not exist, and if it does, it might not have all the "usual" properties. In such cases, it is questionable whether the group should be endowed with the "Lie" qualifier. The literature is not uniform. For the explicit examples, the relevant structures are supposedly in place.
Definition
Lie algebra extensions are formalized in terms of short
exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context ...
s.
A short exact sequence is an exact sequence of length three,
such that is a
monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y.
In the more general setting of category theory, a monomorphism ...
, is an
epimorphism
In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms ,
: g_1 \circ f = g_2 \circ f ...
, and . From these properties of exact sequences, it follows that (the image of) is an
ideal
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Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
in . Moreover,
:
but it is not necessarily the case that is isomorphic to a subalgebra of . This construction mirrors the analogous constructions in the closely related concept of
group extension
In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence
:1\to N\;\ove ...
s.
If the situation in prevails, non-trivially and for Lie algebras over the same
field
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* Battlefield
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* Meadow, a grass ...
, then one says that is an extension of by .
Properties
The defining property may be reformulated. The Lie algebra is an extension of by if
is exact. Here the zeros on the ends represent the zero Lie algebra (containing the
null vector
In mathematics, given a vector space ''X'' with an associated quadratic form ''q'', written , a null vector or isotropic vector is a non-zero element ''x'' of ''X'' for which .
In the theory of real bilinear forms, definite quadratic forms an ...
only) and the maps are the obvious ones; maps to and maps all elements of to . With this definition, it follows automatically that is a monomorphism and is an epimorphism.
An extension of by is not necessarily unique. Let denote two extensions and let the primes below have the obvious interpretation. Then, if there exists a Lie algebra isomorphism such that
:
then the extensions and are said to be equivalent extensions. Equivalence of extensions is an
equivalence relation.
Extension types
Trivial
A Lie algebra extension
:
is trivial if there is a subspace such that and is an
ideal
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Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
in .
[
]
Split
A Lie algebra extension
:
is split if there is a subspace such that as a vector space and is a subalgebra in .
An ideal is a subalgebra, but a subalgebra is not necessarily an ideal. A trivial extension is thus a split extension.
Central
Central extensions of a Lie algebra by an abelian Lie algebra can be obtained with the help of a so-called (nontrivial) 2-cocycle (background
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Record ...
) on . Non-trivial 2-cocycles occur in the context of projective representation In the field of representation theory in mathematics, a projective representation of a group ''G'' on a vector space ''V'' over a field ''F'' is a group homomorphism from ''G'' to the projective linear group
\mathrm(V) = \mathrm(V) / F^*,
where G ...
s (background
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) of Lie groups. This is alluded to further down.
A Lie algebra extension
:
is a central extension if is contained in the center
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*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentrici ...
of .
Properties
*Since the center commutes with everything, in this case is abelian.
*Given a central extension of , one may construct a 2-cocycle on . Suppose is a central extension of by . Let be a linear map from to with the property that , i.e. is a section of . Use this section to define by
:
The map satisfies
:
To see this, use the definition of on the left hand side, then use the linearity of . Use Jacobi identity on to get rid of half of the six terms. Use the definition of again on terms sitting inside three Lie brackets, bilinearity of Lie brackets, and the Jacobi identity on , and then finally use on the three remaining terms that and that so that brackets to zero with everything.
It then follows that satisfies the corresponding relation, and if in addition is one-dimensional, then is a 2-cocycle on (via a trivial correspondence of with the underlying field).
A central extension
:
is universal if for every other central extension
:
there exist ''unique'' homomorphisms and such that the diagram
commutes, i.e. and . By universality, it is easy to conclude that such universal central extensions are unique up to isomorphism.
Construction
By direct sum
Let , be Lie algebras over the same field . Define
:
and define addition pointwise on . Scalar multiplication is defined by
:
With these definitions, is a vector space over . With the Lie bracket:
is a Lie algebra. Define further
:
It is clear that holds as an exact sequence. This extension of by is called a trivial extension. It is, of course, nothing else than the Lie algebra direct sum. By symmetry of definitions, is an extension of by as well, but . It is clear from that the subalgebra is an ideal (Lie algebra). This property of the direct sum of Lie algebras is promoted to the definition of a trivial extension.
