On tensor products of non-unitaryrepresentations of

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Karlstads universitet 651 88 Karlstad Faculty of Technology and Science

Physics

Carl Stigner

On tensor products of non-unitary representations of

Physics D-level thesis

Date: 2007-06-21 Supervisor: Jürgen Fuchs Examiner: Jens Fjelstad

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tries is often carried out in the form of Lie algebras and their representations. Knowing the representation theory of a Lie algebra includes knowing how tensor products of representations behave. In this thesis two methods to study and decompose tensor products of representations of non-compact Lie algebras are presented and applied to sl(2, R). We focus on products containing non-unitary representations, especially the product of a unitary highest weight representation and a non-unitary finite dimensional. Such products are not necessarily decomposable. Following the theory of B. Kostant we use infinitesimal characters to show that this kind of tensor product is fully reducible iff the sum of the highest weights in the two modules is not a positive integer or zero. The same result is obtained by looking for an invariant coupling between the product module and the contragredient module of some possible submodule. This is done in the formulation by Barut & Fronsdal. From the latter method we also obtain a basis for the submodules consisting of vectors from the product module. The described methods could be used to study more complicated semisimple Lie algebras.

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CONTENTS

1 Introduction

The universe is full of symmetries. In early days scientists studied nature, derived some set of equations describing the system under consideration and discovered that these equations displayed some kind of symmetry. Nowadays the roles are somewhat interchanged. The study of symmetries provides one of the most important tools to investigate and gain more information about the fundamental physics. By symmetry we roughly mean some transfor- mation of a physical system leaving the dynamics invariant. We can divide the symmetries into discrete and continuous ones. Discrete symmetries arise e.g. in the study of crystal lattices and are described by discrete groups. The topic of this thesis is related to continuous symmetries which are characterized by the fact that they can be made arbitrary small, which leads us to the concept of infinitesimal symmetries. Lie groups, Lie algebras and their representations are important tools used to describe these symmetries.

The continuous symmetries are often described as the action of some Lie group G on a manifold describing the physical system. The manifold could be our three-dimensional space, the Minkowski space-time, the configuration space of a classical mechanical system etc. Well known examples of Lie groups are the group SO(3) of all rotations of a three-dimensional space or the group SO(3, 1) of Lorentz transformations in Minkowski space-time. To every finite dimensional real Lie group we can associate an underlying Lie algebra g. Since a Lie group is also a differentiable manifold there is always a tangent space to any point in the Lie group manifold. The Lie algebra can be identified with the tangent space at the identity of G and at least locally (i.e. in the vicinity of the unit element) any element γ in G can be written with the help of a map Exp : g → G as γ = Exp(x) for some x in g. If G is a matrix Lie group the map Exp is nothing but the usual exponential power series. Since the Lie algebra is a vector space the analysis is often simplified by considering the underlying algebra rather than the entire group. Roughly speaking the Lie algebra corresponds to infinitesimal symmetries and the Lie group to finite symmetries. It’s important to emphasize though, that not all symmetries are described by groups. There are Lie algebras that do not correspond to a Lie group. Nevertheless they describe some infinitesimal symmetry.

The study of Lie algebras in physics is mostly related to quantum theory, though there are situations in classical physics where a Lie algebra structure arises. For example the Poisson bracket equips the functions on phase space in the Hamiltonian formulation of classical me- chanics with the structure of a Lie algebra. The Lie algebra itself is an abstract vectors space and the contact with physical states is obtained by its representations. Representations of Lie algebras typically appear as transformations on a vector space of wave functions. For more information on Lie groups, Lie algebras and their applications see e.g. [1, 2, 3].

We will study the representation theory of one of the simplest non-compact algebras sl(2, R), the underlying algebra of SL(2, R). One reason why the group SL(2, R) is interesting in physics is is that it preserves a Lorentzian metric with signature (−1, 1, 1). As an example the isometry group, as well as the boundary conformal group of AdS3, the three dimensional Lorentzian signature anti- de Sitter space AdS3is SL(2, R)L× SL(2, R)R. The underlying algebra of this group is spanned by two commuting sl(2, R)-algebras. Since sl(2, R) is non-compact all unitary representations are necessarily infinite dimensional. Following

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1 INTRODUCTION

theories from [4] and [5] we will study the tensor product of a unitary highest weight representation and a finite dimensional non-unitary representation in detail. Tensor products of non-unitary representations of non-compact real Lie algebras arise e.g. in the study of the solution space of the field equations of conformal gravity and supergravity (see e.g. [6, 7]).

Non-unitary representations of sl(2, R) appear e.g. in [8] where some aspects of the description of certain black holes as D-branes in string theory is treated. Non-normalizable modes propagate in the non-unitary representations. Their purpose is to tune the boundary conditions in order to associate to the string theory in the bulk of AdS3 a conformal field theory on its boundary. The ability to tune the boundary conditions is necessary to describe the appropriate correlation functions of the conformal filed theory.

The main purpose of this thesis is to gain more understanding regarding representation theory in general and in particular the representation theory of non-compact algebras. We start in section 2 by reviewing the construction of the representations of sl(2, R) as performed by Barut & Fronsdal in [4]. We construct all irreducible representation of the algebra and derive unitarity conditions. We will also construct normalized bases of these representations. We will consider only the mathematical aspects of the representations, which will be realized as abstract vector spaces without any physical system in mind. Since we describe the representations in normalized states the results could easily be applied to any situation where normalized states of the representations under consideration appear. The contragredient representation is studied and a map from an irreducible representation to its contragredient is constructed. It’s not obvious whether a tensor product of a unitary and a non- unitary representation is reducible or not. In section 3 we study tensor products of arbitrary highest weight modules, not necessarily unitary ones. We follow the theory by Kostant in [5] and use infinitesimal characters in order to determine under which conditions the tensor product of a unitary highest weight module and a non-unitary is fully reducible. In section 4 we study the decomposition of three kinds of tensor productsD1 ⊗ D2. The main topic is the product of a unitary irreducible infinite dimensional highest weight module and a finite dimensional non-unitary. We use methods from [4] in order to find the submodules of the product module. This is done by looking for an invariant coupling between the contragredient representation to some representationD and the product module D1⊗ D2. We are also able to derive a basis for the submodules consisting of vectors in the product module.

In appendix A we study the product of a unitary highest weight module and a two dimensional module by using nothing but the properties of the two representations in the product.

This is the simplest non-trivial case of a product between a unitary and a non unitary module. By starting from the highest weight state of the module we investigate the reducibility and derive a basis for the entire product module. We will see that the results coincide with what we find in section 3 and 4. In appendix B there is some additional information con- cerning concepts of the text. The reader is assumed to have basic knowledge regarding Lie algebras and representation theory. For convenience some of the more lengthy calculations are presented in detail in appendix C.

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2 Construction of representations

We start by reviewing the construction of the irreducible representations of sl(2, R) as performed in [4]. The formulation obtained here turns out to be quite useful for our purposes.

The representations constructed are representations of so(2, 1). Since this algebra is isomorphic to sl(2, R) these representations are also representations of latter. The starting point is the fundamental representation of SO(2, 1) spanned by matrices of the form

α β β^∗ α^∗

, |α|²− |β|² = 1. (2.1)

This set of matrices forms a group, which is often referred to as the spinor group. This group is locally isomorphic to both SO(2, 1) and SL(2, R). The underlying algebra of the two groups is spanned by the operators L12, L₁₃, L₂₃with the commutators







[L₁₂, L₁₃] = −iL23, [L₁₂, L₂₃] = iL₁₃, [L₁₃, L₂₃] = iL₁₂.

(2.2)

To see that SL(2, R) can be obtained from the algebra, we start by realizing the algebra as 2×2-matrices with the help of the Pauli matrices by setting







L⁰₁₂= ¹₂σ2, L⁰₁₃= ₂ⁱσ₃, L⁰₂₃= −₂ⁱσ₁.

(2.3)

with

σ₁=

0 1 1 0

, σ₂=

0 −i i 0

, σ₃ =

1 0 0 −1

. (2.4)

We can now write a general element of the algebra sl(2, R) as L⁰ = a₁₂L⁰₁₂+a₁₃L⁰₁₃+a₂₃L⁰₂₃ with aij ∈ R. The group SL(2, R), realized as the group of real 2 × 2-matrices with unit determinant, is obtained by the map

L⁰ 7→ exp(iL⁰). (2.5)

The representations are constructed by choosing a basis such that the two algebras sl(2, R) and so(2, 1) are obtained by restricting the allowed linear combinations into different sets.

By constructing representations of so(2, 1) we obtain representations of both algebras. The starting point is the fundamental representation (2.1) of SO(2, 1). Let the generators Lµν be

realized as 





L₁₂= ¹₂σ₃, L₁₃= −₂ⁱσ₁, L23= ₂ⁱσ2.

(2.6)

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2 CONSTRUCTION OF REPRESENTATIONS

A common basis for the two algebras is given by the generators¹ (H := 2L₁₂,

E± := iL₁₃± L23, (2.7)

satisfying the commutators

[H, E±] = ±2E^±, [E₊, E−] = H. (2.8) Consider a general linear combination of the generators:

x = _HH + ₊E₊+ ₋E₋. (2.9)

By imposing different restrictions on the coefficients H and ± we can get no less than 3 different algebras, so(3), so(2, 1) and sl(2, R). As already mentioned the two latter are isomorphic. The restrictions are the following:

so(2, 1) _H ∈ R −= −^∗+

sl(2, R) _H ∈ R ⁻, ₊∈ R so(3) _H ∈ R −= ^∗₊

As already mentioned, so(2, 1) will be kept in mind when constructing the representations.

Since so(2, 1) and sl(2, R) have the same representations, most of what we say in the following regarding irreducible representations will be true for both algebras. If we use − = −^∗+

and consider a general element of the algebra x = HH + +E++ −E− and its hermitian conjugate in some representation we have

x^†= _HH^†− ⁻E₊^† − +E₋^†. (2.10) In a unitary representation of so(2, 1) (and sl(2, R)) we have

H^†= H, E₊^† = −E−. (2.11)

Therefore we have x^†= x for an arbitrary element of so(2, 1), but not of sl(2, R) however.

Let ξ := (ξ1, ξ2) be a spinor, i.e. basis for the two valued fundamental representation of the group SO(2, 1). For our purpose we can think of ξ1, ξ₂just as formal variables. The action of the operators{H, E±} on the basis of the two dimensional representation (2.1) can be written as

H =

1 0 0 −1

= ξ₁ ∂

∂ξ₁ − ξ2

∂

∂ξ₂, E₊=

0 1 0 0

= ξ₁ ∂

∂ξ₂, E₋=

0 0 1 0

= ξ₂ ∂

∂ξ1

. (2.12)

1We use a slightly different normalization than in [4].