By semidirect sum
Inspired by the construction of a semidirect product (background
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) of groups using a homomorphism , one can make the corresponding construct for Lie algebras.
If is a Lie algebra homomorphism, then define a Lie bracket on by
With this Lie bracket, the Lie algebra so obtained is denoted and is called the semidirect sum of and .
By inspection of one sees that is a subalgebra of and is an ideal in . Define by and by . It is clear that . Thus is a Lie algebra extension of by .
As with the trivial extension, this property generalizes to the definition of a split extension.
Example
Let be the Lorentz group
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
and let denote the translation group
In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted as the addition of a constant vector to every ...
in 4 dimensions, isomorphic to , and consider the multiplication rule of the Poincaré group
The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...
:
(where and are identified with their images in ). From it follows immediately that, in the Poincaré group, . Thus every Lorentz transformation corresponds to an automorphism of with inverse and is clearly a homomorphism. Now define
:
endowed with multiplication given by . Unwinding the definitions one finds that the multiplication is the same as the multiplication one started with and it follows that . From follows that and then from it follows that .
By derivation
Let be a derivation (background
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) of and denote by the one-dimensional Lie algebra spanned by . Define the Lie bracket on by[To show that the ]Jacobi identity
In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associ ...
holds, one writes everything out, uses the fact that the underlying Lie algebras have a Lie product satisfying the Jacobi identity, and that .
:
It is obvious from the definition of the bracket that is and ideal in in and that is a subalgebra of . Furthermore, is complementary to in . Let be given by and by . It is clear that . Thus is a split extension of by . Such an extension is called extension by a derivation.
If is defined by , then is a Lie algebra homomorphism into . Hence this construction is a special case of a semidirect sum, for when starting from and using the construction in the preceding section, the same Lie brackets result.
By 2-cocycle
If is a 2-cocycle (background
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) on a Lie algebra and is any one-dimensional vector space, let (vector space direct sum) and define a Lie bracket on by
:
Here is an arbitrary but fixed element of . Antisymmetry follows from antisymmetry of the Lie bracket on and antisymmetry of the 2-cocycle. The Jacobi identity follows from the corresponding properties of and of . Thus is a Lie algebra. Put and it follows that . Also, it follows with and that . Hence is a central extension of by . It is called extension by a 2-cocycle.
Theorems
Below follows some results regarding central extensions and 2-cocycles.
Theorem
Let and be cohomologous 2-cocycles on a Lie algebra and let and be respectively the central extensions constructed with these 2-cocycles. Then the central extensions and are equivalent extensions.
Proof
By definition, . Define
:
It follows from the definitions that is a Lie algebra isomorphism and holds.
Corollary
A cohomology class defines a central extension of which is unique up to isomorphism.
The trivial 2-cocycle gives the trivial extension, and since a 2-coboundary is cohomologous with the trivial 2-cocycle, one has
Corollary
A central extension defined by a coboundary is equivalent with a trivial central extension.
Theorem
A finite-dimensional simple Lie algebra has only trivial central extensions.
Proof
Since every central extension comes from a 2-cocycle , it suffices to show that every 2-cocycle is a coboundary. Suppose is a 2-cocycle on . The task is to use this 2-cocycle to manufacture a 1-cochain such that .
The first step is to, for each , use to define a linear map . But the linear maps are elements of . This suffices to express in terms of , using the isomorphism . Next, a linear map is defined that turns out to be a derivation. Since all derivations are inner, one has for some . An expression for in terms of and is obtained. Thus set, trusting that is a derivation,
:
Let be the 1-cochain defined by
:
Then
:
showing that is a coboundary. By the previous results, any central extension is trivial.
To verify that actually is a derivation, first note that it is linear since is, then compute
:
By appeal to the non-degeneracy of , the left arguments of are equal on the far left and far right.
The observation that one can define a derivation , given a symmetric non-degenerate associative form and a 2-cocycle , by
:
or using the symmetry of and the antisymmetry of ,
:
leads to a corollary.
Corollary
Let be a non-degenerate symmetric associative bilinear form and let be a derivation satisfying
:
then defined by
:
is a 2-cocycle.
Proof
The condition on ensures the antisymmetry of . The Jacobi identity for 2-cocycles follows starting with
:
using symmetry of the form, the antisymmetry of the bracket, and once again the definition of in terms of .
If is the Lie algebra of a Lie group and is a central extension of , one may ask whether there is a Lie group with Lie algebra . The answer is, by Lie's third theorem In the mathematics of Lie theory, Lie's third theorem states that every finite-dimensional Lie algebra \mathfrak over the real numbers is associated to a Lie group ''G''. The theorem is part of the Lie group–Lie algebra correspondence.
Histori ...
affirmative. But is there a ''central extension'' of with Lie algebra ? The answer to this question requires some machinery, and can be found in .
Applications
The "negative" result of the preceding theorem indicates that one must, at least for semisimple Lie algebras, go to infinite-dimensional Lie algebras to find useful applications of central extensions. There are indeed such. Here will be presented affine Kac–Moody algebras and Virasoro algebras. These are extensions of polynomial loop-algebras and the Witt algebra respectively.
Polynomial loop-algebra
Let be a polynomial loop algebra (background
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),
:
where is a complex finite-dimensional simple Lie algebra. The goal is to find a central extension of this algebra. Two of the theorems apply. On the one hand, if there is a 2-cocycle on , then a central extension may be defined. On the other hand, if this 2-cocycle is acting on the part (only), then the resulting extension is trivial. Moreover, derivations acting on (only) cannot be used for definition of a 2-cocycle either because these derivations are all inner and the same problem results. One therefore looks for derivations on . One such set of derivations is
:
In order to manufacture a non-degenerate bilinear associative antisymmetric form on , attention is focused first on restrictions on the arguments, with fixed. It is a theorem that ''every'' form satisfying the requirements is a multiple of the Killing form on . This requires
:
Symmetry of implies
:
and associativity yields
:
With one sees that . This last condition implies the former. Using this fact, define . The defining equation then becomes
:
For every the definition
:
does define a symmetric associative bilinear form
:
These span a vector space of forms which have the right properties.
Returning to the derivations at hand and the condition
:
one sees, using the definitions, that
:
or, with ,
:
This (and the antisymmetry condition) holds if , in particular it holds when .
Thus choose and . With these choices, the premises in the corollary are satisfied. The 2-cocycle defined by
:
is finally employed to define a central extension of ,
:
with Lie bracket
:
For basis elements, suitably normalized and with antisymmetric structure constants, one has
:
This is a universal central extension of the polynomial loop algebra.
;A note on terminology
In physics terminology, the algebra of above might pass for a Kac–Moody algebra, whilst it will probably not in mathematics terminology. An additional dimension, an extension by a derivation is required for this. Nonetheless, if, in a physical application, the eigenvalues of or its representative are interpreted as (ordinary) quantum numbers, the additional superscript on the generators is referred to as the level. It is an additional quantum number. An additional operator whose eigenvalues are precisely the levels is introduced further below.
Current algebra
As an application of a central extension of polynomial loop algebra, a current algebra
Certain commutation relations among the current density operators in quantum field theories define an infinite-dimensional Lie algebra called a current algebra. Mathematically these are Lie algebras consisting of smooth maps from a manifold into a ...
of a quantum field theory is considered (background
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). Suppose one has a current algebra, with the interesting commutator being
with a Schwinger term. To construct this algebra mathematically, let be the centrally extended polynomial loop algebra of the previous section with
:
as one of the commutation relations, or, with a switch of notation () with a factor of under the physics convention,[
:
Define using elements of ,
:
One notes that
:
so that it is defined on a circle. Now compute the commutator,
:
For simplicity, switch coordinates so that and use the commutation relations,
:
Now employ the ]Poisson summation formula
In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of ...
,
:
for in the interval and differentiate it to yield
:
and finally
:
or
:
since the delta functions arguments only ensure that the arguments of the left and right arguments of the commutator are equal (formally ).
By comparison with , this is a current algebra in two spacetime dimensions, ''including a Schwinger term'', with the space dimension curled up into a circle. In the classical setting of quantum field theory, this is perhaps of little use, but with the advent of string theory where fields live on world sheets of strings, and spatial dimensions are curled up, there may be relevant applications.