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Whenever it’s clear from the context we will write xv instead of R(x)v for the action of a Lie algebra-element x on a vector in a representation R. We recognize the matrices as the ordinary matrix realization of sl(2, R). The modules are created by considering the linear space spanned by the formal expressions

|a, bi := ξ1^aξ^b₂, (2.13)

where we allow the exponents a, b to take arbitrary complex values. The representations may be regarded as products of the fundamental representation with non-integral numbers of factors. By acting with the differential operator form on the states |a, bi we obtain an induced action on the states|a, bi:







(2.14)

By imposing appropriate restrictions on the values of a and b different sets of invariant subspaces, and with that the irreducible representations are formed. We see that E+(E₋) raises (lowers) the H- eigenvalue by steps of 2. Each representation is characterized by two invariants. The first is

Φ := a + b, (2.15)

which as we will see later is related to the Casimir operator as well as the highest and lowest weights in case they exist. Since the action of the algebra only affects the H-eigenvalue by raising or lowering it by steps of 2 we can define an invariant E0 and write the spectrum of H as

a − b = E0− 2m, (2.16)

where m is an integer. The Casimir operator is

L²= H(H + 2) + 4E₋E₊. (2.17)

To simplify notation we leave out the factor of 1/4 commonly inserted in the Casimir operator. The eigenvalue of the Casimir operator on the a state|a, bi is

L²|a, bi = (a + b)(a + b + 2)|a, bi = Φ(Φ + 2)|a, bi. (2.18)

2.1 Irreducible representations

By imposing restrictions on the parameters a and b different invariant subspaces are found.

This way we get all irreducible representations. The representations can be divided into different series. In table 2.1 all irreducible representations are listed together with their defining properties.

An interesting subset of representations is obtained if we restrict E0to integer values and require the representations to be unitary. All derived representations on the algebra of unitary representations of the group SL(2, R) has integer weights (see e.g. [9]). This means that only

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unitary representations of the algebra with integral weights can be exponentiated to unitary group representations. There are reasons to consider the whole set of representations of the algebra (see e.g. [8]); thus this complete set will be included. The irreducible representations are obtained in the following way:

Discrete highest weight series D⁻(Φ): b positive integer or zero but a not positive inte- ger. We see that if b ∈ Z the subspace in which b ≥ 0 is invariant under the action of the algebra since E+|Φ, 0i = 0. This subspace constitutes an irreducible highest weight representation,D⁻(Φ). Since a is not a positive integer or zero we have Φ /∈ Z+∪ {0} . The highest weight state is |Φ, 0i and the H-eigenvalues are Φ, Φ−2, ...This means setting E0 = Φ and m = 0, 1, 2, .... Restricting E0 to be integer means restricting a to negative integers. In this case Φ is necessarily a negative integer.

Discrete lowest weight seriesD⁺(Φ): a positive integer or zero but b not positive integer.

This representation is similar toD⁻(Φ). In this case we have a lowest weight representation, D⁺(Φ), of the algebra in the subspace a ≥ 0, since E−|0, Φi = 0. Also in this case we have Φ /∈ Z⁺ ∪ {0}. The lowest weight state is |0, Φi and the H-eigenvalues are −Φ, −Φ+2, ....

This means setting E0 = −Φ and m = 0, −1, −2, .... If we restrict E0 to integer values b is a negative integer and the possible values of Φ are again the negative integers.

Finite dimensional seriesD(Φ): a, b positive integers or zero. If both a and b are integers the subspace in which a, b ≥ 0 is invariant under the action of the algebra. In this case we have a finite dimensional representationD(Φ), since E+|Φ, 0i = 0 = E−|0, Φi. Since both a and b are positive integers Φ is a positive integer. We have E0 = Φ and m = 0, 1, ...Φ.

Continuous seriesD(L², E₀): a and b not integers. In this case there is no highest or low- est weight state. Thus there is no loss of generality to assume −1 < Re(E⁰) ≤ 1. Two representations in the continuous series are equivalent only if the spectra of H are the same and the eigenvalue of the Casimir operator L² is the same. Hence the representations with Φ and Φ⁰ = −Φ−2 are equivalent. Therefore it’s natural to instead label the representations by the eigenvalue of the Casimir operator. Since a and b ar not positive integers we have Φ ± E0 ∈ 2Z. If we restrict E/ 0 to integer values it’s enough to consider E0∈ {0, 1}.

Representation E₀ Allowed values of Φ Spectrum of H

D⁻(Φ) Φ Φ /∈ Z⁺∪ {0} Φ, Φ−2, ...

D⁺(Φ) −Φ Φ /∈ Z⁺∪ {0} −Φ, −Φ+2, ...

D(Φ) Φ Φ ∈ Z −Φ, −Φ+2, ..., Φ

D(L², E₀) −1 < <(E0) ≤ 1 Φ±E0 ∈ 2Z/ ..., E₀−2, E0, E₀+2, ...

Table 1: Irreducible representations

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Note that we only get the irreducible representations. There are for example also reducible highest weight and lowest weight representations (e.g. Verma modules with positive integer highest weight). These will appear as submodules of products of the formD(Φ1)^±⊗ D(Φ2) with Φ1+Φ2 ∈ Z⁺.

2.2 Unitary representations and normalized states

It’s of great interest to know under which conditions the representations are unitarizable. In the case of sl(2, R),D^±(Φ) is unitarizable but D(Φ) is in general not unless it is the trivial representation. D(L², E0) is unitarizable under certain conditions. We introduce the normalized states

|Φ, mi := N^mξ^a₁ξ₂^b = Nm ξ1ξ2Φ/2 ξ1

ξ₂

E0/2−m

, (2.19)

with Φ and m given by (2.15) and (2.16). The algebra acts on these states according to







H|Φ, mi = E₀− 2m

|Φ, mi, E₊|Φ, mi = ¹₂(Φ − E0) + m _Nm

Nm−1|Φ, m − 1i, E₋|Φ, mi = ¹₂(Φ + E₀) − m _N_m

Nm+1|Φ, m + 1i.

(2.20)

For unitary representations we can define an inner producth·|·i such that:

hΦ, m|Φ, m⁰i = δmm⁰. (2.21)

First of all we notice that in a unitary representation we have E0 ∈ R, since H^†= H. We can derive the recursion formula (see appendix C.1)

N_m N_m−1

2

= m −¹₂ −¹₂E₀−¹₂(Φ + 1)

m −¹₂− ¹₂E₀+¹₂(Φ^∗+ 1), (2.22) for the normalization constants Nm. In order for a representation to be unitary we need the right hand side to be real and positive for all values of m occurring in the representation.