Kac–Moody algebra
The derivation used in the construction of the 2-cocycle in the previous section can be extended to a derivation on the centrally extended polynomial loop algebra, here denoted by in order to realize a Kac–Moody algebra (background
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). Simply set
:
Next, define as a vector space
:
The Lie bracket on is, according to the standard construction with a derivation, given on a basis by
:
For convenience, define
:
In addition, assume the basis on the underlying finite-dimensional simple Lie algebra has been chosen so that the structure coefficients are antisymmetric in all indices and that the basis is appropriately normalized. Then one immediately through the definitions verifies the following commutation relations.
:
These are precisely the short-hand description of an untwisted affine Kac–Moody algebra. To recapitulate, begin with a finite-dimensional simple Lie algebra. 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. Extend the derivation to this new space, use the standard prescription for a split extension by a derivation and an untwisted affine Kac–Moody algebra obtains.
Virasoro algebra
The purpose is to construct the Virasoro algebra
In mathematics, the Virasoro algebra (named after the physicist Miguel Ángel Virasoro) is a complex Lie algebra and the unique central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string the ...
(named after Miguel Angel Virasoro)[ Miguel Angel Virasoro, born 1940 is an Argentine physicist. The Virasoro algebra, named after him, was first published in ] as a central extension by a 2-cocycle of the Witt algebra (background
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). The Jacobi identity for 2-cocycles yields
Letting and using antisymmetry of one obtains
:
In the extension, the commutation relations for the element are
:
It is desirable to get rid of the central charge
In theoretical physics, a central charge is an operator ''Z'' that commutes with all the other symmetry operators. The adjective "central" refers to the center of the symmetry group—the subgroup of elements that commute with all other elemen ...
on the right hand side. To do this define
:
Then, using as a 1-cochain,
:
so with this 2-cocycle, equivalent to the previous one, one has[The same effect can be obtained by a change of basis in .]
:
With this new 2-cocycle (skip the prime) the condition becomes
:
and thus
:
where the last condition is due to the antisymmetry of the Lie bracket. With this, and with (cutting out a "plane" in ), yields
:
that with (cutting out a "line" in ) becomes
:
This is a difference equation generally solved by
:
The commutator in the extension on elements of is then
:
With it is possible to change basis (or modify the 2-cocycle by a 2-coboundary) so that
:
with the central charge absent altogether, and the extension is hence trivial. (This was not (generally) the case with the previous modification, where only obtained the original relations.) With the following change of basis,
:
the commutation relations take the form
:
showing that the part linear in is trivial. It also shows that is one-dimensional (corresponding to the choice of ). The conventional choice is to take and still retaining freedom by absorbing an arbitrary factor in the arbitrary object . The Virasoro algebra is then
:
with commutation relations
Bosonic open strings
The relativistic classical open string (background
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) is subject to quantization. This roughly amounts to taking the position and the momentum of the string and promoting them to operators on the space of states of open strings. Since strings are extended objects, this results in a continuum of operators depending on the parameter . The following commutation relations are postulated in the Heisenberg picture
In physics, the Heisenberg picture (also called the Heisenberg representation) is a formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which the operators (observables and others) incorporate a dependency on time, but ...
.
:
All other commutators vanish.
Because of the continuum of operators, and because of the delta functions, it is desirable to express these relations instead in terms of the quantized versions of the Virasoro modes, the Virasoro operators. These are calculated to satisfy
:
They are interpreted as creation and annihilation operators
Creation operators and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator (usually d ...
acting on Hilbert space, increasing or decreasing the quantum of their respective modes. If the index is negative, the operator is a creation operator, otherwise it is an annihilation operator. (If it is zero, it is proportional to the total momentum operator.) In view of the fact that the light cone plus and minus modes were expressed in terms of the transverse Virasoro modes, one must consider the commutation relations between the Virasoro operators. These were classically defined (then modes) as
:
Since, in the quantized theory, the alphas are operators, the ordering of the factors matter. In view of the commutation relation between the mode operators, it will only matter for the operator (for which ). is chosen normal order
In quantum field theory a product of quantum fields, or equivalently their creation and annihilation operators, is usually said to be normal ordered (also called Wick order) when all creation operators are to the left of all annihilation operator ...
ed,
:
where is a possible ordering constant. One obtains after a somewhat lengthy calculation the relations
:
If one would allow for above, then one has precisely the commutation relations of the Witt algebra. Instead one has
:
upon identification of the generic central term as times the identity operator, this is the Virasoro algebra, the universal central extension of the Witt algebra.