In the finite dimensional case the right hand side has positive solution for all m only for the trivial representation with Φ = 0. Therefore none of the non-trivial finite dimensional representations are unitary. In the case ofD^±(Φ) we have Φ ∈ R since Φ = E0. We also see that we have positive solutions for all values of m iff Φ < 0. The normalization constants

N_m :=

"

m − 1 −¹₂(E₀+ Φ)

! m +¹₂(Φ − E⁰)

!

#1/2

(2.23)

satisfy the recursion formula above. Here x! = Γ(x + 1) where Γ is the Gamma function In the case ofD(L², E₀) we can have Φ real or complex. If Φ is complex we need Φ = −1+λi with λ∈ R in order for the right hand side of (2.22) to be real. This condition is also sufficient for the right hand side to be positive for all values of m. In this case we have Φ(Φ+2) < −1.

This series is often referred to as the principal continuous series,DP(L², E₀). If Φ is real the

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restrictions for unitarity means |Φ + 1| < 1 − |E⁰|. This series is often referred to as the supplementary continuous series, DS(L², E₀). If we restrict E0 to integer values only the case E0= 0 is possible in a unitary representation and therefore we have −2 < Φ < 0. In the case Φ ∈ R we can use N^mas given by (2.23) and for Φ =−1+λi we can choose N^m = 1 for all m to normalize the basis. We summarize the conditions for unitary representations:

E₀∈ R,









D(Φ) : Φ = 0, D^±(Φ) : Φ < 0, DP(L², E₀) : Φ = −1 + λi, DS(L², E₀) : |Φ + 1| < 1 − |E0|.

(2.24)

By in addition require E0 to be integer we obtain exactly the derived representations on unitary representations of the group SL(2, R) obtained in [9].

Another method to derive unitarity conditions is presented in [10] where highest weight modules of different real forms of a complex semisimple Lie algebra g are discussed. The method uses a certain sesquilinear form on the universal enveloping algebra U (g). The sesquilinear form is dependent on a choice of a root of the algebra. Any Verma-module is as vector space isomorphic to U (g−), where g−is the subalgebra spanned by stepoperators corresponding to negative roots (g− = E₋in the case of sl(2)). Thus the sesquilinear form induces a sesquilinear form on the Verma module, and if the module has a nontrivial submodule, the quotient module of a reducible Verma module and it’s irreducible submodule.

The highest weight of the Verma module is the weight corresponding to the root on which the sesquilinear form depend (in the case of sl(2) the roots and weights are just numbers and in this case the weight is equal to the root as numbers). Under certain conditions this sesquilinear form is nondegenerate and positive definite and therefore induces a positive definite scalar product on the Verma module. This way the representations are unitarized.

The result for su(1, 1) is that highest weight modules are unitary iff the highest weight is negative. Since su(1, 1) is isomorphic to sl(2, R) this result is valid for both algebras and thus it confirms result obtained above for highest weight modules.

If we look again atD(Φ) we have already seen that the representation is unitary iff Φ = 0. In any case we can insert normalization constants Kmaccording to

|Φ, mi := Kmξ^a₁ξ₂^b = K_m ξ₁ξ₂Φ/2 ξ₁ ξ2

E0/2−m

, (2.25)

such that the normalized basis satisfy conditions analogous to (2.20). To keep (E+)^†= −E⁻ we define instead the indefinite scalar product

hΦ, m|Φ, m⁰i = (−1)^mδ_m,m⁰. (2.26) Following an analogous calculation as in appendix C.1 we can see that

Km

K_m−1

2

= 1 + Φ − m

m . (2.27)

The right hand side is positive for all m = 0, 1, ...Φ and we can define the normalization constants

K_m :=

1

(m)!(Φ − m)!

1/2

, (2.28)

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satisfying the recursion formula.

2.3 The contragredient representation

In section 4 the contragredient representation will be used in order to decompose tensor product. Therefore we investigate the representation contragredient to a representation of sl(2)²in order to gain some useful information.

Consider a Lie algebra g and some representation R acting on a module V . Assume we have a basis{v^a|a = 1, 2, ... dim V } for V and a dual basis {f^a|a = 1, 2, ... dim V } for V^∗satisfying

f^b(v_a) = δ^b_a. (2.29)

The contragredient representation R⁺, acting on the dual space V^∗is defined by the equality R⁺(x)f

(v) = −f R(x)v

(2.30) for all v ∈ V , f ∈ V^∗and x∈ g. The construction of R⁺is often expressed by stating that the matrices are of the form R⁺(x) = −R(x)^t ∀x ∈ g. These descriptions are equivalent as can be seen by calculating the matrix elements. For arbitrary f^b ∈ V^∗and va∈ V we have, using linearity of the functions f^b, for all x∈ g

R⁺(x)f^b

(v_a) = −f R(x)v_a

= −f^b R(x)c

av_c

= − R(x)c

aδ_c^b = − R(x)b

a. (2.31) We use the Einstein summation convention throughout the section. Expanding the left hand side we get

R⁺(x)f^b

(v_a) =

R⁺(x)b cf^c

(v_a) = R⁺(x)b

cδ^c_a= R⁺(x)b

a. (2.32)

Since the indices for rows and columns in matrices are interchanged when going from V to it’s dual we see that

R⁺(x) = −R(x)^t, ∀x ∈ g. (2.33)

2.3.1 Irreducible modules

Consider now sl(2) and an irreducible module V , spanned by eigenvectors of R(H). For now it could be a highest weight representation, a lowest weight representation or neither.

Label the basis vectors by their R(H)−eigenvalues such that {vλ} is a basis for V where R(H)v_λ= λv_λ. Take again a basis for V^∗satisfying

f^λ⁰(v_λ) = δ_λ^λ⁰. (2.34)

First of all R⁺(H) is diagonal, since its transpose is diagonal. Therefore the basis vectors f^λ are eigenvectors of R⁺(H) with some eigenvalue eλ. Using (2.30) we have for f = f^λ and v = v_λ

eλ = R⁺(H)f^λ

(v_λ) = −f^λ R(H)v_λ

= −λf^λ(v_λ) = −λ. (2.35)

2the arguments is the same for sl(2) and any of it’s real forms

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We see that eλ = −λ and therefore all eigenvalues of R occur in R⁺but with opposite sign:

R⁺(H)f^λ = −λf^λ. (2.36)

We introduce the notation

R(E±)vλ = α^±_λvλ±2, (2.37)

where the coefficients α^±_λ are nothing but the non-zero matrix elements of R(E±). Note that in an irreducible highest (lowest) weight module, all α^±_λ except α⁺_Λ (α⁻_Λ) where Λ is the highest (lowest) weight are non zero. To see how R⁺(E±) act on V^∗ we calculate

R⁺(E−)f^λ−2

v_λ = −f^λ−2 R(E−)v_λ

= −α⁻λf^λ−2(v_λ−2) = −α⁻λ 6= 0, (2.38) if Λ is not a a lowest weight. If we expand R⁺(E−)f^λ−2= dµf^µwe see that dλ= −α⁻λ. Since R⁺(E−) = − R(E−)t

we also know that R⁺(E−) is a matrix where all elements except R⁺(E₋)i+1

i , i = 0, 1, ... are zero. Therefore all other coefficients dµ with µ 6= λ are zero.

Thus if R(E−)v_λ = α⁻_λv_λ−2 we can write R⁺(E₋)f^λ−2 = −α⁻_λf^λ. A similar calculation shows that that R⁺(E+)f_λ+2 = −α⁺λf^λ. Hence if λ±2 is not a highest or lowest weight the action of E±on V^∗ is given by

R⁺(E±)f^λ±2= −α^±λf^λ. (2.39) If Λ is a highest weight we have R⁺(E−)f^Λ = 0 since there exist no f^Λ+2. We see that the contragredient representation to a highest weight representation is a lowest weight representation and a similar argument shows that the contragredient representation to a lowest weight representation is a highest weight representation. We can also see that the module contragredient to an irreducible representation is again irreducible. In the description of the formal variables ξ1, ξ₂the state|Φ, mi = Nm(ξ₁ξ₂)^Φ/2(^ξ_ξ¹

2)^E⁰^/2−mis mapped to the state:

hΦ, m| = 1

N_m ξ₁ξ₂Φ/2ξ₂ ξ₁

E0/2−m

. (2.40)

2.3.2 Verma module

Apart from the irreducible modules it’s interesting to know the structure of the module contragredient to a Verma module with a non trivial submodule, i.e. a Verma module with highest weight Λ∈ Z⁺∪ {0}. In such a module we have







R(E₋)v_λ = −α⁻_λv_λ−2, α_λ⁻6= 0, ∀λ ∈ {Λ, Λ−2, ...}, R(E+)v_λ = −α⁺λv_λ+2, α_λ⁺6= 0, λ /∈ {Λ, −Λ−2},

R(E₊)v_λ = 0 λ ∈ {Λ, −Λ−2}.

(2.41)

The module can be illustrated schematically in a weight diagram as

. . . v_−Λ−4 v_−Λ−2 v_−Λ . . . v_Λ−2 v_Λ

−→←− • ^−→←− • ^←− • ^−→←−. . .^−→_←− • ^−→←− •

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where left arrows correspond to action of E−and right arrows to the action of E+. For the contragredient module of a Verma module with positive highest weight the construction above and especially (2.39) is still valid. The difference from the irreducible modules is that α⁺_−Λ−2= 0. As a consequence we have

R⁺(E₊)f^−Λ= 0. (2.42)

Thus there is a finite dimensional submodule in the contragredient representation spanned by the vectors

f^Λ, f^Λ+2, ..., f^−Λ

. However since R⁺(E−)f^−Λ−2 = −α⁻−Λf^−λwith α⁻_−Λ6= 0 the module is not reducible. Schematically we can picture the contragredient module in the following way:

f^Λ f^Λ−2 . . . f^−Λ+2 f^−Λ f^−Λ−2 . . .

• ^−→←− • ^−→←−. . ._←−^−→ • ^−→←− • ^←− • ^−→←−

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3 DECOMPOSINGD⁻(Λ) ⊗ D(Λ⁰) BY USING INFINITESIMAL CHARACTERS

3 Decomposing D

⁻

(Λ) ⊗ D(Λ

⁰

) by using infinitesimal characters

Consider the tensor productD⁻(Λ) ⊗ D(Λ⁰), where D⁻(Λ) is the unitary irreducible highest weight module with highest weight Λ andD(Λ⁰) is the finite dimensional irreducible module with highest weight Λ⁰. SinceD(Λ⁰) is not unitary it’s not possible to say directly whether the product module is fully reducible or not. Let{e^(Λ)_λ |λ = Λ, Λ−2, ...} be a basis for D⁻(Λ) and{e^(Λ_λ⁰⁰⁾|λ⁰ = −Λ, −Λ+2, ..., Λ} be a basis for D(Λ⁰) The highest weight state of the tensor product is e^(Λ)_Λ ⊗ e^(ΛΛ⁰⁰⁾. As a vector space, the product space is a direct sum

V =M

k≥0

V_k, (3.1)

of subspaces of the kind V_k:= spann

e^(Λ)_{Λ−2(k−i)}⊗ e^(ΛΛ⁰−2i⁰⁾

i ∈

0, min(Λ⁰, k)o

, k = 0, 1, 2, ..., (3.2) such that the H-eigenvalue is Λ+Λ⁰−2k on these subspaces. We also see that the dimensions of these subspaces are dim(Vk) = min(k + 1, Λ⁰+ 1). The action of the operators E±takes us from Vkto Vk∓1. The dimensions of the subspaces are related by







dim(V₀) = 1,

dim(V_k+1) = dim(V_k) + 1, k < Λ⁰, dim(V_k+1) = dim(V_k), k ≥ Λ⁰.

(3.3)

Each subspace Vk, with k ≤ Λ⁰ contain a unique (up to normalization) highest weight state v_k. These are also the only highest weight in the module. Since none of the basis vectors except e^(Λ)_Λ ⊗ e^(ΛΛ⁰⁰⁾is a highest weight state the existence of a highest weight always implies a lowering of the dimension when going from Vk to Vk−1. Thus the states vk are the only possible highest weight states. In what follows we refer to the eigenvalues of these highest weight states as highest weights, simply because these states are possible candidates to highest weights in highest weight submodules. There are no lowest weight states in the product, because that would mean that the dimension is lowered when acting with E−on some subspace Vk. This is due to the fact that there are no lowest weight states in the basis. Therefore there can be no finite dimensional submodules. Hence if the module is fully reducible it is a direct sum of infinite dimensional modules with highest weights Λ + Λ⁰, Λ + Λ⁰−2, ..., Λ −Λ⁰. In order to find conditions for full reducibility of the tensor product module we introduce the concept of infinitesimal characters as presented in [5]. We say that a highest weight module R_Λwith highest weight Λ admits an infinitesimal character ξΛ(u) if for any u in the center Z, of the universal enveloping algebra there exist some scalar ξΛ(u) such that

R_Λ(u) = ξ_Λ(u)I. (3.4)

The theory is formulated for representations of the universal enveloping algebra U (g). Since there is a one-to-one correspondence between representations of g and representations of U (g) we can apply the results regarding reducibility to the g-modules directly. In the case of

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sl(2, R) Z consists only of the Casimir operator L²and we have (see appendix B.2 for more information):

ξ_Λ(L²) = Λ(Λ + 2). (3.5)

From [5] we know the following. An infinitesimal character occurring in the tensor product R(Λ) ⊗ D(Λ⁰), where R(Λ) is an arbitrary highest weight module with highest weight Λ, is necessarily of the form ξΛ+λi(L²) with λi ∈ {−Λ⁰, −Λ⁰+2, ..., Λ⁰}. Here R(Λ) = D⁻(Λ). We also know that if all infinitesimal characters are distinct the product module is a direct sum of modules of the form:

Y_i :=

v ∈ R(Λ) ⊗ D(Λ⁰)|L²v = ξ_Λ+λi(L²)v

. (3.6)

If all infinitesimal characters are distinct we have for the unique highest weight state in each V_k, k ≤ Λ⁰, L²v_k = ξ_Λ+Λ0−2k(L²)v_k, since L² = H(H + 2) + 4E₋E₊and vkis a highest weight state which means ξΛ(L²)v_k= H(H + 2)v_k. Since [L², E±] = 0, Λ⁰+1 disjoint highest weight modules are generated from the highest weight states. This is already the entire module.

Thus if all infinitesimal characters are distinct, the module is a direct sum of Λ⁰+1 highest weight modules.

Two infinitesimal characters ξµ, ξ_λ are equal iff they are conjugated under the action of the translated Weyl group, ˜W . For sl(2, R), the translated Weyl group is of order two and the action of the non-trivial element is given by ˜σ(λ) = −λ − 2 for any weight λ. The possible infinitesimal characters and also the possible L²-eigenvalues for the tensor product are:

(Λ+Λ⁰)(Λ+Λ⁰+2), (Λ+Λ⁰−2)(Λ+Λ⁰), . . . , (Λ−Λ⁰)(Λ−Λ⁰+2). (3.7) Consider now two conjugated highest weights Λ+λ and Λ+λ⁰. Let Λ+λ ≥ 0 and let vΛ+λ

be the corresponding highest weight state. By acting repeatedly with E− on this state we reach some state uΛ+λ⁰. Since vΛ+λ is L²-eigenstate also uΛ+λ⁰ is eigenstate with the same eigenvalue. Since H(H + 2) has the same eigenvalue on two conjugated weights we have E−E₊u_Λ+λ0 = 0. Since there are no lowest weight states in the product module uΛ+λ⁰ is the unique highest weight state vΛ+λ⁰, with weight Λ+λ⁰. Hence the module³ D(Λ+λˆ ⁰) is em- bedded into ˆD(Λ+λ) and therefore the tensor product is not fully reducible. This means that the problem of finding under which circumstances the product module is fully reducible is reduced to determining whether there is a pair of conjugated weights in the set {Λ−Λ⁰, Λ−Λ⁰+2, ..., Λ+Λ⁰}. Since D⁻(Λ) is irreducible we have Λ /∈ Z⁺∪ {0}. We include all possible highest weight representations and consider three cases separately.

Λ is not integer. If Λ is not integer, this is true also for Λ+λi for all i since λi is integer.

Therefore Λ+λi− ˜σ(Λ+λⁱ) = 2(Λ+λi+1) is not not in 2Z and thus ˜σ(Λ+λi) is not in the set of weights. This means that the infinitesimal character ξ_σ(Λ+λ_˜ i)(L²) does not occur in the tensor product. Hence all infinitesimal characters are distinct and the module is fully reducible.

3We use the notation ˆD to indicate that the submodule is not necessarily one of the reducible ones. By ˆD(Λ) we mean some highest weight module with highest weight Λ.

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3 DECOMPOSINGD⁻(Λ) ⊗ D(Λ⁰) BY USING INFINITESIMAL CHARACTERS

|Λ| > Λ⁰and Λ∈ Z⁻. If|Λ| > Λ⁰the sequence of infinitesimal characters is strictly increasing and thus all infinitesimal characters are distinct. All highest weight are negative and therefore the product module is a direct sum of irreducible highest weight modules of the formD⁻(Λ+Λ⁰−2k), k ∈ [0, 1, ..., Λ⁰].

|Λ| ≤ Λ⁰and Λ∈ {Z−}. The last case occurs if |Λ| ≤ Λ⁰and Λ∈ {Z−}. Let Λ + Λ⁰ = ˜Λ ∈ N be the highest weight of the product module. This means ˜σ(Λ+Λ⁰) = −Λ−Λ⁰−2 = −˜Λ−2.

We also have Λ−Λ⁰ = 2Λ−˜Λ ≤ −˜Λ−2 since Λ ≤ −1. Hence ˜σ(Λ+Λ⁰) is in the set of highest weights. Therefore if|Λ| ≤ Λ⁰ and Λ is a negative integer there is always at least one pair of highest weights conjugated under the action of ˜W . Similarly one can see that such a pair is in the set of highest weights iff the nonnegative weight in the pair is a highest weight. I.e. the number of pairs of conjugated weights equals the number of nonnegative highest weights.

We have seen that all characters are distinct iff|Λ| > Λ⁰or Λ is not an integer, i.e. if Λ+Λ⁰ ∈/ Z∪ {0}. Thus we can write

D⁻(Λ) ⊗ D(Λ⁰) =

Λ⁰

M

k=0

D⁻(Λ + Λ⁰− 2k), iff Λ+Λ⁰∈ Z ∪ {0}./ (3.8)

We also know that L² can be written on Jordan form where none of the blocks have dimension larger than 2 (see e.g. [11]). As a consequence, in the case when the product module is not fully reducible, it can be written as a direct sum of submodules where each H-eigenspace has dimension no greater than 2. Hence the product module can always be decomposed into at least

Λ⁰+1 /2

parts, wherebac is the integer part of a. Remember that the lack of reducibility is a consequence of the existence of conjugated highest weights. This means that if we start in a 2-dimensional H-eigenspace and act with E+, we will eventually reach two conjugated highest weight states. I.e. if a submodule is not irreducible it is a module with two highest weight states in which the weights are conjugated under the action of ˜W . The method of infinitesimal characters may seem to be a bit of a ”heavy artillery” in the case of sl(2). The same result could have been obtained by simpler arguments. Nevertheless the method described is a nice way to obtain at least some information about reducibility also for more complicated algebras. As we have seen not all information regarding reducibility is obtained by just considering the infinitesimal characters.

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4 Reduction of product representation

Consider the tensor product D1 ⊗ D2 of two arbitrary sl(2, R)-representations . For know D1 andD2 can be any irreducible representations of the algebra. We can again realize the representations in formal expressions according to

D¹ : η^c₁η₂^d, Φ1= c + d, (4.1) D² : χ^e₁χ^f₂, Φ2 = e + f. (4.2) The action of the algebra is given by (2.12) with ξ replaced by η or χ. To find the representations occurring in the decomposition ofD1⊗ D2we look for representationsD having the property that an invariant can be formed from the basis vectors ofD^∗, D1 andD2 whereD^∗ is the representation contragredient toD. If such an invariant exist D is contained in D1⊗D2. LetD^∗be realized as

D^∗: ξ₁^aξ₂^b, Φ = a + b. (4.3)

We look for an invariant coupling betweenD¹, D² andD^∗. That means an invariant of the form

I = X

a,b,c,d,e,f

C_abcdefξ^a₁ξ₂^bη₁^cη₂^dχ^e₁χ^f₂. (4.4)

The terms in the invariant is to be understood as ξ₁^aξ^b₂

⊗ η^c₁η^d₂

⊗ χ^e₁χ^f₂

. The exponents a, b, c, d, e, f satisfy (4.1),(4.2) and (4.3). Since HI = 0 we also have

a + c + e = b + d + f. (4.5)

The only invariants possible to construct from two spinors in a product of two of the modules are:

(ξη) := ξ1η2− ξ²η1, (ξχ), (ηχ). (4.6) Hence, an invariant in the productD^∗⊗ D1⊗ D2has to be on the form

I ∝ ξηα

ξχβ

ηχγ

= ξ₁η₂− ξ2η₁α

ξ₁χ₂− ξ2χ₁β

η₁χ₂− η2χ₁γ

. (4.7)

If we compare the values of the exponents in (4.4) with (4.7) we find







α + β = a + b = Φ, α + γ = c + d = Φ₁, β + γ = e + f = Φ₂.

(4.8)

This system of equations has the unique solution







α = ¹₂(Φ + Φ₁− Φ2), β = ¹₂(Φ − Φ1+ Φ₂), γ = ¹₂(−Φ + Φ1+ Φ₂).

(4.9)

When appropriate restrictions are imposed on the parameters α, β, γ the possible values of Φ are determined. However, note that the value of Φ does not alone determine the representation D. One value of Φ can occur in different classes of representations. We will have to study the invariant I in a bit more detail.

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4 REDUCTION OF PRODUCT REPRESENTATION

Let f^λ be a basis forD^∗. A convenient way to write the invariant is I =X

λ

f^λ⊗ vλ, (4.10)

where we sum over all weights λ occurring in D and vλ ∈ D1 ⊗ D2. When the invariant is written this way the vectors vλ constitute a basis for D as a submodule of the product moduleD1⊗ D2. This is quite easily realized for any semi-simple Lie algebra by acting with some step-operator on the invariant and using linear independence of the vectors f^µ⊗ vλ. See appendix B.3 for more information.

4.1 The product of two finite dimensional representations

Consider a product of two finite dimensional modules with highest weights Φ1 and Φ2. The finite dimensional representations are unitary in the case of so(3) and non-unitary in the case of so(2, 1) and sl(2, R). In any case the product module is fully reducible. In the case of so(3) this is simply because any tensor product of two finite dimensional unitary modules is fully reducible. For sl(2, R) the reducibility can be deduced from infinitesimal characters. Thus most of the analysis of this product will be the same for sl(2, R) and so(3).

The decomposition of finite dimensional representations of so(3) is a well-known standard example, often performed for su(2) ∼= so(3).

The exponents a, b, c, d, e, f takes only non-negative integral values since D is necessarily finite dimensional. Therefore also α, β, γ are positive integers. γ ≥ 0 implies Φ ≤ Φ¹ + Φ2

and α, β ≥ 0 implies Φ ≥ |Φ1− Φ2|. Thus we arrive at the well known result

|Φ1−Φ2| ≤ Φ ≤ Φ1+Φ₂. (4.11)

where Φ runs from |Φ1−Φ2| to Φ1+Φ₂ in even integer steps. Without loss of generality we can assume Φ1 ≤ Φ2. SinceD(Φ1) ⊗ D(Φ2) is isomorphic to D(Φ2) ⊗ D(Φ1) the analysis for Φ₁ > Φ₂ would be completely equivalent with the roles of Φ1 and Φ2 interchanged. The Φ₁+ 1 possible values for α are 0, 1, 2, ..., Φ1 and we have

Φ = Φ2− Φ¹+ 2α. (4.12)

Letting α run from 0 to Φ1 we get all irreducible representations contained in the tensor product, precisely once. From section 3 we know that the infinitesimal characters occurring in the product module are

(Φ₂−Φ1)(Φ₂−Φ1+2), (Φ₂−Φ1+2)(Φ₂−Φ1+4), ..., (Φ₂+Φ₁)(Φ₂+Φ₁+2). (4.13) The sequence of infinitesimal characters is strictly increasing and therefore they are all distinct. Thus the product module is a direct sum of Φ1+1 modules. These modules are exactly the Φ1+1 submodules D(Φ) with Φ given by (4.12). Thus we have

D(Φ1) ⊗ D(Φ2) =

Φ1

M

α=0

D(Φ2−Φ1+2α). (4.14)

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D(1) ⊗ D(1)

We start by considering the simplest non-trivial well-known caseD(1) ⊗ D(1). Here we have two cases, α = 0 or α = 1. If α = 0 we have γ = 1 and therefore β = 0. Hence the invariant is

I₀ = η₁χ₂− η2χ₁ = ξ₁⁰ξ₂⁰ η₁χ₂− η2χ₁

. (4.15)

I₀corresponds to the antisymmetric trivial representation. A basis for D(0)∗

is{ξ1⁰ξ₂⁰}. A basis forD(0) in the product module D(1) ⊗ D(1) is {η¹χ2− η²χ1}.

The other invariant is found for α = β = 1 and γ = 0. The invariant is I₁= ξ₁²η₂χ₂− ξ1ξ₂ η₂χ₁+η₁χ₂

+ ξ²₂η₁χ₁. (4.16) We see that I1corresponds to the symmetric three dimensional representation where a basis for D(2)∗

is{ξ1², ξ₁ξ₂, ξ₂²}. A basis for D(2) as a submodule of the product module is given by{η1χ₁, η₁χ₂+η₂χ₁, η₂χ₂}.

The general case: D(Φ₁) ⊗ D(Φ₂)

If we consider the general case with Φ1≤ Φ2we have as already mentioned Φ1+1 invariants.

The parameters α, β, γ satisfy: 





α ∈ {0, 1, ..., Φ1}, β = Φ − α, γ = Φ₁− α,

(4.17)

where Φ = Φ2− Φ¹+ 2α. We can write down the invariants by using α as a single parameter.

The Φ1+1 different invariants are calculated to be (see appendix C.2):

I_α = XΦ ν=0

ξ^Φ−ν₁ ξ₂^ν

min(ν,Φ−α)X

j=0

Φ1X−α+j p=j

ν−jα _Φ−α

j

_Φ₁_−α

p−j

(−1)^p+ν−jη₁^Φ¹^−α−p+νη₂^α+p−νχ^p₁χ^Φ₂²^−p,

(4.18) for α = 0, 1, 2, ...Φ1. We see that the terms in the summation over p and j all have H- eigenvalue−Φ + 2ν. We also see that, since the vectors ξ^Φ−ν1 ξ₂^ν with ν = 0, 1, 2, ...Φ is a basis forD^∗(Φ), the invariant is on the form (4.10). Thus by the summing over p and j with fixed value of ν we get the eigenvector with eigenvalue−Φ+2ν in D(Φ). So we can write for the submoduleD(Φ) of D(Φ1) ⊗ D(Φ2) with Φ = Φ2− Φ1+ 2α and α ∈ {0, 1, 2, ..., Φ1}:

D(Φ) = span ψ^(α)_ν

ν = 0, 1, 2, ..., Φ

, (4.19)

where the states ψ^(α)ν are defined as ψ^(α)_ν :=

min(ν,Φ−α)X

j=0

Φ1X−α+j p=j

ν−jα _Φ−α

j

_Φ₁_−α

p−j

(−1)^p−jη₁^Φ¹^−α−p+νη₂^α+p−νχ^p₁χ^Φ₂²^−p. (4.20)

Worth to notice is that by expressing the invariants in the normalized basis of the modules given by (2.19) and (2.40) one can read of the Clebsch-Gordan coefficients.

On tensor products of non-unitaryrepresentations of

Carl Stigner

On tensor products of non-unitary representations of

Physics D-level thesis

Contents

1 Introduction

2 Construction of representations

3 Decomposing D

(Λ) ⊗ D(Λ

) by using infinitesimal characters

4 Reduction of product representation