The operator enters the theory as the Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
, modulo an additive constant. Moreover, the Virasoro operators enter into the definition of the Lorentz generators of the theory. It is perhaps the most important algebra in string theory. The consistency of the Lorentz generators, by the way, fixes the spacetime dimensionality to 26. While this theory presented here (for relative simplicity of exposition) is unphysical, or at the very least incomplete (it has, for instance, no fermions) the Virasoro algebra arises in the same way in the more viable superstring theory
Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modeling them as vibrations of tiny supersymmetric strings.
'Superstring theory' is a shorthand for supersymmetric string t ...
and M-theory
M-theory is a theory in physics that unifies all consistent versions of superstring theory. Edward Witten first conjectured the existence of such a theory at a string theory conference at the University of Southern California in 1995. Witten's ...
.
Group extension
A projective representation of a Lie group (background
Background may refer to:
Performing arts and stagecraft
* Background actor
* Background artist
* Background light
* Background music
* Background story
* Background vocals
* ''Background'' (play), a 1950 play by Warren Chetham-Strode
Record ...
) can be used to define a so-called group extension
In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence
:1\to N\;\ove ...
.
In quantum mechanics, Wigner's theorem
Wigner's theorem, proved by Eugene Wigner in 1931, is a cornerstone of the mathematical formulation of quantum mechanics. The theorem specifies how physical symmetries such as rotations, translations, and CPT are represented on the Hilbert sp ...
asserts that if is a symmetry group, then it will be represented projectively on Hilbert space by unitary or antiunitary operators. This is often dealt with by passing to the universal covering group of and take it as the symmetry group. This works nicely for the rotation group and the Lorentz group
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
, but it does not work when the symmetry group is the Galilean group
In physics, a Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. These transformations together with spatial rotati ...
. In this case one has to pass to its central extension, the Bargmann group, which is the symmetry group of the Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
. Likewise, if , the group of translations in position and momentum space, one has to pass to its central extension, the Heisenberg group
In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form
::\begin
1 & a & c\\
0 & 1 & b\\
0 & 0 & 1\\
\end
under the operation of matrix multiplication. Elements ...
.
Let be the 2-cocycle on induced by . Define[If the 2-cocycle takes its values in the abelian group , i. e. it is a phase factor, which will always be the case in the contezt of ]Wigner's theorem
Wigner's theorem, proved by Eugene Wigner in 1931, is a cornerstone of the mathematical formulation of quantum mechanics. The theorem specifies how physical symmetries such as rotations, translations, and CPT are represented on the Hilbert sp ...
, then may be replaced with in the construction.
:
as a set and let the multiplication be defined by
:
Associativity holds since is a 2-cocycle on . One has for the unit element
:
and for the inverse
:
The set is an abelian subgroup of . This means that is not semisimple. The center
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentrici ...
of , includes this subgroup. The center may be larger.
At the level of Lie algebras it can be shown that the Lie algebra of is given by
:
as a vector space and endowed with the Lie bracket
:
Here is a 2-cocycle on . This 2-cocycle can be obtained from albeit in a highly nontrivial way.[ The reference states the fact and that it is difficult to show. No further references are given. Expressions on a slightly different form can be found though in and .]
Now by using the projective representation one may define a map by
:
It has the properties
:
so is a bona fide representation of .
In the context of Wigner's theorem, the situation may be depicted as such (replace by ); let denote the unit sphere in Hilbert space , and let be its inner product. Let denote ray space and the ray product. Let moreover a wiggly arrow denote a group action
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
. Then the diagram
commutes, i.e.
:
Moreover, in the same way that is a symmetry of preserving , is a symmetry of preserving . The fibers of are all circles. These circles are left invariant under the action of . The action of on these fibers is transitive with no fixed point. The conclusion is that is a principal fiber bundle over with structure group .[
]
Background material
In order to adequately discuss extensions, structure that goes beyond the defining properties of a Lie algebra is needed. Rudimentary facts about these are collected here for quick reference.
Derivations
A derivation on a Lie algebra is a map
:
such that the Leibniz rule
: