The Algebraic Approach to Duality: An Introduction
aa r X i v : . [ m a t h . P R ] F e b The Algebraic Approach to Duality:An Introduction
Anja Sturm ∗ Jan M. Swart † Florian V¨ollering ‡ February 21, 2018
Abstract
This survey article gives an elementary introduction to the algebraicapproach to Markov process duality, as opposed to the pathwise ap-proach. In the algebraic approach, a Markov generator is written asthe sum of products of simpler operators, which each have a dual withrespect to some duality function. We discuss at length the recent sug-gestion by Giardin`a, Redig, and others, that it may be a good ideato choose these simpler operators in such a way that they form anirreducible representation of some known Lie algebra. In particular,we collect the necessary background on representations of Lie algebrasthat is crucial for this approach. We also discuss older work by Lloydand Sudbury on duality functions of product form and the relationbetween intertwining and duality.
MSC 2010.
Primary: 82C22, Secondary: 60K35, 17B10, 22E46.
Keywords.
Interacting particle system, duality, intertwining, representationsof Lie algebras
Acknowledgement.
Work sponsored by grant 16-15238S of the Czech ScienceFoundation (GA CR). ∗ Institute for Mathematical Stochastics, Georg-August-Universit¨at G¨ottingen, Gold-schmidtstr. 7, 37077 G¨ottingen, Germany; [email protected] † The Czech Academy of Sciences, Institute of Information Theory and Automation.Pod vod´arenskou vˇeˇz´ı 4, 18208 Prague 8, Czech Republic; [email protected] ‡ Department of Mathematics, University of Bath, Claverton Down, Bath BA2 7AY,United Kingdom; [email protected] ontents A A crash course in Lie algebras 41
A.1 Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41A.2 Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43A.3 Relation between Lie groups and Lie algebras . . . . . . . . . 46A.4 Relation between algebras and Lie algebras . . . . . . . . . . . 49A.5 Adjoints and unitary representations . . . . . . . . . . . . . . 50A.6 Dual, quotient, sum, and product spaces . . . . . . . . . . . . 54A.7 Irreducible representations . . . . . . . . . . . . . . . . . . . . 60A.8 Semisimple Lie algebras . . . . . . . . . . . . . . . . . . . . . 62A.9 Some basic matrix Lie groups . . . . . . . . . . . . . . . . . . 64A.10 The Lie group SU(1,1) . . . . . . . . . . . . . . . . . . . . . . 65A.11 The Heisenberg group . . . . . . . . . . . . . . . . . . . . . . 672
Introduction
The aim of the present text is to give an introduction to the algebraic ap-proach to the theory of duality of Markov processes. In particular, we presentsome of the pioneering work done by Lloyd and Sudbury [LS95, LS97, Sud00]and spend a lot of time explaining the more recent work of Giardin`a, Redig,and others [GKRV09, CGGR15]. The algebraic approach differs fundamen-tally from the pathwise approach propagated in e.g., [JK14, SS16]. In prin-ciple, the algebraic approach is able to find a wider class of dualities, but theprice we pay for this is that it may suggest dual operators that turn out notto be Markov generators.In the remainder of this section, we quickly introduce the basic ideasbehind the algebraic approach. In Subsection 1.2, we explain how Markovprocess duality can algebraically be viewed as an intertwining relation be-tween the generator of one Markov process and the adjoint of the generatorof another Markov process. As explained in Subsection 1.3, it is then naturalto view a Markov generator as being built up out of sums and products ofother, simpler operators. If all these building blocks have duals with respectto a duality function, then so has the original Markov generator.A central idea of of Giardin`a, Redig, et al. [GKRV09, CGGR15] is tochoose these building blocks so that they form a representation of a Liealgebra. To understand why that may be a good idea, one needs quite a bitof background on Lie algebras. Since probabilists may not be familiar withthis, after a small detour to pathwise duality in Subsection 1.4, we devote allof Section 2 to providing this background.The study of Lie algebras and their representations is a huge subject witha venerable history. Although there exist good introductory texts, we willneed some theory that is considered too advanced for the usual textbooks. Inparticular, this refers to the representation theory of non-compact Lie groupslike SU(1,1) or the Heisenberg group. In order to squeeze the essential factsthat we need for our purposes into little over 10 pages, we had to cut somecorners and in some cases resign on full mathematical rigour. We also leaveout a lot of background material (e.g., Lie groups, as opposed to Lie algebras,stay almost completely out of the picture). To partly compensate for this,we have added Appendix A which gives a somewhat more complete, but stillsketchy picture.After our little excursion into Lie algebras, in Section 3, we come tothe core of our text. In Subsections 3.1, 3.3, and 3.4 we demonstrate theapproach via Lie algebras on three examples, which are based on representa-3ion theory for the Heisenberg algebra, SU(2), and SU(1,1), respectively. InSubsection 3.1, we formulate a general principle and apply it to discover aself-duality of the Wright-Fisher diffusion from (1.8). After Subsection 3.2,which is needed to deal with infinite state spaces, in Subsection 3.3, we usethe well-known representation theory of SU(2) to derive a duality for thesymmetric exclusion process. This duality is not very interesting on its own,but serves as a preparation for the symmetric inclusion process in Section 3.4which turns out to be very similar to the former, except that SU(2) is replacedby SU(1,1).In Sections 3.5–3.7 we present results of Lloyd and Sudbury [LS95, LS97,Sud00] that do not require knowledge of Lie algebras, but do use some factsabout tensor products from Section 2.6. In particular, in Section 3.5 wediscuss duality functions of product form, including q-duality, while in Sec-tion 3.6 we discuss intertwining of Markov processes, and in particular thin-ning relations which are closely connected to q-duality.In Sections 3.8 and 3.9, finally, we discuss another observation from[GKRV09], who show that nontrivial dualities can sometimes be found bystarting from a “trivial” duality which is based on reversibility, and thenusing a symmetry of the model to transform such a duality into a nontrivialone. Although Lie algebras are not strictly needed in this approach, writinggenerators in terms of the basis elements of a representation of a Lie algebracan help finding suitable symmetries.
In Section 1, for technical simplicity, we mostly restrict ourselves to Markovprocesses with finite state spaces. As we will see in Section 3, many of thebasic ideas discussed here can with some care be made to work also in infinitedimensional settings. How to do this is in part discussed in Section 3.2, butfor brevity, we will not always go into the technical details and sometimesuse the calculations of the present section merely as an inspiration.The generator of a continuous-time Markov process with finite state spaceΩ is a matrix L such that L ( x, y ) ≥ x = y ) and X y L ( x, y ) = 0 . (1.1)Equivalently, we can identify L with the linear operator L : R Ω → R Ω definedby Lf ( x ) := X y ∈ Ω L ( x, y ) f ( y ) ( x ∈ Ω) . (1.2)4 linear operator L : R Ω → R Ω is a Markov generator (i.e., satisfies (1.1)) ifand only if the semigroup of operators ( P t ) t ≥ defined by P t := e tL = ∞ X n =0 n ! t n L n is a Markov semigroup, i.e., P t is a probability kernel for each t ≥
0. If L isa Markov generator, then ( P t ) t ≥ are the transition kernels of some Ω-valuedMarkov process ( X t ) t ≥ .Let Ω and ˆΩ be finite sets. We can view a function D : Ω × ˆΩ → R as amatrix ( D ( x, y )) x ∈ Ω , y ∈ ˆΩ that as in (1.2) corresponds to a linear operator D : R ˆΩ → R Ω .Let L and ˆ L be generators of Markov processes ( X t ) t ≥ and ( Y t ) t ≥ withstate spaces Ω and ˆΩ and semigroups ( P t ) t ≥ and ( ˆ P t ) t ≥ , and let D : Ω × ˆΩ → R be a function. We make the following simple observation. Below, we let A † ( x, y ) := A ( y, x ) (or A † ( x, y ) := A ( y, x ) for matrices over the complexnumbers) denote the adjoint of a matrix A . Lemma 1 (Duality)
The following conditions are equivalent. (i) LD = D ˆ L † , (ii) P t D = D ˆ P † t for all t ≥ , (iii) E x [ D ( X t , y )] = E y [ D ( x, Y t )] for all x ∈ Ω , y ∈ ˆΩ , and t ≥ . Proof
If (i) holds for L , then it also holds for any linear combination ofpowers of L . In particular, filling in the definition of P t , we see that (i)implies (ii). Conversely, differentiating with respect to t , we see that (ii)implies (i). Condition (iii) is just a rewrite of (ii).If the conditions of Lemma 1 are satisfied, then we say that ( X t ) t ≥ and( Y t ) t ≥ are dual with duality function D . If L = ˆ L , then we speak of self-duality . Condition (i) can also be written as LD ( · , y )( x ) = ˆ LD ( x, · )( y ) ( x ∈ Ω , y ∈ ˆΩ) . (1.3)Under suitable assumptions, the equivalence of (iii) and (1.3) can often alsobe established for Markov processes with infinite state space. The semigroup property says that P = I and P s P t = P s + t . In other words, h A † f | g i := h f | Ag i where h f | g i := P x ∈ Ω f ( x ) g ( x ) denotes the usualinner product on C Ω . For adjoints with respect to a general inner product on finite orinfinite dimensional spaces we write A ∗ .
5n algebraic relation of the form AB = BC is called an intertwiningrelation between operators A and C . The operator B is called the intertwiner .Thus, Lemma 1 says that two Markov processes are dual if and only if thereexists an intertwiner between the generator of one Markov process, and theadjoint of the generator of another Markov process. Note that if L is dualto ˆ L with duality function D , then ˆ L is dual to L with duality function D † .Thus, duality is a symmetric concept.Closely related to Markov process duality is the concept of intertwiningof Markov processes , which has a more narrow meaning than the algebraicconcept of intertwining. Let, again, L and ˆ L be generators of Markov pro-cesses ( X t ) t ≥ and ( Y t ) t ≥ with state spaces Ω and ˆΩ and semigroups ( P t ) t ≥ and ( ˆ P t ) t ≥ . Let K : Ω × ˆΩ → R be a function. In what follows, we assumethat K is a probability kernel, i.e., K ( x, y ) ≥ ∀ x, y and P y K ( x, y ) = 1 foreach x . Lemma 2 (Intertwining of Markov processes)
The the following con-ditions are equivalent. (i) LK = K ˆ L . (ii) P t K = K ˆ P t ( t ≥ . (iii) µ K = ν implies µ P t K = ν ˆ P t ( t ≥ . Proof
The equivalence of (i) and (ii) follows by the same argument as inLemma 1. Condition (ii) implies µ P t K = ( µ K ) ˆ P t ( t ≥ µ = δ x we see that (iii) implies (ii).In condition (iii), note that µ P t and ν ˆ P t describe the laws at time t ofthe Markov processes ( X t ) t ≥ and ( Y t ) t ≥ started in initial laws µ and ν ,respectively. If the conditions of Lemma 2 are satisfied, then we say that theMarkov processes ( X t ) t ≥ and ( Y t ) t ≥ are intertwined .If K is invertible as a matrix, then LK = K ˆ L implies ˆ LK − = K − L ;however, K − will in general not be a probability kernel. In view of this,in an intertwining relation between Markov processes, the two processes donot play symmetric roles. To stress the different roles of X and Y , following[Swa13], it is convenient to say that Y is an intertwined Markov process ontop of X .If the conditions of Lemma 2 are satisfied, then the Markov processes X and Y can actually be coupled such that ( X t , Y t ) t ≥ is a Markov process and P [ Y t ∈ · | ( X s ) ≤ s ≤ t ] = K ( X s , · ) a . s . ( t ≥ , see [Fil92, Swa13]. Note that this strengthens condition (iii) of Lemma 2.6 .3 The algebraic approach We make the following simple observation. Below, R Ω denotes the space ofall functions f : Ω → R . Lemma 3 (Duality of building blocks)
Let Ω , ˆΩ be finite spaces and let A i : R Ω → R Ω , B i : R ˆΩ → R ˆΩ ( i = 1 , , and D : R ˆΩ → R Ω be linearoperators such that A i D = DB † i ( i = 1 , . (1.4) Then ( r A + r A ) D = D ( r B + r B ) † and ( A A ) D = D ( B B ) † . (1.5)Lemma 3 implies that if we can write a Markov generator L as a linearcombination of products of “simpler” operators A i , for example, (denotingthe identity operator by I ), L = r ∅ I + r A + r A A + r A A , (1.6)and these “building blocks” satisfy A i D = DB † i for some duality function D ,the L will be dual to the operatorˆ L = r ∅ I + r B + r B B + r B B . (1.7)Note that in each term, we have not only replaced A i by B i but also reversedthe order of the factors. If we are lucky, ˆ L is a Markov generator and wehave discovered a Markov duality.We demonstrate this approach on the Wright-Fisher diffusion with selec-tion parameter s ∈ R , which is the diffusion in [0 ,
1] with generator Lf ( x ) = x (1 − x ) ∂ ∂x + sx (1 − x ) ∂∂x . (1.8)We are immediately cheating here, since L is not a linear operator acting ona finite dimensional space. Ignoring the difficulties associated with infinitedimension, we can write L in terms of simpler “building blocks” as follows.We set A − f ( x ) := (1 − x ) f ( x ) and A + f ( x ) := ∂∂x f ( x ) , (1.9)and we write L in terms of these building blocks as L = A − ( I − A − ) A + ( sI + A + ) . (1.10)As our dual space, we choose N = { , , . . . } and as our duality function wechoose the function D : [0 , × N → R given by D ( x, n ) := (1 − x ) n ( x ∈ [0 , , n ∈ N ) . (1.11)7et B ± be operators acting on functions f : N → R as B − f ( n ) := f ( n + 1) and B + f ( n ) := − nf ( n − . (1.12)Then B ± are dual to A ± in the sense of (1.3), i.e., A ± D ( · , n )( x ) = B ± D ( x, · )( n ) ( x ∈ [0 , , n ∈ N ) . (1.13)Therefore, in view of Lemma 3, the following operator should be dual to L :ˆ L = ( sI + B + ) B + ( I − B − ) B − . (1.14)(Note that we have replaced A ± by B ± and reversed the order of the factors.)A little calculation reveals thatˆ Lf ( n ) = n ( n − (cid:8) f ( n − − f ( n ) } + sn (cid:8) f ( n + 1) − f ( n ) (cid:9) . (1.15)This is not, in general, a Markov generator. For s ≥
0, however, it is thegenerator of a Markov process in N that jumps from n to n − n ( n −
1) and from n to n + 1 with rate sn .Recall that the commutator of two operators A, B is defined as [
A, B ] := AB − BA . For our operators A ± , it is easy to check that[ A − , A + ] = I. (1.16)This is similar to the commutation relation between the position and momen-tum operators in quantum physics. Indeed, the operators A ± can be used todefine a representation of the Heisenberg algebra, which is a particular Liealgebra. The connection to Lie algebras can help us to choose good buildingblocks and can sometimes also suggest duality functions. To explain this,we need some theory about representations of Lie algebras, which will bepresented in the next section. In the remainder of this section, we point out some differences and similaritiesbetween the algebraic and pathwise approaches to Markov process duality. A random mapping representation of a probability kernel K is a random map M such that K ( x, d y ) = P [ M ( x ) ∈ d y ] . (1.17)A stochastic flow is a collection ( X s,u ) s ≤ u of random maps X s,u : Ω → Ω suchthat X s,s = I and X t,u ◦ X s,t = X s,u . We say that ( X s,u ) s ≤ u has independentincrements if X t ,t , . . . , X t n − ,t n (1.18)8re independent for any t < · · · < t n . If ( X s,u ) s ≤ u is a stochastic flowwith independent increments such that the law of X s,u depends only on thedifference u − s , and X is an independent Ω-valued random variable, thensetting X t := X ,t ( X ) ( t ≥
0) (1.19)defines a Markov process with transition kernels P u − s ( x, d y ) := P [ X s,u ( x ) ∈ d y ] ( s ≤ u ) . (1.20)Note that this formula says that X s,u is a random mapping representation of P u − s .Markov processes can often be constructed from stochastic flows. Forexample, if a stochastic differential equation has unique strong solutions,then these solutions (for different initial states) define a stochastic flow withindependent increments that can be used to construct a diffusion process. If L is the generator of a Markov process with finite state space Ω, then L canalways be written in the form Lf ( x ) = X m ∈G r m (cid:8) f (cid:0) m ( x ) (cid:1) − f (cid:0) x (cid:1)(cid:9) , (1.21)where G is a finite collection of maps m : Ω → Ω. We say that two maps m, ˆ m are dual with respect to a duality function D if D (cid:0) m ( x ) , y (cid:1) = D (cid:0) x, ˆ m ( y ) (cid:1) ( x ∈ Ω , y ∈ ˆΩ) . (1.22)Two stochastic flows ( X s,u ) s ≤ u and ( Y s,u ) s ≤ u are dual if for each s ≤ u , a.s., Y − u, − s is dual to X s,u . If two stochastic flows are dual, then we say thattheir associated Markov processes are pathwise dual . It is easy to see thatthis implies Markov process duality.We recall that in the algebraic approach, there may be many ways inwhich a given Markov generator can be written in terms of more elementary“building blocks” as in (1.6). Similarly, in the pathwise approach, there areusually many different ways in which a Markov generator can be written interms of maps as in (1.21). In the algebraic approach we have seen thatif all building blocks have duals with respect to a given duality function,then a Markov generator built up from these building blocks also has a dualˆ L . Similarly, in the pathwise approach, if all maps m occurring in (1.21) The definition of duality for stochastic flows that we give here is a weak one. It isoften natural to give a somewhat stronger definition, see [SS16]. m with respect to some duality function D , then the process withgenerator L is pathwise dual to the process with generatorˆ Lf ( x ) := X m ∈G r m (cid:8) f (cid:0) ˆ m ( x ) (cid:1) − f (cid:0) x (cid:1)(cid:9) . (1.23)An advantage of the pathwise approach is that an operator ˆ L of this form isguaranteed to me a Markov generator. On the other hand, not all dualitiescan be constructed as pathwise dualities, so the algebraic approach is moregeneral. Nevertheless, many known dualities, including the duality for theWright-Fisher diffusion discussed in the previous subsection, can be obtainedin a pathwise way or as limits of such pathwise dualities, see [Swa06, AH07].There are more analogies between the algebraic and pathwise approaches.In Subsection 3.8, we will see that in the algebraic approach, nontrivial du-alities can sometimes be found by starting with a “trivial” duality obtainedfrom reversibility and then applying a symmetry transformation. In [SS16],it is shown that nontrivial pathwise dualities can be found by starting witha “trivial” duality to the inverse image map and then looking for invariantsubspaces of the dual process. A complex (resp. real) Lie algebra is a finite-dimensional linear space g over C (resp. R ) together with a map [ · , · ] : g × g → g called Lie bracket suchthat(i) ( x , y ) [ x , y ] is bilinear,(ii) [ x , y ] = − [ y , x ] (skew symmetry),(iii) [ x , [ y , z ]] + [ y , [ z , x ]] + [ z , [ x , y ]] = 0 (Jacobi identity).An adjoint operation on a Lie algebra g is a map x x ∗ such that(i) x x ∗ is conjugate linear,(ii) ( x ∗ ) ∗ = x , In this section, we mostly focus on complex Lie algebras. Some results stated in thepresent section (in particular, part (b) of Schur’s lemma) are true for complex Lie algebrasonly. See Appendix A for a more detailed discussion. x ∗ , y ∗ ] = [ y , x ] ∗ .If g is a complex Lie algebra, then the space of its skew symmetric elements h := { x ∈ g : x ∗ = − x } forms a real Lie algebra. Conversely, starting from areal Lie algebra h , we can always find a complex Lie algebra g equipped witha adjoint operation such that h is the space of skew symmetric elements of g . Then g is called the complexification of h .If { x , . . . , x n } is a basis for g , then the Lie bracket on g is uniquelycharacterized by the commutation relations [ x i , x j ] = n X k =1 c ijk x k ( i < j ) . The constants c ijk are called the structure constants . If g is equipped withan adjoint operation, then the latter is uniquely characterized by the adjointrelations x ∗ i = X j d ij x j . Example
Let V be a finite dimensional complex linear space, let L ( V ) denotethe space of all linear operators A : V → V , and let tr( A ) denote the traceof an operator A . Then g := { A ∈ L ( V ) : tr( A ) = 0 } with [ A, B ] := AB − BA is a Lie algebra. Note that tr([ A, B ]) = tr( AB ) − tr( BA ) = 0 by the basicproperty of the trace, which shows that [ A, B ] ∈ g for all A, B ∈ g . Note alsothat g is in general not an algebra, i.e., A, B ∈ g does not imply AB ∈ g . If V is equipped with an inner product h · | · i (which we always take conjugatelinear in its first argument and linear in its second argument) and A ∗ denotesthe adjoint of A with respect to this inner product, i.e., h A ∗ v | w i := h v | Aw i , then one can check that A A ∗ is an adjoint operation on g .By definition, a Lie algebra homomorphism is a map φ : g → h from oneLie algebra into another that preserves the structure of the Lie algebra, i.e., φ is linear and φ ([ A, B ]) = [ φ ( A ) , φ ( B )] . If φ is invertible, then its inverse is also a Lie algebra homomorphism. Inthis case we call φ a Lie algebra isomorphism . We say that a Lie algebra11omomorphism φ is unitary if it moreover preserves the structure of theadjoint operation, i.e., φ ( A ∗ ) = φ ( A ) ∗ . If g is a Lie algebra, then we can define a conjugate of g , which is a Liealgebra g together with a conjugate linear bijection g ∋ x x ∈ g such that[ x , y ] = [ y , x ] . It is easy to see that such a conjugate Lie algebra is unique up to naturalisomorphisms, and that the g is naturally isomorphic to g . If g is equippedwith an adjoint operation, then we can define an adjoint operation on g by x ∗ := ( x ∗ ). Example
Let V be a complex linear space on which an inner product isdefined and let g ⊂ L ( V ) be a linear subspace such that A, B ∈ g implies[ A, B ] ∈ g . Then g is a sub-Lie-algebra of L ( V ). Now g := { A ∗ : A ∈ g } ,together with the map A := A ∗ is a realization of the conjugate Lie algebraof g . If V is a finite dimensional linear space, then the space L ( V ) of linear oper-ators A : V → V , equipped with the commutator [ A, B ] := AB − BA is a Lie algebra. By definition, a representation of a complex Lie algebra g isa pair ( V, π ) where V is a complex linear space of dimension dim( V ) ≥ π : g → L ( V ) is a Lie algebra homomorphism. A representation is unitary if this homomorphism is unitary and faithful if π is an isomorphism to itsimage π ( g ) := { π ( x ) : x ∈ g } .There is another way of looking at representations that is often useful. If( V, π ) is a representation, then we can define a map g × V ∋ ( x , v ) x v ∈ V by x v := π ( x ) v . Such a map satisfies(i) ( x , v ) Av is bilinear (i.e., linear in both arguments),(ii) [ x , y ] v = x ( y v ) − y ( x v ). 12ny map with these properties is called a left action of g on V . It is easyto see that if V is a complex linear space that is equipped with a left actionof g , then setting π ( x ) v := x v defines a Lie algebra homomorphism from g to L ( V ). Thus, we can view representations as linear spaces on which a leftaction of g is defined. Example
For any Lie algebra, we may set V := g . Then, using the Jacobiidentity, one can verify that the map ( x , y ) [ x , y ] is a left action of g on V . (See Lemma 15 in the appendix.) In this way, every Lie algebra can berepresented on itself. This representation is not always faithful, but for manyLie algebras of interest, it is.Yet another way to look at representations is in terms of commutationrelations. Let g be a Lie algebra with basis elements x , . . . , x n , which satisfythe commutation relations[ x i , x j ] = n X k =1 c ijk x k ( i < j ) . Let V be a complex linear space with dim( V ) ≥ X , . . . , X n ∈ L ( V )satisfy [ X i , X j ] = n X k =1 c ijk X k ( i < j ) . Then there exists a unique Lie algebra homomorphism π : g → L ( V ) suchthat π ( x i ) = X i ( i = 1 , . . . , n ). Thus, any collection of linear operators thatsatisfies the commutation relations of g defines a representation of g . Such arepresentation is faithful if and only if X , . . . , X n are linearly independent.If g is equipped with an adjoint operation and V is equipped with an innerproduct, then the representation ( V, π ) is unitary if and only if X , . . . , X n satisfy the adjoint relations of g , i.e., x ∗ i = X j d ij x j and X ∗ i = X j d ij X j . Let V be a representation of a Lie algebra g . By definition, an invariantsubspace of V is a linear subspace W ⊂ V such that x w ∈ W for all w ∈ W and x ∈ g . A representation is irreducible if its only invariant subspaces are W = { } and W = V .Let V, W be two representations of the same Lie algebra g . By definition,an intertwiner of representations is a linear map φ : V → W that preservesthe structure of a representation, i.e., φ ( x v ) = x φ ( v ) . φ is a bijection then its inverse is also an intertwiner. In this case wecall φ an isomorphism and say that the representations are equivalent (or isomorphic ).The following result can be found in, e.g., [Hal03, Thm 4.29]. Below andin what follows, we let I ∈ L ( V ) denote the identity operator Iv := v . Proposition 4 (Schur’s lemma)(a)
Let V and W be irreducible representations of the same Lie algebraand let φ : V → W be an intertwiner. Then either φ = 0 or φ is anisomorphism. (b) Let V be an irreducible representation of a Lie algebra and let φ : V → V be an intertwiner. Then φ = λI for some λ ∈ C . For us, the following simple consequence of Schur’s lemma will be impor-tant.
Corollary 5 (Unique intertwiner)
Let ( V, π V ) and ( W, π W ) be equiva-lent irreducible representations of some Lie algebra. Then there exists anintertwiner φ : V → W that is unique up to a multiplicative constant, suchthat φπ V ( x ) = π W ( x ) φ. Proof
By assumption, V and W are equivalent, so there exists an isomor-phism φ : V → W . Assume that ψ : V → W is another intertwiner. Then φ − ◦ ψ is an intertwiner from V into itself, so by part (b) of Schur’s lemma, φ − ◦ ψ = λI and hence ψ = λφ .If V is a complex linear space, then we can define a conjugate of V , whichis a complex linear space V together with a conjugate linear bijection φ φ . Example
Let V be a complex linear space with inner product h · | · i . Let V ′ denote the dual space of V , i.e., the space of all linear forms l : V → C . Forany v ∈ V , we can define a linear form h v | ∈ V ′ by h v | w := h v | w i . Then V ′ ,together with the map v
7→ h v | , is a realization of the conjugate of V .If ( V, π ) is a representation of a Lie algebra g , then we can equip theconjugate space V with the structure of a representation of the conjugateLie algebra g by putting x v := x v. It is easy to see that this defines a left action of g on V . We call V , equippedwith this left action of g , the conjugate of the representation V .There is a close relation between Lie algebras and Lie groups. Roughlyspeaking, a Lie group is a smooth differentiable manifold that is equipped14ith a group structure. In particular, a matrix Lie group G is a group whoseelements are invertible linear operators acting on some finite dimensionallinear space V . The Lie algebra of G is then defined as h := { A ∈ L ( V ) : e tA ∈ G ∀ t ≥ } . In general, this is a real Lie algebra. More generally, one can associate aLie algebra to each Lie group (not necessarily a matrix Lie group) and provethat each Lie algebra is the Lie algebra of some Lie group. Under a certaincondition (simple connectedness), the Lie algebra determines its associatedLie group uniquely. A finite dimensional representation of a Lie group G isa pair ( V, Π) where V is a finite dimensional linear space and Π : G → L ( V )is a group homomorphism. Each representation ( V, π ) of a real Lie algebra h gives rise to a representation ( V, Π) of the associated Lie group such thatΠ( e tA ) = e tπ ( A ) . If g is the complexification of h and ( V, π ) is a unitaryrepresentation of g , then ( V, Π) is a unitary representation of G in the sensethat Π( A ) is a unitary operator for each A ∈ G . All his is explained in moredetail in Appendix A. The Lie algebra su (2) is the three dimensional complex Lie algebra definedby the commutation relations between its basis elements[ s x , s y ] = 2 i s z , [ s y , s z ] = 2 i s x , [ s z , s x ] = 2 i s y . (2.1)It is customary to equip su (2) with an adjoint operation that is defined by s ∗ x = s x , s ∗ y = s y , s ∗ z = s z . (2.2)A faithful unitary representation of su (2) is defined by the Pauli matrices S x := (cid:18) (cid:19) , S y := (cid:18) − ii (cid:19) , and S z := (cid:18) − (cid:19) . (2.3)It is straightforward to check that these matrices are linearly independent andsatisfy the commutation and adjoint relations (2.1) and (2.2). In particular,this shows that su (2) is well-defined. Not every set of commutation relations that one can write down defines a bona fide Liealgebra. By linearity and skew symmetry, specifying [ x i , x j ] for all i < j uniquely definesa bilinear map [ · , · ], but such a map may fail to satisfy the Jacobi identity. Similarly, it isnot a priori clear that (2.2) defines a bona fide adjoint operation, but the faithful unitaryrepresentation defined by the Pauli matrices shows that it does.
15n general, if S x , S y , S z are linear operators on some complex linear space V that satisfy the commutation relations (2.1), and hence define a represen-tation ( V, π ) of su (2), then the so-called Casimir operator is defined as C := S + S + S . The operator C is in general not an element of { π ( x ) : x ∈ su (2) } , i.e., C doesnot correspond to an element of the Lie algebra su (2). It does correspond,however, to an element of the so-called universal enveloping algebra of su (2);see Appendix A.4 below.The finite-dimensional irreducible representations of su (2) are well un-derstood. Part (a) of the following proposition follows from Theorem 22in the appendix, using the compactness of the Lie group SU(2). Parts (b)and (c), and also Proposition 7 below, follow from [Hal03, Thm 4.32] and acalculation of the Casimir operator for the representation in Proposition 7. Proposition 6 (Irreducible representations of su (2) ) Let S x , S y , S z belinear operators on a finite dimensional complex linear space V , that satisfythe commutation relations (2.1) and hence define a representation ( V, π ) of su (2) . Then: (a) There exists an inner product h · | · i on V , which is unique up to amultiplicative constant, such that with respect to this inner product therepresentation ( V, π ) is unitary. (b) If the representation ( V, π ) is irreducible, then there exists an integer n ≥ , which we call the index of ( V, π ) , such that the Casimir operator C is given by C = n ( n + 2) I . (c) Two irreducible representations
V, W of su (2) are equivalent if and onlyif they have the same index. Proposition 6 says that the finite dimensional irreducible representationsof su (2), up to isomorphism, can be labeled by their index n , which is a natu-ral number n ≥
1. We next describe what an irreducible representation withindex n looks like. In spite of the beautiful symmetry of the commutationrelations (2.1), it will be useful to work with a different, less symmetric basis { j − , j + , j } defined as j − := ( s x − i s y ) , j + := ( s x + i s y ) , and j := s z , (2.4)which satisfies the commutation and adjoint relations:[ j , j ± ] = ± j ± , [ j − , j + ] = − j , ( j − ) ∗ = j + , ( j ) ∗ = j . (2.5)The next proposition describes what an irreducible representation of su (2)with index n looks like. 16 roposition 7 (Raising and lowering operators) Let V be a finite di-mensional complex linear space that is equipped with an inner product andlet J ± , J be linear operators on V that satisfy the commutation and adjointrelations (2.5) and hence define a unitary representation ( V, π ) of su (2) . As-sume that ( V, π ) is irreducible and has index n . Then V has dimension n + 1 and there exists an orthonormal basis { φ ( − n/ , φ ( − n/ , . . . , φ ( n/ } such that J φ ( k ) = kφ ( k ) ,J − φ ( k ) = p ( n/ − k + 1)( n/ k ) φ ( k − ,J + φ ( k ) = p ( n/ − k )( n/ k + 1) φ ( k + 1) (2.6) for k = − n/ , − n/ , . . . , n/ , with the conventions J − φ ( − n/
2) := 0 and J + φ ( n/
2) := 0 . We see from (2.6) that φ ( k ) is an eigenvector of J with eigenvalue k ,and that the operators J ± maps such an eigenvector into an eigenvectorwith eigenvalue k ±
1, respectively. In view of this, J ± are called raising and lowering operators, or also creation and annihilation operators. It isinstructive to see how this property of J ± follows rather easily from thecommutation relations (2.5). Indeed, if φ ( k ) is an eigenvector of J witheigenvalue k , then the commutation relations imply that J J + φ ( k ) = (cid:0) J + J + [ J , J + ] (cid:1) φ ( k ) = (cid:0) J + J + J + (cid:1) φ ( k ) = ( k + 1) J + φ ( k ) , which shows that J + φ ( k ) is a (possibly zero) multiple of φ ( k + 1). Theconcept of raising and lowering operators can be generalized to other Liealgebras. The Lie algebra su (1 ,
1) is defined by the commutation relations[ t x , t y ] = 2 i t z , [ t y , t z ] = − i t x , [ t z , t x ] = 2 i t y . (2.7)Note that this is the same as (2.1) except for the minus sign in the secondequality. A faithful representation is defined by the matrices T x := (cid:18) − (cid:19) , T y := (cid:18) ii (cid:19) , T z := (cid:18) − (cid:19) . (2.8)17t is customary to equip su (1 ,
1) with an adjoint operation such that t ∗ x = t x , t ∗ y = t y , t ∗ z = t z . (2.9)Note however, that the matrices in (2.8) are not self-adjoint and hence do notdefine a unitary representation of su (1 , su (1 ,
1) are infinite dimensional. In a given representationof su (1 , Casimir operator is defined as C := ( T x ) − ( T y ) − ( T z ) . (2.10)Again, it is useful to introduce raising and lowering operators, defined as k := t x and k ± := ( t y ± i t z ) , which satisfy the commutation and adjoint relations[ k , k ± ] = ± k ± , [ k − , k + ] = 2 k , ( k − ) ∗ = k + , ( k ) ∗ = k , (2.11)The following proposition is rewritten from [Nov04, formulas (8) and (9)],where this is stated without proof or reference. The constant r > Bargmann index [Bar47, Bar61].
Proposition 8 (Representations of su (1 , ) For each real constant r > , there exists an irreducible unitary representation of su (1 , on a Hilbertspace with orthonormal basis { φ (0) , φ (1) , . . . } on which the operators K , K ± act as K φ ( k ) = ( k + r ) φ ( k ) ,K − φ ( k ) = 1 { k ≥ } p k ( k − r ) φ ( k − ,K + φ ( k ) = p ( k + 1)( k + 2 r ) φ ( k + 1) . (2.12) In this representation, the Casimir operator is given by C = r ( r − I . In what follows, we will need one more representation of su (1 , α > f : [0 , ∞ ) → R : K − f ( z ) = z ∂ ∂z f ( z ) + α ∂∂z f ( z ) , K + f ( z ) = zf ( z ) , K f ( z ) = z ∂∂z f ( z ) + αf ( z ) . (2.13) Since su (1 ,
1) is simple, all representations are faithful. As explained in Subsection A.3,each Lie algebra is the Lie algebra of a unique simply connected Lie group. In the case of su (1 , H of the Lie group SU(1 ,
1) (the latter itself not beingsimply connected). By Theorem 18 in the appendix, each representation of su (1 ,
1) givesrise to a representation of H . Since H is not compact, the existence of a finite dimensionalunitary representation would now contradict Lemma 23 in the appendix. su (1 , C , i.e.,[ K , K ± ] = ±K ± and [ K − , K + ] = 2 K , (2.14)and hence define a representation of su (1 , C = α ( α − I and hence theBargmann index is r = α/ α > f : N → R : K − f ( x ) = xf ( x − ,K + f ( x ) = ( α + x ) f ( x + 1) ,K f ( x ) = ( α + x ) f ( x ) . (2.15)One can check that these operators satisfy the commutation relations[ K ± , K ] = ± K ± and [ K + , K − ] = 2 K . (2.16)This is similar to (2.11), except that the order of the elements inside thecommutator is reversed. In view of the remarks at the end of Section 2.1,this means that the operators K , K ± define a representation of the conjugateLie algebra associated with su (1 , α .A complete classification of all irreducible representations of su (1 , The
Heisenberg algebra h is the three dimensional complex Lie algebra definedby the commutation relations[ a − , a + ] = a , [ a − , a ] = 0 , [ a + , a ] = 0 . (2.17)It is customary to equip h with an adjoint operation that is defined by( a ± ) ∗ = ± a ± , ( a ) ∗ = a . (2.18) The monumental encyclopedic book [VK91] is written in a style that some readersmay need to get used to, since it does not use the usual theorem-proof layout but ratherstates an enormous amount of facts in the main text while leaving a lot of detail to befilled in by the reader.
Schr¨odinger representation of h is defined by A − f ( x ) = ∂∂x f ( x ) , A + f ( x ) = xf ( x ) , A f ( x ) = f ( x ) , (2.19)which are interpreted as operators on the Hilbert space L ( R , d x ) of com-plex functions on R that are square integrable with respect to the Lebesguemeasure. Note in this representation, A is the identity operator. Any rep-resentation of h with this property is called a central representation. TheSchr¨odinger representation is a unitary representation, i.e., A − is skew sym-metric and A + and A are self-adjoint, viewed as linear operators on theHilbert space L ( R , d x ).Since iA − and A + are self-adjoint, by Stone’s theorem, one can definecollections of unitary operators ( U − t ) t ∈ R and ( U + t ) t ∈ R by U − s := e tA − and U + t := e itA + . (2.20)These operators form one-parameter groups in the sense that U ± = I and U ± s U ± t = U ± s + t ( s, t ∈ R ). Note that we have a factor i in the definition of U + t but not in the definition of U − s , because A + is self-adjoint but A − is skewsymmetric. The commutation relations (2.17) lead, at least formally, to thefollowing commutation relation between U − s and U + t U − s U + t = e ist U + t U − s ( s, t ∈ R ) . (2.21)Indeed, for small ε , we have U − εs U + εt = (cid:0) I + εsA − + ε s ( A − ) + O ( ε ) (cid:1)(cid:0) I + iεtA + − ε t ( A + ) + O ( ε ) (cid:1) = I + εsA − + ε s ( A − ) + iεtA + − ε t ( A + ) + iε stA − A + + O ( ε )= I + εsA − + ε s ( A − ) + iεtA + − ε t ( A + ) + iε stA + A − + iε st [ A − , A + ] + O ( ε )= (cid:0) iε st + O ( ε ) (cid:1) U + εt U − εs + O ( ε ) . (2.22)The commutation relation (2.21) then follows formally by writing U − s U + t = ( U − s/n ) n ( U + t/n ) n = (cid:0) in − st + O ( n − ) (cid:1) n ( U + t/n ) n ( U − s/n ) n −→ n →∞ e ist U + t U − s . (2.23) More generally, the center of a Lie algebra g is the linear space c := { c ∈ g : [ x , c ] =0 ∀ x ∈ g } . A central representation of a Lie algebra is then a representation ( V, π )such that for each c ∈ c , there exists a c ∈ C such that π ( c ) = cI . Note that with thisdefinition, if ( V, π ) is a faithful central representation of h , then we can always “normalize”it by multiplying π with a constant so that π ( a ) = I . Stone-von Neumann theorem states that all unitary, central representa-tions of the Heisenberg algebra that satisfy (2.21) are equivalent [Ros04]. If V is a linear space and V , . . . , V n are linear subspaces of V such that everyelement v ∈ V can uniquely be written as v = v + · · · + v n with v i ∈ V i , then we say that V is the direct sum of V , . . . , V n and write V = V ⊕ · · · ⊕ V n . If Ω , Ω are finite sets and C Ω denotes the linear spaceof all functions f : Ω i → C , then we have the natural isomorphism C Ω ⊎ Ω ∼ = C Ω ⊕ C Ω , where Ω ⊎ Ω denotes the disjoint union of Ω and Ω .If g , . . . , g n are Lie algebras, then we equip g ⊕· · ·⊕ g n with the structureof a Lie algebra by putting, for x i , y i ∈ g i , (cid:2) x + · · · + x n , y + · · · + y n (cid:3) := [ x i , y i ] + · · · + [ x n , y n ] . (2.24)Note that this has the effect that elements of diffent Lie algebras g , . . . , g n mutually commute. In particular, if { x , x , x } and { x , x , x } are basesfor g and g , respectively, then { x , x , x , x , x , x } is a basis for g ⊕ g and [ x ki , x lj ] = 0 whenever i = j .By definition, a bilinear map of two variables is a function that is linearin each of its arguments. If V and W are finite dimensional linear spaces,then their tensor product is a linear space V ⊗ W together with a bilinearmap V × W ∋ ( v, w ) v ⊗ w ∈ V ⊗ W that has the property:If F is another linear space and b : V × W → F is bilinear, thenthere exists a unique linear map b : V ⊗ W → F such that b ( v ⊗ w ) = b ( v, w ) ( v ∈ V, w ∈ W ) . { e (1) , . . . , e ( n ) } and { f (1) , . . . , f ( m ) } are bases for V and W , then one canprove that (cid:8) e ( i ) ⊗ f ( j ) : 1 ≤ i ≤ n, ≤ j ≤ m (cid:9) (2.25)is a basis for V ⊗ W . In particular, this means that one has the naturalisomorphism C Ω × Ω ∼ = C Ω ⊗ C Ω . (2.26)If A ∈ L ( V ) and B ∈ L ( V ), then one defines A ⊗ B ∈ L ( V ⊗ W ) by( A ⊗ B )( v ⊗ w ) := ( Av ) ⊗ ( Bw ) . (2.27)We note that not every element of V ⊗ W is of the form v ⊗ w for some v ∈ V and w ∈ W . Nevertheless, since the right-hand side of (2.27) is bilinear in v and w , the defining property of the tensor product tells us that this formulaunambiguously defines a linear operator on V ⊗ W .One can check that the notation A ⊗ B is good notation in the sense thatthe space L ( V ⊗ W ) together with the bilinear map ( A, B ) A ⊗ B is arealization of the tensor product L ( V ) ⊗ L ( W ). Thus, one has the naturalisomorphism L ( V ⊗ W ) ∼ = L ( V ) ⊗ L ( W ) . If V and W are equipped with inner products, then we equip V ⊗ W withan inner product by putting h v ⊗ w | η ⊗ ξ i := h v | η ih w | ξ i , (2.28)which has the effect that if { e (1) , . . . , e ( n ) } and { f (1) , . . . , f ( m ) } are or-thonormal bases for V and W , then the basis for V ⊗ W in (2.25) is alsoorthonormal. Again, one needs the defining property of the tensor productto see that (2.28) is a good definition.If V, W are representations of Lie algebras g , h , respectively, then we cannaturally equip the tensor product V ⊗ W with the structure of a represen-tation of g ⊕ h by putting( x + y )( v ⊗ w ) := ( x v ) ⊗ ( y w ) . (2.29)Again, since the right-hand side is bilinear, using the defining property ofthe tensor product, one can see that this is a good definition.Let V , V be representations of some Lie algebra g , and let W , W berepresentations of another Lie algebra h . Let φ : V → V and ψ : W → W be intertwiners. Then one can check that φ ⊗ ψ : V ⊗ W → V ⊗ W (2.30)22s also an intertwiner.If h , . . . , h n are n copies of the Heisenberg algebra, and a − i , a + i , a i arebasis elements of h i that satisfy the commutation relations (2.17), then abasis for h ⊕ · · · ⊕ h n is formed by all elements a ± i , a i with i = 1 , . . . , n , andthese satisfy [ a − i a + j ] = δ ij a i and [ a ± i , a j ] = 0 . Since the center of h ⊕· · ·⊕ h n is spanned by the elements a i with i = 1 , . . . , n ,a central representation of h ⊕ · · · ⊕ h n must map all these elements tomultiples of the identity. In particular, a central representation of h ⊕· · ·⊕ h n is never faithful (unless n = 1). The Lie algebra h ( n ) is the 2 n +1 dimensionalLie algebra with basis elements a ± i ( i = 1 , . . . , n ) and a , which satisfy thecommutation relations[ a − i a + j ] = δ ij a and [ a ± i , a ] = 0 . A central representation of h ( n ) is a representation ( V, π ) such that π ( a ) = I . The Schr¨odinger representation of the “ n -dimensional” Heisenberg algebrais the central representation of h ( n ) on L ( R n , d x ) given by A − f ( x ) = ∂∂x i f ( x ) and A + f ( x ) := x i f ( x ) . (2.31) After our excursion into the theory of Lie algebras, we return to our maintopic, which is the algebraic approach to Markov process duality. We recallfrom Lemma 3 that if a Markov generator L can be written in terms of“building blocks” A i that each have a dual B i with respect to some dualityfunction D , then also L has a dual ˆ L with respect to D . As mentioned atthe end of Section 1.3, it may be a good idea to choose the A i ’s so that theydefine a representation of some Lie algebra. The next proposition says thatin such a situation, other, equivalent representations of the same Lie algebramay lead to dual Markov processes.Recall the definition of a conjugate Lie algebra g from Section 2.1. If Y , . . . , Y n are matrices that define a representation of g , then their adjoints Y † , . . . , Y † n define a representation of the original Lie algebra g . Proposition 9 (Intertwiners as duality functions)
Let L be the gener-ator of a Markov process with finite state space Ω . Let X , . . . , X n be linearoperators on C Ω that form a representation of some Lie algebra g . Assumethat L can be written as a linear combination of products of the operators , . . . , X n L = X ( i ,...,i k ) ∈I r i ,...,i k X i · · · X i k , (3.1) where I is some finite set whose elements are sequences ( i , . . . , i k ) with k ≥ and ≤ i m ≤ n for each m . Assume that Y , . . . , Y n are linear operators on C ˆΩ that define a representation of the conjugate Lie algebra g . Assume thatthe representations of g defined by Y † , . . . , Y † n and X , . . . , X n are equivalent.Then there is a bijective intertwiner D , i.e., X i D = DY † i for each i , and L is dual w.r.t. the duality function D to the operator ˆ L := X ( i ,...,i k ) ∈I r i ,...,i k Y i k · · · Y i . (3.2) Proof
By definition, two representations are equivalent if and only if thereexists a bijective intertwiner. The fact that L is dual to the operator in (3.2)is then immediate from Lemma 3.At first sight, it may seem unlikely that Proposition 9 could be of muchuse. Even if we can write a generator in terms of a basis of a representationof some Lie algebra g , and we also find some representation of the conjugateLie algebra g , we still have to be lucky in the sense that the representationsof g defined by Y † , . . . , Y † n and X , . . . , X n are equivalent, and there is noguarantee that the operator in (3.2) is a Markov generator. Nevertheless,in what follows, we will see that Proposition 9 can help us find nontrivialdualities. In the next subsection, we demonstrate this on the operator L from (1.8), which is the generator of a Wright-Fisher diffusion with selection. In Subsection 1.3, we have seen that the operator L from (1.8) can as in(1.10) be written in terms of the “building blocks” A ± from (1.9). As wehave seen in (1.16), these operators satisfy[ A − , A + ] = I, (3.3)and hence define a central representation of the Heisenberg algebra h , asdefined in Subsection 2.5.It will be convenient to find a way of writing L in a more symmetric waythan in (1.10). To this aim, we change the definitions of A ± to A − f ( x ) := − √ s ∂∂x f ( x ) and A + f ( x ) := √ sxf ( x ) , (3.4)24hich again satisfy (3.3), and we write L in terms of these new buildingblocks as L = − A + ( √ s − A + ) A − ( √ s − A − ) . (3.5)We observe from (3.3) that setting B − := A + and B + := A − defines operatorssuch that [ B − , B + ] = − I , i.e., B − , B + define a central representation of theconjugate Heisenberg algebra h .We recall from Section 2.5 that the Stone-von Neumann theorem statesthat, more or less, all central representations of the Heisenberg algebra areequivalent. In view of this and Proposition 9, we may expect that the oper-ator ˆ L = − ( √ s − B − ) B − ( √ s − B + ) B + (3.6)is dual to L with respect to some (so far unknown) duality function D .(Here (3.6) is obtained from (3.5) by replacing A ± by B ± and reversing theorder of the factors.) Since B ± = A ∓ , we observe that in fact ˆ L = L , so ourcalculations lead us to suspect that the Wright-Fisher diffusion with selectionparameter s > D . This function must satisfy − √ s ∂∂x D ( x, y ) = A − D ( · , y )( x ) = B − D ( x, · )( y ) = √ syD ( x, y ) , (3.7)which says that ∂∂x D ( x, y ) = − syD ( x, y ) and leads to the requirement that D ( x, y ) = D (0 , y ) e − syx . In a similar way, the requirement A + D = DB + yields D ( x, y ) = D ( x, e − syx and in particular D (0 , y ) = D (0 , D ( x, y ) = e − sxy (cid:0) x, y ∈ [0 , (cid:1) , (3.8)and we conclude that the Wright-Fisher diffusion with selection parameter s > L = L and L is a Markov generator, this turned outright as well. 25 emark It is possible to “discover” the moment dual (1.15) of the Wright-Fisher duality along similar lines as we have discovered its self-duality here,by considering a suitable representation of the conjugate Heisenberg algebra h on functions f : N → R and applying Propositions 9 and 10. Such aderivation is less natural, however, since it requires choosing a rather peculiarrepresentation of h that more or less has the duality function from (1.11)tacitly built into it. In the previous subsection, just before (3.6) we appealed to Proposition 9.In doing so, we cheated in the sense that the operators A ± from (3.4) do notact on a finite-dimensional space. The most obvious consequence of this isthat it is not clear how the adjoint operators B † i from Proposition 9 shouldbe defined. Closely related to this is that in the infinite dimensional setting,it is not immediately clear that duality functions define intertwiners and viceversa. In this subsection we show that these difficulties can be resolved byintroducing a suitable inner product on the spaces of complex functions onΩ and ˆΩ, respectively.Assume that X , . . . , X n and Y , . . . , Y n are linear operators on L -spaces L (Ω , µ ) and L ( ˆΩ , ν ), respectively, that define representations of a Lie al-gebra g and its conjugate g . Let Y ∗ i denote the adjoint of Y i with respectto the inner product on L ( ˆΩ , ν ). Assume that Φ : L ( ˆΩ , ν ) → L (Ω , µ ) is alinear operator of the formΦ g ( x ) = Z g ( y ) D ( x, y ) ν (d y ) , (3.9)for some function D : Ω × ˆΩ → C such that the expressions in (3.10) beloware well-defined. Proposition 10 (Intertwiners and duality functions)
The operator Φ is an intertwiner of the representations defined by X , . . . , X n and Y ∗ , . . . , Y ∗ n ,i.e., X i Φ = Φ Y ∗ i ( i = 1 , . . . , n ) , if and only if D is a duality function, in the sense that X i D ( · , y )( x ) = Y i D ( x, · )( y ) ( i = 1 , . . . , n ) (3.10) for a.e. x, y with respect to the product measure µ ⊗ ν . roof We observe that Z f ( x ) µ (d x ) Z g ( y ) ν (d y ) X i D ( · , y )( x ) = Z g ( y ) ν (d y ) h f | X i D ( · , y ) i µ = Z g ( y ) ν (d y ) h X ∗ i f | D ( · , y ) i µ = Z X ∗ i f ( x ) µ (d x ) Z g ( y ) ν (d y ) D ( x, y )= h X ∗ i f | Φ g i µ = h f | X i Φ g i µ and Z f ( x ) µ (d x ) Z g ( y ) ν (d y ) Y i D ( x, · )( y ) = Z f ( x ) µ (d x ) h g | Y i D ( x, · ) i ν = Z f ( x ) µ (d x ) h Y ∗ i g | D ( x, · ) i ν = Z f ( x ) µ (d x ) Z Y ∗ i g ( y ) ν (d y ) D ( x, y )= h f | Φ Y ∗ i g i µ . Since this holds for all f, g , the statement follows.
Remark
Proposition 10 allows us to obtain an intertwiner from a dualityfunction. Conversely, if Φ : L ( ˆΩ , ν ) → L (Ω , µ ) is a bounded linear operator,then setting ∆( f ⊗ g ) := Z f ( x )Φ g ( x ) µ (d x )defines a linear form on the linear span of all functions of the form f ⊗ g . If∆ is bounded, then it can uniquely be extended to a bounded linear formon L (Ω , µ ) ⊗ L ( ˆΩ , ν ) ∼ = L (Ω × ˆΩ , µ ⊗ ν ) , so that by the Riesz representation theorem there exists a D ∈ L (Ω × ˆΩ , µ ⊗ ν ) such that ∆( f ⊗ g ) := Z f ( x ) D ( x, y ) g ( x ) µ (d x ) ν (d y ) , proving that Φ is of the form (3.9) (although there is no guarantee that D ( · , y ) and D ( x, · ) are in the domains of X i and Y i , resp., if these areunbounded operators). Using Cauchy-Schwarz, it is easy to see that | ∆( f ⊗ g ) | ≤ k Φ k k f ⊗ g k , proving that∆ is bounded on functions of the form f ⊗ g . Nevertheless, ∆ may fail to be bounded onthe linear span of such functions. A counterexample is Ω = ˆΩ = [0 , µ = ν = Lebesguemeasure, and Φ the identity map, which gives ∆( F ) = R F ( x, x ) d x . Since the Lebesguemeasure on the diagonal { ( x, y ) : x = y } does not have a density w.r.t. µ ⊗ ν , this doesnot correspond to a bounded linear form on L (Ω × ˆΩ , µ ⊗ ν ). .3 The symmetric exclusion process In this subsection, we demonstrate Proposition 9 on a simple example, whichinvolves the simple exclusion process and the Lie algebra su (2). In the end,we find a self-duality that is not entirely trivial, but also not very useful. Thepresent subsection serves mainly as a warm-up for Subsection 3.4 where wewill replace su (2) by su (1 , S be a finite set and let r : S × S → [0 , ∞ ) be a function that issymmetric in the sense that r ( i, j ) = r ( j, i ). Consider the Markov processwith state space Ω = { , } S and generator Lf ( x ) := X ij r ( i, j )1 { ( x i ,x j )=(1 , } (cid:8) f ( x − δ i + δ j ) − f ( x ) (cid:9) , (3.11)where δ i ∈ Ω is defined as δ i ( j ) := 1 { i = j } . Then L is the generator of a symmetric exclusion process or SEP . We define operators J ± i and J i by J − i f ( x ) := 1 { x i =0 } f ( x + δ i ) , J + i f ( x ) := 1 { x i =1 } f ( x − δ i ) , and J i f ( x ) := ( x i − ) f ( x ) . (3.12)It is straightforward to check that[ J i , J ± j ] = ± δ ij J ± i and [ J − i , J + j ] = − δ ij J i . (3.13)It follows that the operators J ± i and J i define a representation of a Liealgebra that consists of a direct sum of copies of su (2), with one copy foreach site i ∈ S . We can write the generator L of the symmetric exclusionprocess in terms of the operators J ± i and J i as L = X { i,j } r ( i, j ) (cid:2) J − i J + j + J − j J + i + 2 J i J j − I (cid:3) , (3.14)where we are summing over all unordered pairs { i, j } . We observe that theoperators K ± i := J ± i , and K i := − J i (3.15)satisfy the same commutation relations as J ± i and J i , except that each com-mutation relation gets an extra minus sign. This shows that the operators K ± i and K i define a representation of the conjugate Lie algebra su (2). More-over, we can alternatively write the generator in (3.14) as L = X { i,j } r ( i, j ) (cid:2) K + j K − i + K + i K − j + 2 K j K i − I (cid:3) . (3.16)28e recall from Subsection 2.3 that two irreducible representations of su (2)with the same dimension are necessarily equivalent. In view of this, we con-jecture that there should exist an intertwiner D , unique up to a multiplicativeconstant, such that J ± i D = D ( K ± i ) † and J i D = D ( K i ) † for all i . By the gen-eral principle in Proposition 9, such an intertwiner is a self-duality functionfor the symmetric exclusion process.We observe that all our operators act on the space of all complex functionson { , } S , which in view of (2.26) is given by C { , } S ∼ = O i ∈ S C { , } . (3.17)For example, if S = { , , } consists of only three sites, then in line with(2.29), J = J ⊗ I ⊗ I, J = I ⊗ J ⊗ I, and J = I ⊗ I ⊗ J , and similarly for J ± , J ± , and J ± . Here J − f = (cid:18) (cid:19) (cid:18) f (1) f (0) (cid:19) = (cid:18) f (1) (cid:19) ,J + f = (cid:18) (cid:19) (cid:18) f (1) f (0) (cid:19) = (cid:18) f (0)0 (cid:19) ,J f = (cid:18) − (cid:19) (cid:18) f (1) f (0) (cid:19) = (cid:18) f (1) − f (0) (cid:19) . (3.18)We equip C { , } and the space in (3.17) with the standard inner product,which has the consequence that A ∗ = A † and( J − i ) ∗ = J + i , ( J + i ) ∗ = J − i , and ( J i ) ∗ = J i , showing that the operators J ± i and J i define a unitary representation of ourLie algebra.According to the general principle (2.30), to find an intertwiner D whichacts on the product space (3.17), it suffices to find an intertwiner for the two-dimensional space corresponding to a single site, and then take the productover all sites. Setting Q := (cid:18) (cid:19) , it is straightforward to check that J ± Q = QJ ∓ = Q ( K ± ) † and J Q = Q ( − J ) = Q ( K ) † . S = { , , } consists of only three sites, then in viewof (2.30) D := Q ⊗ Q ⊗ Q satisfies J ± i D = D ( K ± i ) † and J i D = D ( K i ) † ( i = 1 , , Q ( x i , y j ) = 1 { x i = y i } andhence the self-duality function of the symmetric exclusion process that wehave found is D ( x, y ) = Y i ∈ S { x i = y i } (cid:0) x, y ∈ { , } S (cid:1) . Let S be a finite set and let α : S → (0 , ∞ ) and q : S × S → [0 , ∞ )be functions such that q ( i, j ) = q ( j, i ) and q ( i, i ) = 0 for each i ∈ S . Bydefinition, the Brownian energy process or BEP with parameters α, q is thediffusion process ( Z t ) t ≥ with state space [0 , ∞ ) S and generator L := X i,j ∈ S q ( i, j ) (cid:2) ( α j z i − α i z j )( ∂∂z j − ∂∂z i ) + z i z j ( ∂∂z j − ∂∂z i ) (cid:3) . (3.19)This diffusion has the property that P i Z t ( i ) is a preserved quantity. Thedrift part of the generator is zero if z i = λα i for some λ >
0. If z i /α i > z j /α j ,then the drift has the tendency to make z i smaller and z j larger.In analogy with (2.13), we define operators acting on smooth functions f : [0 , ∞ ) S → R by: K − i f ( z ) = z i ∂ ∂z i f ( z ) + α i ∂∂z i f ( z ) , K + i f ( z ) = z i f ( z ) , K i f ( z ) = z i ∂∂z i f ( z ) + α i f ( z ) . (3.20)By (2.14), these operators satisfy the commutation relations[ K i , K ± j ] = ± δ ij K ± i and [ K − i , K + j ] = 2 δ ij K i . It follows that these operators define a representation of the Lie algebra M i ∈ S g i , where each g i is a copy of su (1 , C [0 , ∞ ) S ∼ = ( C [0 , ∞ ) ) ⊗ S , | S | copies of C [0 , ∞ ) .We can express the generator (3.19) of the Brownian energy process interms of the operators from (3.20) as L = X i,j ∈ S q ( i, j ) (cid:2) K + i K − j + K − i K + j − K i K j + α i α j (cid:3) . (3.21)Note that this is very similar to the expression for the symmetric exclusionprocess in (3.14).We define operators acting on functions f : N S → R by K − i f ( x ) = x i f ( x − δ i ) ,K + i f ( x ) = ( α i + x i ) f ( x + δ i ) ,K i f ( x ) = ( α i + x i ) f ( x ) . (3.22)In view of (2.16), these operators define a representation of the conjugateof our Lie algebra. It turns out that the conjugate of this representation isequivalent to the representation defined by the operators in (3.20). This is anontrivial statement that depends crucially on the fact that the parameters α i are the same in both expressions. Indeed, we have seen in Subsection 2.4that α is twice the Bargmann index and that representations with a differentBargmann index have a different Casimir operator and hence are not equiv-alent. Letting Φ denote the intertwiner of K ± i and ( K ± i ) † , we can write Φ inthe form (3.9), where by Proposition 10 D is a duality function. Similar towhat we did at the end of Subsection 3.3, we will choose a duality functionof product form: D ( z, x ) = Y i ∈ S Q ( z i , x i ) ( z ∈ [0 , ∞ ) S , x ∈ N S ) , (3.23)where Q is a duality function for the single-site operators, i.e., K ± Q ( · , x )( z ) = K ± Q ( z, · )( x ) , K Q ( · , x )( z ) = K Q ( z, · )( x ) (3.24)( z ∈ [0 , ∞ ) , x ∈ N ). It turns out that Q ( z, x ) := Γ( α + x )Γ( α ) z x = z x x − Y k =0 ( α + k ) . (3.25)does the trick. This may look a bit complicated but the form of this dualityfunction can in fact quite easily be guessed from the inductive relation zQ ( z, x ) = K + Q ( · , x )( z ) = K + Q ( z, · )( x ) = ( α + x ) Q ( z, x + 1) . L = X i,j ∈ S q ( i, j ) (cid:2) K − j K + i + K + j K − i − K j K i + α j α i (cid:3) . (3.26)It turns out that we are lucky in the sense that this is a Markov generator. Inview of the similarity with (3.14) (with the role of su (2) replaced by su (1 , symmetric inclu-sion process or SIP . The fact that ˆ L is a Markov generator can be seen byrewriting it asˆ L := X i,j ∈ S q ( i, j ) h α j x i (cid:8) f (cid:0) x − δ i + δ j (cid:1) − f (cid:0) x (cid:1)(cid:9) + x i x j (cid:8) f (cid:0) x − δ i + δ j (cid:1) − f (cid:0) x (cid:1)(cid:9)i . (3.27)The Markov process ( X t ) t ≥ with generator ˆ L has the property that P i X t ( i )is a preserved quantity. The terms in the generator involving the constants α j describe a system of independent random walks, where each particle at i jumps with rate α j to the site j . A reversible law for this part of the dynamicsis a Poisson field with local intensity λα i for some λ >
0. The remainingterms in the generator describe a dynamics where particles at i jump to j with a rate that is proportional to the number x ( j ) of particles at j . Thispart of the dynamics causes an attraction between particles. In the previous two subsections, we have seen that for a Markov processwhose state space is a Carthesian product of other spaces, it is often naturalto choose duality functions of product form as in (3.23). This idea does notdepend on Lie algebras and is in fact older than the use of Lie algebras induality theory.In a series of papers [LS95, LS97, Sud00], Lloyd and Sudbury have system-atically searched for dualities in a large class of interacting particle systems,which contains many well-known systems such as the voter model, contactprocess, and symmetric exclusion process. Let S be a finite set and let q : S → [0 , ∞ ) be a function such that q ( i, j ) = q ( j, i ) and q ( i, i ) = 0 for all i ∈ S . Let L = L ( a, b, c, d, e ) be the Markov generator, acting on functions32 : { , } S → R , as Lf ( x ) = X i,j ∈ S q ( i, j ) h a { ( x ( i ) ,x ( j ))=(1 , } (cid:8) f ( x − δ i − δ j ) − f ( x ) (cid:9) b { ( x ( i ) ,x ( j ))=(0 , } (cid:8) f ( x + δ i ) − f ( x ) (cid:9) c { ( x ( i ) ,x ( j ))=(1 , } (cid:8) f ( x − δ i ) − f ( x ) (cid:9) d { ( x ( i ) ,x ( j ))=(0 , } (cid:8) f ( x − δ j ) − f ( x ) (cid:9) e { ( x ( i ) ,x ( j ))=(0 , } (cid:8) f ( x + δ i − δ j ) − f ( x ) (cid:9)i . (3.28)The dynamics of the Markov process with generator L can be described bysaying that for each pair of sites i, j , the configuration of the process at thesesites makes the following transitions with the following rates:11
00 with rate aq ( i, j ) (annihilation) ,
11 with rate bq ( i, j ) (branching) ,
01 with rate cq ( i, j ) (coalecence) ,
00 with rate dq ( i, j ) (death) ,
10 with rate eq ( i, j ) (exclusion dynamics) . Note that the factor in front of a disappears since the total rate of thistransition is a ( q ( i, j ) + q ( j, i )) = aq ( i, j ). A lot of well-known interactingparticle systems fall into this class. For examplevoter model b = d = 1 , other parameters 0 , contact process b = λ, c = d = 1 , other parameters 0 , symmetric exclusion e = 1 , other parameters 0 . As we have already seen in (3.17), the class of all functions f : { , } S → R can be written as the tensor product R { , } S ∼ = O i ∈ S R { , } , with one ‘factor’ R { , } for each site i ∈ S . Moreover, duality functions D on the space { , } S × { , } S can be viewed as matrices corresponding tolinear operators that act on R { , } S . Based on various arguments that are notvery important at this point, Lloyd and Sudbury decided to look for dualityfunctions of product form D ( x, y ) = Y i ∈ S Q ( x i , y i ) , (3.29)33here Q is a 2 × Q , Lloyd and Sudbury find a rich class of dualities for matrices ofthe form (cid:18) Q q (0 , Q q (0 , Q q (1 , Q q (1 , (cid:19) = (cid:18) q (cid:19) , (3.30)where q ∈ R \{ } is a constant. This choice of Q yields the duality function D q ( x, y ) := Y i ∈ S Q q ( x i , y i ) = q P i ∈ S x i y j (cid:0) x, y ∈ { , } S (cid:1) . (3.31)In particular, setting q = 0 yields D ( x, y ) = 1 { P i ∈ S x i y j =0 } , which corresponds to the well-known additive systems duality , while q = − cancellative systems duality . For these special values of q , theduality can in fact be upgraded to a pathwise duality as in Subsection 1.4,using a construction in terms of open paths in a graphical representation. In-terestingly, for other values of q , there seems to be no pathwise interpretationof the duality with duality function D q .We cite the following theorem from [LS95, Sud00]. A somewhat more gen-eral version of this theorem which drops the symmetry assumption q ( i, j ) = q ( j, i ) at the cost of replacing (3.32) by a somewhat more complicated set ofconditions can be found in [Swa06, Appendix A in the version on the ArXiv]. Theorem 11 (q-duality)
The generators L ( a, b, c, d, e ) and L ( a ′ , b ′ , c ′ , d ′ , e ′ ) from (3.28) are dual with respect to the duality function D q from (3.31) ifand only if a ′ = a +2 qγ, b ′ = b + γ, c ′ = c − (1+ q ) γ, d ′ = d + γ, e ′ = e − γ, (3.32) where γ := ( a + c − d + qb ) / (1 − q ) . In Subsection 1.2, when we introduced Markov process duality, we also de-fined the similar concept of intertwining of Markov processes. So far, we havenot discussed this second concept very much, but it turns out that the twoare closely related. In particular, as Lloyd and Sudbury already observed[LS95, Sud00], there is a close connection between q-duality and thinningrelations. To explain this, we start with a general principle, that says that iftwo Markov processes are both dual to a third Markov process, then we canexpect an intertwining relation between the first two processes.34 emma 12 (Duality and intertwining)
Let Ω and ˆΩ be finite sets, andlet L i : R Ω → R Ω , ˆ L : R ˆΩ → R ˆΩ , and D i : R ˆΩ → R Ω be linear operators suchthat L i D i = D i ˆ L † ( i = 1 , . (3.33) Assume that D and D are invertible. Then L ( D D − ) = ( D D − ) L . (3.34) Proof
This follows by writing D − L D = ˆ L † = D − L D .We have seen that for interacting particle systems, there are good reasonsto look for duality functions of product form as in (3.29). Likewise, it isnatural to look for intertwining probability kernels of product form. If thestate space is of the form { , } S , this means that we are looking for kernelsof the form K ( x, y ) = Y i ∈ S M ( x i , y i ) (cid:0) x, y ∈ { , } S (cid:1) , where M is a probability kernel on { , } . If we moreover require that M (0 ,
0) = 1 (which is natural for interacting particle systems for which theall zero state is a trap), then there is only a one-parameter family of suchkernels. For p ∈ [0 , M p be the probability kernel on { , } given by M p = (cid:18) M p (0 , M p (0 , M p (1 , M p (1 , (cid:19) := (cid:18) − p p (cid:19) , (3.35)and let K p ( x, y ) := Y i ∈ S M p ( x i , y i ) (cid:0) x, y ∈ { , } S (cid:1) (3.36)the corresponding kernel on { , } S of product form. We can interpret aconfiguration of particles, where x i = 1 if the site i is occupied by a particle,and x i = 0 otherwise. Then K p is a thinning kernel that independently foreach site throws away particles with probability 1 − p or keeps them withprobability p . It is easy to see that K p K p ′ = K pp ′ , i.e., first thinning with p and then with p ′ is the same as thinning with pp ′ .There is a close relation between Lloyd and Sudbury’s duality function D q from (3.31) and thinning kernels of the form (3.36). We claim that D q D − q ′ = K p with p = 1 − q − q ′ ( q, q ′ ∈ R , q ′ = 1) . (3.37)35ince both D q and K p are of product form, i.e., D q = O i ∈ S Q q and K p = O i ∈ S M p with Q q and M p as in (3.30) and (3.35), it suffices to check that Q q Q − q ′ = M p with p = 1 − q − q ′ . Indeed, one can check that Q − q = (cid:18) q (cid:19) − = (1 − q ) − (cid:18) − q − (cid:19) ( q = 1) , and that Q q Q − q ′ = (1 − q ′ ) − (cid:18) q (cid:19) (cid:18) − q ′ − (cid:19) = (cid:18) q − q ′ − q ′ − q − q ′ (cid:19) = M p , as claimed. Proposition 13 (Thinning and q -duality) Let L and L be generatorsof Markov processes with state space { , } S . Assume that there exists anoperator ˆ L such that L i D q i = D q i ˆ L † ( i = 1 ,
2) (3.38) for some q , q ∈ R such that q = 1 and p := (1 − q ) / (1 − q ) ∈ [0 , . Then L K p = K p L . (3.39) Proof
This follows from (3.37) and Lemma 12. Note that in general, thereis no guarantee that the operator D D − from Lemma 12 is a probabilitykernel. In a way, Proposition 13 explains why the q-duality function D q isnatural, because it is closely linked to the natural concept of thinning. In this section, we demonstrate Lloyd-Sudbury theory on the example of the biased voter model with selection parameter s >
0, which is the interactingparticle system with generator L ( a, b, c, d, e ) = L (0 , s, , ,
0) =: L bias .
36e apply Theorem 11 to find q -duals of the biased voter model. For simplic-ity, we restrict ourselves here to dual generators of the form L ( a ′ , b ′ , c ′ , d ′ , e ′ )with a ′ = 0, which means that we must choose the parameter q as q = 0 or q = (1 + s ) − . For q = 0 we find the dual generator L ( a ′ , b ′ , c ′ , d ′ , e ′ ) = L (0 , s, , ,
1) =: L braco , which describes a system of branching and coalescing random walks withbranching parameter s . For q = (1 + s ) − , we find a self-duality, i.e., in thiscase L ( a ′ , b ′ , c ′ , d ′ , e ′ ) = L ( a, b, c, d, e ) = L bias .Since L bias and L braco are both q -dual to ˆ L = L bias , Proposition 13 tellsus that there is a thinning relation between biased voter models and systemsof branching and coalescing random walks of the form L bias K p = K p L braco with p = 1 − (1 + s ) − − s s . As explained in Subsection 3.6, this implies that if we start a biased votermodel ( X t ) t ≥ and a system of branching and coalescing random walks ( Y t ) t ≥ in initial states µ bias t and µ braco t denote the laws of X t and Y t , then µ bias0 K p = µ braco0 implies µ bias t K p = µ braco t ( t ≥ . In other words, the following two procedures are equivalent:(i) Evolve a particle configuration for time t according to biased votermodel dynamics, then thin with p .(ii) Thin a particle configuration with p , then evolve for time t accordingto branching coalescing random walk dynamics.In particular, if we start X in the initial state X ( i ) = 1 for all i ∈ S , thenbecause of the nature of the voter model, we will have X t ( i ) = 1 for all i ∈ S and t ≥
0. Applying the thinning relation now shows that product measurewith intensity p is an invariant law for branching coalescing random walkdynamics. Thus, there is a close connection between:I. q -duality,II. thinning relations,III. invariant laws of product form.Although Lloyd-Sudbury theory is restricted to Markov processes with statespace of the form { , } S , many other dualities, including the self-dualityof the Wright-Fisher diffusion from Section 3.1, can be derived from Lloyd-Sudbury duals by taking a suitable limit [Swa06].37 .8 Time-reversal and symmetry In this subsection we present an idea from [GKRV09], which says that non-trivial dualities can sometimes be found by starting from a “trivial” dualitywhich is based on time reversal, and then using a symmetry of the modelto transform such a duality into a nontrivial one. Although Lie algebrasare not strictly needed in this approach, writing generators in terms of thebasis elements of a representation of a Lie algebra can help finding suitablesymmetries.Each irreducible Markov process with finite state space Ω has a uniqueinvariant measure, i.e., a probability measure µ such that µL = 0 or equivalently µP t = µ ( t ≥ , where L denotes the generator and ( P t ) t ≥ the semigroup of the Markovprocess. Irreducibility implies that µ ( x ) > x ∈ Ω. Letting ( X t ) t ∈ R denote the stationary process, we see that the semigroup ( ˜ P t ) t ≥ of the time-reversed process is given by˜ P t ( x, y ) = P [ X = y, X t = x ] P [ X t = x ]= µ ( y ) P t ( y, x ) µ ( x ) = µ ( y ) P t ( y, x ) µ ( x ) − ( t ≥ . Differentiating shows that the generator ˜ L of the time-reversed process isgiven by ˜ L ( x, y ) = µ ( y ) L ( y, x ) µ ( x ) − . Let R denote the diagonal matrix R ( x, y ) := δ x,y µ ( x ) − . Then L ( y, x ) µ ( x ) − = ˜ L ( x, y ) µ ( y ) − = µ ( y ) − ˜ L † ( y, x ) can be rewritten as LR = R ˜ L † , which shows that ˜ L is dual to L with duality function R . In particular,reversible processes (for which ˜ L = L ) are always self-dual with dualityfunction R ( x, y ). Note that since R is diagonal, it is invertible with R − ( x, y ) := δ x,y µ ( x ) ( x, y ∈ Ω) . This formula is wrong in [GKRV09, below (12)]. V be a finite dimensional complex linear space and let L : V → V beany linear operator (not necessarily a Markov generator). Then it is knownthat there exists an invertible matrix Q ∈ L ( V ) such that LQ = QL † or equivalently L † Q − = Q − L (3.40)Thus, every finite dimensional linear operator is self-dual and the self-dualityfunction Q can be chosen such that it is invertible, viewed as a matrix. Let C L := { A ∈ L ( V ) : AL = LA } be the algebra of all elements of L ( V ) that commute with L . We call thisthe space of symmetries of L . In [GKRV09, Thm 2.6], the following simpleobservation is made. Lemma 14 (Self-duality functions)
Let L be a linear operator on somefinite dimensional linear space V . Fix some Q as in (3.40). Then the set ofall self-duality functions of L is given by { SQ : S ∈ C L } . Proof
Clearly, if S ∈ C L , then LSQ = SLQ = SQL † , showing that SQ is a self-duality function. Conversely, if D is a self-dualityfunction, then we can write D = SQ with S = DQ − . Now, since D is aself-duality function, SL = DQ − L = DL † Q − = LDQ − = LS, which shows that S ∈ C L .For dualities, we can play a similar game. Once we have two operators L, ˆ L that are dual with duality function D , i.e., LD = D ˆ L † , we have that for any S ∈ C L , the operators L, ˆ L are also dual with dualityfunction SD , as follows by writing LSD = SLD = SD ˆ L † . If D is invertible, then every duality function of L and ˆ L is of this form.Indeed, if ˜ D is any duality function, then we can write ˜ D = SD with S =˜ DD − . Now SL = ˜ DD − L = ˜ DL † D − = L ˜ DD − = LS, proving that S ∈ C L . See also [GKRV09, Thm 2.10].39 .9 The symmetric exclusion process revisited Following [GKRV09, Sect. 3.1], we demonstrate the principles explained inthe previous subsections to derive a self-duality of the symmetric exclusionprocess. Our starting point is formula (3.14), which expresses the generator L in terms of operators J ± i , J i that define a representation ( V, π ) of a Liealgebra g that is the direct sum of finitely many copies of the Lie algebra su (2), with one copy for each site i ∈ S . Since r ( i, j ) = r ( j, i ), we can rewritethis formula as L = X i,j r ( i, j ) (cid:2) J − i J + j + J − j J + i + 2 J i J j − I (cid:3) . (3.41)A straightforward calculation shows that X k [ J ± k , L ] = 0 and X k [ J k , L ] = 0 ( k ∈ S ) . (3.42)We need a bit of general theory. If U, V, W are representations of thesame Lie algebra g , then we can equip their tensor product U ⊗ V ⊗ W withthe structure of a representation of g by putting A ( u ⊗ v ⊗ w ) := Au ⊗ v ⊗ w + u ⊗ Av ⊗ w + u ⊗ v ⊗ Aw ( A ∈ g ) , (3.43)and similar for the tensor product of any finite number of representations,see formula (A.13) in the appendix. This definition also naturally equips U ⊗ V ⊗ W with the structure of a representation of the Lie group G associatedwith g , in such a way that e tA ( u ⊗ v ⊗ w ) = e tA u ⊗ e tA v ⊗ e tA w ( A ∈ g , t ≥ , where for each A ∈ g , the operator e tA is an element of the Lie group G associated with g . Thus, the representation (3.43) corresponds to letting theLie group act in the same way on each space in the tensor product.In our specific set-up, this means that the operators K − , K + , K definedby K − := X k J − k , K + := X k J + k , K := X k J k (3.44)define a representation of su (2) on the product space C { , } S ∼ = O i ∈ S C { , } . K − , K + , K satisfy the commutation relations of su (2).) Let c − K − + c + K + + c K be an operator in the linear space spannedby K − , K + , K . Then e t ( c − K − + c + K + + c K ) = O i ∈ S e t ( c − J − + c + J + + c J ) ( t ≥ , (3.45)i.e., a natural group of symmetries of the generator L is formed by all oper-ators of the form (3.45) and their products, and this actually corresponds toa representation of the Lie group SU(2).We take this as our motivation to look at one specific operator of theform (3.45), which is e K + . One can check that the uniform distribution is aninvariant law for the exclusion process, so by the principle of Subsection 3.8,the function D ( x, y ) = 1 { x = y } = Y i ∈ S { x i = y i } is a trivial self-duality function. Applying Lemma 14 to the symmetry S = e K + , we see that SD = SI = S is also a self-duality function. Since S factorizes over the sites, it suffices to calculate S for a single site, and thentake the product. We recall from (3.18) that J + f (cid:18) (cid:19) (cid:18) f (1) f (0) (cid:19) = (cid:18) f (0)0 (cid:19) , which gives e J + = ∞ X n =0 n ! ( J + ) n = I + J + = (cid:18) (cid:19) and finally yields the duality function S ( x, y ) = Y i ∈ S { x i ≥ y i } (cid:0) x, y ∈ { , } S (cid:1) . A A crash course in Lie algebras
A.1 Lie groups
In the present appendix, we give a bit more background on Lie algebras. Inparticular, we explain how Lie algebras are closely linked to Lie groups, andhow every Lie algebra can naturally be embedded in an algebra, called theuniversal enveloping algebra. We also explain how properties of the Lie group41in particular, compactness) are related to representations of its associatedLie algebra.A group is a set G which contains a special element I , called the identity ,and on which a group product ( A, B ) AB and inverse operation A A − are defined such that(i) IA = AI = A (ii) ( AB ) C = A ( BC )(iii) A − A = AA − = I .A group is abelian (also called commutative ) if AB = BA for all A, B ∈ G .A group homomorphism is a map Φ from one group G into another group H that preserves the group structure, i.e.,(i) Φ( I ) = I ,(ii) Φ( AB ) = Φ( A )Φ( B ),(iii) Φ( A − ) = Φ( A ) − .If Φ is a bijection, then Φ − is also a group homomorphism. In this case, wecall Φ a group isomorphism . A subgroup of a group G is a subset H ⊂ G suchthat I ∈ H and H is closed under the product and inverse, i.e., A, B ∈ H imply AB ∈ H and A ∈ H implies A − ∈ H . A subgroup is in a natural wayitself a group.A Lie group is a smooth manifold G which is also a group such that thegroup product and inverse functions G × G ∋ ( A, B ) AB ∈ G and G ∋ A A − ∈ G are smooth. A finite-dimensional representation of G is a finite-dimensionallinear space V over R or C together with a map G × V ∋ ( A, v ) Av ∈ V such that(i) v Av is linear,(ii) Iv = v ,(iii) A ( Bv ) = ( AB ) v . 42etting L ( V ) denote the space of all linear operators A : V → V , theseconditions are equivalent to saying that the map Π : G → L ( V ) defined byΠ( A ) v := Av is a group homomorphism from G into the general linear group GL( V ) ofall invertible linear maps A : V → V . A representation is faithful if Π isone-to-one, i.e., if A Π( A ) is a group isomorphism between G and thesubgroup Π( G ) := { Π( A ) : A ∈ G } of GL( V ).One can prove that if G is a Lie group and V is a faithful finite-dimensionalrepresentation, then Π( G ) is a closed subset of GL( V ) and Π : G → Π( G )is a homeomorhism. Conversely, each closed subgroup of GL( V ) is a Liegroup. Such Lie groups are called matrix Lie groups . Not every Lie grouphas a finite dimensional faithful representation, so not every Lie group is amatrix Lie group, but many important Lie groups are matrix Lie groups andfollowing [Hal03] we will mostly focus on them from now on. A.2 Lie algebras An algebra is a finite-dimensional linear space a over R or C with a specialelement I called unit element and on which there is defined a product a × a ∋ ( A, B ) AB ∈ a such that(i) ( A, B ) AB is bilinear,(ii) IA = AI = A ,(iii) ( AB ) C = A ( BC ).In some textbooks, algebras are not required to contain a unit element. Wespeak of a real resp. complex algebra depending on whether a is a linearspace over R or C . An algebra is abelian if AB = BA for all A, B ∈ G .In any algebra, the commutator of two elements A, B is defined as [
A, B ] = AB − BA . If V is a linear space, then L ( V ) is an algebra.An algebra homomorphism is a map φ : a → b from one algebra intoanother that preserves the structure, i.e.,(i) φ is linear,(ii) φ ( I ) = I , 43iii) φ ( AB ) = φ ( A ) φ ( B ).Algebra homomorphisms that are bijections have the property that φ − isalso a homomorphism; these are called algebra isomorphisms. A subalgebra of an algebra a is a linear subspace b ⊂ a that contains I and is closed underthe product. Lie algebras , Lie algebra homomorphisms , and isomorphisms have alreadybeen defined in Section 2.1. A sub-Lie-algebra is a linear subspace h ⊂ g suchthat A, B ∈ h implies [ A, B ] ∈ h . If g is an algebra, then g , equipped with the commutator map [ · , · ], is a Liealgebra. As the example in Section 2.1 shows. Lie algebras need not be analgebras.A representation of an algebra a is a linear space V together with a map a × V → V that satisfies(i) ( A, v ) Av is bilinear,(ii) Iv = v ,(iii) A ( Bv ) = ( AB ) v .If a is a complex algebra, then we require V to be a linear space over C , buteven when a is a real algebra, it is often useful to allow for the case that V isa linear space over C . In this case, bilinearity means real linearity in the firstargument and complex linearity in the second argument. We speak of real or complex representations depending on whether V is a linear space over R or C .A representation V of an algebra a gives in a natural way rise to analgebra homomorphism π : a → L ( V ) defined as π ( A ) v := Av ( A ∈ a , v ∈ V ) . Conversely, given an algebra homomorphism π : a → L ( V ) we can equip V with the structure of a representation by defining Av := π ( A ) v . Thus, arepresentation V of an algebra a is equivalent to a pair ( V, π ) where V is alinear space and π : a → L ( V ) is an algebra homomorphism. A representa-tion ( V, π ) is faithful if π is an isomorphism between a and the subalgebra π ( a ) = { π ( A ) : A ∈ a } of L ( V ).Representations of Lie algebras have already been defined in Section 2.2.If V is a complex representation of a real algebra or Lie algebra a , thenthe image of a under π is only a real subspace of L ( V ). We can define a44omplex algebra or Lie algebra a C whose elements can formally be written as A + iB with A, B ∈ a ; this is called the complexification of a . Then π extendsuniquely to a homomorphism from a C to L ( V ), see [Hal03, Prop. 3.39], so V is also a representation of a C .Every algebra has a faithful representation. Indeed, a together with themap ( A, B ) AB is a representation of itself, and it is not hard to see (usingour assumption that I ∈ a ) that this representation is faithful. Lie algebrascan be represented on themselves in a construction that is very similar to theone for algebras. Lemma 15 (Lie algebra represented on itself )
A Lie algebra g , equippedwith the map ( A, B ) [ A, B ] , is a representation of itself. Proof
It will be convenient to use somewhat different notation for the Liebracket. If g is a Lie algebra and X ∈ g , then we define ad X : g → g byad X ( A ) := [ X, A ] . We need to show that g ∋ X ad X ∈ L ( g ) is a Lie algebra homomor-phism. Bilinearity follows immediately from the bilinear property (i) of theLie bracket, so it remains to show thatad [ X,Y ] ( Z ) = ad X (ad Y ( Z )) − ad Y (ad X ( Z )) . This can be rewritten as[[
X, Y ] , Z ] = [ X, [ Y, Z ]] − [ Y, [ X, Z ]] . Using also the skew symmetric property (ii) of the Lie bracket, this can berewritten as 0 = [ Z, [ X, Y ]] + [ X, [ Y, Z ]] + [ Y, [ Z, X ]] , which is the Jacobi identity.In general, representing a Lie algebra on itself as in Lemma 15 need notyield a faithful representation. (For example, any abelian algebra is also aLie algebra and for such Lie algebras ad X = 0 for each X .) By definition,the center of a Lie algebra g is the set { X ∈ g : [ X, A ] = 0 ∀ A ∈ g } . (A.1)We say that the center is trivial if it contains only the zero element. If g hasa trivial center, then the representation X ad X of g on itself is faithful.Indeed, ad X = ad Y implies [ X, A ] = [
Y, A ] for all A ∈ g and hence X − Y isan element of the center of g . If the center is trivial, this implies X = Y .45 .3 Relation between Lie groups and Lie algebras Let V be a linear space and let G ⊂ GL( V ) be a matrix Lie group. Bydefinition, the Lie algebra g of G is the space of all matrices A such thatthere exists a smooth curve γ in G with γ (0) = I and ∂∂t γ ( t ) (cid:12)(cid:12) t =0 = A. In manifold terminology, this says that g is the tangent space to G at I . Forany matrix A , we define e A := ∞ X k =0 n ! A n . (A.2)The following lemma follows from [Hal03, Cor. 3.46]. The main idea be-hind this lemma is that the elements of the Lie algebra act as “infinitesimalgenerators” of the Lie group. Lemma 16 (Exponential formula)
Let g be the Lie algebra of a Lie group G ⊂ GL( V ) . Then the following conditions are equivalent. (i) A ∈ g (ii) e tA ∈ G for all t ∈ R . The following lemma (a precise proof of which can be found in [Hal03,Thm 3.20]) says that our terminology is justified.
Lemma 17 (Lie algebra property)
The Lie algebra of any matrix Liegroup is a real Lie algebra.
Proof (sketch)
Let λ ∈ R and A ∈ g . By assumption, there exists a smoothcurve γ such that γ (0) = I and ∂∂t γ ( t ) (cid:12)(cid:12) t =0 = A . But now t γ ( λt ) is alsosmooth and ∂∂t γ ( λt ) (cid:12)(cid:12) t =0 = λA , showing that g is closed under multiplicationwith real scalars.Also, if A, B ∈ g , then in the limit as t → e tA e tB = (cid:0) ( I + tA + O ( t ) (cid:1)(cid:0) ( I + tB + O ( t ) (cid:1) = I + ( A + B ) t + O ( t ) , which suggests that A + B lies in the tangent space to G at I ; making thisidea precise proves that indeed A + B ∈ g , so g is a real linear space.To complete the proof, we must show that [ A, B ] ∈ g for all A, B ∈ g . Itis easy to see that for any A, B ∈ g , as t → e tA , e tB ] = t [ A, B ] + O ( t ) , e tA e tB e − tA e − tB = e tA { e − tA e tB + [ e tB , e − tA ] } e − tB = I + t [ A, B ] + O ( t ) . Since e tA e tB e − tA e − tB ∈ G , this suggests that [ A, B ] lies in the tangent spaceto G at I .By [Hal03, Cor. 3.47], if g is the Lie algebra of a Lie group G , then thereexist open neighbourhoods 0 ∈ O ⊂ g and I ∈ U ⊂ G such that the map O ∋ A e A ∈ U is a homeomorphism (a continuous bijection whose inverse is also continuous).The identity component G of a Lie group G is the connected component thatcontains the identity. By [Hal03, Prop. 1.10], G is a subgroup of G . If U is an open neighbourhood of I , then each element of G can be written asthe product of finitely many elements of U . In particular, if G is connected,then U generates G . Therefore (see [Hal03, Cor. 3.47]), if G is a connectedLie group, then each element X ∈ G can be written as X = e A · · · e A n (A.3)for some A , . . . , A n ∈ g . As [Hal03, Example 3.41] shows, even if G isconnected, it is in general not true that for each A, B ∈ g there exists a C ∈ g such that e A e B = e C and hence in general { e A : A ∈ g } need not be agroup; in particular, this is not always G .Anyway, the Lie algebra uniquely characterizes the local structure of a Liegroup, so it should be true that if two Lie groups G and H are isomorphic,then their Lie algebras g and h are also isomorphic. Indeed, by [Hal03,Thm. 3.28], each Lie group homomorphism Φ : G → H gives rise to a uniquehomomorphism φ : g → h of Lie algebras such thatΦ( e A ) = e φ ( A ) ( A ∈ g ) . (A.4)In general, the converse conclusion cannot be drawn, i.e., two different Liegroups may have the same Lie algebra. By definition, a Lie group G is simplyconnected if it is connected and “has no holes”, i.e., every continuous loopcan be continuously shrunk to a point. (E.g., the surface of a ball is simplyconnected but a torus is not.) We cite the following theorem from [Hal03,Thm. 5.6]. In fact, G is a normal subgroup -see formula (A.9) below for the definition of a normalsubgroup. heorem 18 (Simply connected Lie groups) Let G and H be matrixLie groups with Lie algebras g and h and let φ : g → h be a homomorphism ofLie algebras. If G is simply connected, then there exists a unique Lie grouphomomorphism Φ : G → H such that (A.4) holds. In particular ([Hal03, Cor. 5.7]), this implies that two simply connectedLie groups are isomorphic if and only if their Lie algebras are isomorphic.Every connected Lie group G has a universal cover ( H, Φ) (this is statedwithout proof in [Hal03, Sect. 5.8]), which is a simply connected Lie group H together with a Lie group homomorphism Φ : H → G such that the asso-ciated Lie algebra homomorphism as in (A.4) is a Lie algebra isomorphism.The following lemma says that such a universal cover is unique up to naturalisomorphisms. Lemma 19 (Uniqueness of the universal cover)
Let G be a connectedLie group and let ( H i , Φ i ) ( i = 1 , be universal covers of G . Then thereexists a unique Lie group isomorphism Ψ : H → H such that Ψ(Φ ( A )) =Φ ( A ) ( A ∈ G ) . Proof
Let φ i : g → h i denote the Lie algebra homomorphism associatedwith Φ i as in (A.4). If a Lie group isomorphism Ψ as in the lemma exists,then the associated Lie algebra isomorphism ψ must satisfy ψ ◦ φ = φ .By assumption, φ i ( i = 1 ,
2) are isomorphisms, so setting ψ := φ ◦ φ − defines a Lie algebra isomorphism from h to h . By assumption, H is simplyconnected, so by Theorem 18, there exists a unique Lie group homomorphismΨ : H → H such that Ψ( e A ) = e ψ ( A ) ( A ∈ h ). Similarly, there exists aunique Lie group homomorphism ˜Ψ : H → H such that ˜Ψ( e A ) = e ψ − ( A ) ( A ∈ h ). Now˜Ψ(Ψ( e A )) = ˜Ψ( e ψ ( A ) ) = e ψ − ◦ ψ ( A ) = e A ( A ∈ h )and similarly Ψ( ˜Ψ( e A )) ( A ∈ h ), which (using the fact that elements of theform e A with A ∈ h i generate H i ) proves that Ψ is invertible and ˜Ψ = Ψ − .Informally, the universal cover H of G is the unique simply connected Liegroup that has the same Lie algebra as G . The universal cover of a matrixLie group need in general not be a matrix Lie group. Lie’s third theorem[Hal03, Thm 5.25] says: Theorem 20 (Lie’s third theorem)
Every real Lie algebra g is the Liealgebra of some connected Lie group G .
48y [Hal03, Conclusion 5.26], we can even take G to be a matrix Lie group,and by restricting to the identity component we can take G to be connected.By going to the universal cover, we can also take G to be simply connected,but in this case we may loose the property that G is a matrix Lie group.Anyway, we can conclude:There is a one-to-one correspondence between Lie algebras andsimply connected Lie groups. Every Lie group has a unique uni-versal cover, which is a simply connected Lie group with the sameLie algebra.Let G be a Lie group with Lie algebra g and let ( V, Π) be a representationof G . Then, by (A.4), there exists a unique Lie algebra homomorphism π : g → L ( V ) such that Π( e A ) = e π ( A ) ( A ∈ g ) . (A.5)More concretely, one has (see [Hal03, Prop. 4.4]) π ( A ) v = ∂∂t Π( e tA ) v (cid:12)(cid:12) t =0 ( A ∈ g , v ∈ V ) . (A.6)We say that ( V, π ) is the representation of g associated with the represen-tation ( V, Π) of G . Conversely, if G is simply connected, then by grace ofTheorem 18, through (A.5), each representation ( V, π ) of g gives rise to aunique associated representation ( V, Π) of G . A.4 Relation between algebras and Lie algebras If a is an algebra and c ⊂ a is any subset of a , then there exists a smallestsubalgebra b ⊂ a such that b contains c . This algebra consists of the linearspan of the unit element I and all finite products of elements of c . We call b the algebra generated by c . If b = a , then we say that c generates a .Let g be a Lie algebra. By definition, an enveloping algebra for g is a pair( a , i ) such that(i) a is an algebra and i : g → a is a Lie algebra homomorphism.(ii) The image i ( g ) of g under i generates a .We cite the following theorem from [Hal03, Thms 9.7 and 9.9]. Theorem 21 (Universal enveloping algebra)
For every Lie algebra g ,there exists an enveloping algebra ( a , i ) with the following properties. If ( b , j ) is an enveloping algebra of g , then there exists a unique algebrahomomorphism φ : a → b such that φ ( i ( A )) = j ( A ) for all A ∈ g . (ii) If { X , . . . , X n } is a basis for g , then a basis for a is formed by allelements of the form i ( X ) k · · · i ( X n ) k n , where k , . . . , k n ≥ are integers. In particular, these elements arelinearly independent. An argument similar to the proof of Lemma 19 shows that the pair ( a , i )from Theorem 21 is unique up to natural isomorphisms. We call ( a , i ) the universal enveloping algebra of g and use the notation U ( g ) := a . By prop-erty (ii), the map i is one-to-one, so we often identify g with its image under i and pretend g is a sub-Lie-algebra of U ( g ).As an immediate consequence of property (i) of Theorem 21, we see that if V is a representation of a Lie algebra g and π : g → L ( V ) is the associated Liealgebra homomorphism, then there exists a unique algebra homomorphism π : U ( g ) → L ( V ) such that π ( A ) = π ( A ) ( A ∈ g ). (Here we view g as asub-Lie-algebra of U ( g ).) Conversely, of course, every representation of U ( g )is also a representation of g .If ( V, π ) is a representation of a Lie algebra g , then we usually denotethe associated representation of U ( g ) also by ( V, π ), i.e., we identify the map π with its extension π . Note, however, that a representation ( V, π ) of a Liealgebra g can be faithful even when the associated representation ( V, π ) of U ( g ) is not. Indeed, by property (ii) of Theorem 21, U ( g ) is always infinitedimensional, even though g is finite dimensional, so finite-dimensional faithfulrepresentations of g are not faithful when viewed as a representation of U ( g ). A.5 Adjoints and unitary representations
Let V be a finite dimensional linear space equipped with an inner product h · | · i , which for linear spaces over C is conjugate linear in its first argumentand linear in its second argument. Each A ∈ L ( V ) has a unique adjoint A ∗ ∈ L ( V ) such that h A ∗ v | w i = h v | Aw i ( v, w ∈ V ) . (A.7)An operator A is self-adjoint (also called hermitian ) if A ∗ = A and skewsymmetric if A ∗ = − A . A positive operator is an operator such that h v | Av i ≥ v . If V, W are linear spaces equipped with inner products, then anoperator U ∈ L ( V, W ) is called unitary if it preserves the inner product, i.e., h U v | U w i = h v | w i ( v, w ∈ V ) . (A.8)50n particular, an operator U ∈ L ( V ) is unitary if and only if it is invertibleand U − = U . If V is a finite dimensional linear space over C , then for v ∈ V we define operators h v | ∈ L ( V, C ) and | v i ∈ L ( C , V ) by h v | w := h v | w i and | v i c := cv. Then h v || w i is an operator in L ( C , C ) which we can identify with the complexnumber h v | w i . Moreover, | v ih w | is an operator in L ( V ). An orthonormal basis { e (1) , . . . , e ( n ) } of V is a basis such that h e ( i ) | e ( j ) i = δ ij . Then A = X ij A ij | e ( i ) ih e ( j ) | , where A ij denotes the matrix of A with respect to the orthonormal basis { e (1) , . . . , e ( n ) } . An operator A ∈ L ( V ) is normal if [ A, A ∗ ] = 0. Anoperator is normal if and only if it is diagonal w.r.t. some orthonormal basis,i.e., if it can be written as A = X i λ i | e ( i ) ih e ( i ) | , where the λ i are the eigenvalues of A . For operators, the following propertiesare equivalent. A is hermitian ⇔ A is normal with real eigenvalues, A is skew symmetric ⇔ A is normal with imaginary eigenvalues, A is positive ⇔ A is normal with nonnegative eigenvalues, A is unitary ⇔ A is normal with eigenvalues of norm 1.By definition, a unitary representation of a Lie group G is a complexrepresentation ( V, Π) where V is equipped with an inner product such thatΠ( A ) is a unitary operator for each A ∈ G . A unitary representation of areal Lie algebra g is a complex representation V that is equipped with aninner product such that π ( A ) is skew symmetric for all A ∈ g . Since e π ( A ) is unitary if and only if π ( A ) is skew symmetric, our definitionsimply that a representation ( V, Π) of a Lie group G is unitary if and only ifthe associated representation ( V, π ) of the real Lie algebra g of G is unitary. Theorem 22 (Compact Lie groups)
Let K be a compact Lie group andlet V be a representation of K . Then it is possible to equip V with an innerproduct so that V becomes a unitary representation of K . roof (sketch) Choose an arbitrary inner product h · | · i on V and define h v | w i K := Z h Π( A ) v | Π( A ) w i d A, where d A denotes the Haar measure on K , which is finite by the assumptionthat K is compact. It is easy to check that h · | · i K is an inner product. Inparticular, since Π( A ) is invertible for each A ∈ K , we have Π( A ) v = 0 andhence h Π( A ) v | Π( A ) v i > v ∈ V and A ∈ K . Now by the fact thatthe Haar measure is invariant under the action of the group h Π( B ) v | Π( B ) w i K = Z h Π( A )Π( B ) v | Π( A )Π( B ) w i d A = Z h Π( AB ) v | Π( AB ) w i d A = Z h Π( C ) v | Π( C ) w i d C = h v | w i K , which proves that V , equipped with the inner product h · | · i K , is a unitaryrepresentation of K .The following lemma is a sort of converse to Theorem 22 since it saysthat noncompact Lie groups do not have faithful unitary representations, atleast when we restrict ourselves to finite-dimensional representations, as wedo here. Lemma 23 (Noncompact Lie groups)
Let K be a noncompact Lie groupand let V be a faithful (finite dimensional) representation of K . Then it isnot possible to equip V with an inner product so that V becomes a unitaryrepresentation of K . Proof
Equip V with an inner product and let U( V ) denote the group of allunitary maps A : V → V . If ( V, Π) is a faithful representation of K , then theimage Π( K ) of K under Π is a closed subset of GL( V ) and Π : K → Π( K ) isa homeomorphism. If ( V, Π) is a unitary representation, then Π( K ) ⊂ U( V )and hence by the compactness of the latter, Π( K ) is compact. Since Π : K → Π( K ) is a homeomorphism, it follows that K is compact.A ∗ -algebra is a complex algebra on which there is defined an adjoint operation A A ∗ such that(i) A A ∗ is conjugate linear,(ii) ( A ∗ ) ∗ = A ,(iii) ( AB ) ∗ = B ∗ A ∗ . 52f V is a complex finite dimensional linear space equipped with an inner prod-uct, then L ( V ), equipped with the adjoint operation (A.7), is a ∗ -algebra.A ∗ -algebra homomorphism is an algebra homomorphism that satisfies φ ( A ∗ ) = φ ( A ) ∗ . A sub- ∗ -algebra of a ∗ -algebra is a subalgebra that is closed under the adjointoperation. By definition, a ∗ -representation of a ∗ -algebra a is a representa-tion ( V, π ) such that V is equipped with an inner product and π is a ∗ -algebrahomomorphism.In general, a ∗ -algebra may fail to have a faithful ∗ -representation. Forfinite dimensional ∗ -algebras, a necessary and sufficient condition for theexistence of a faithful representation is that A ∗ A = 0 implies A = 0 , but it is rather difficult to prove this; see [Swa17] and references therein.In infinite dimensions, one needs the theory of C ∗ -algebras, which are ∗ -algebras equipped with a norm that in faithful representations correspondsto the operator norm k A k = sup k v k≤ k Av k .Recall the definition of an adjoint operation on a complex Lie algebra g from Section 2.1. Recall also that we called a Lie algebra homomorphism unitary if φ ( A ∗ ) = φ ( A ) ∗ , and that a unitary representation is a representa-tion ( V, π ) such that V is equipped with an inner product and π is a unitaryLie algebra homomorphism. Lemma 24 (Universal enveloping ∗ -algebra) Let g be a Lie- ∗ -algebra.Then there exists a unique adjoint operation on its universal enveloping al-gebra U ( g ) that coincides with the adjoint operation on g . Proof
Recall from Sections 2.2 that every complex linear space V has a conjugate space which is a linear space V together with a conjugate linearbijection V ∋ v v ∈ V . If a is a complex algebra, then we can equip a with the structure of an algebra by putting A B := BA.
It is not hard to see that a map A A ∗ defined on some algebra a is anadjoint operation if and only if the map A A ∗ from a into a is an algebrahomomorphism. By the definition of an adjoint operation on a Lie algebra,[ A ∗ , B ∗ ] = − [ A, B ] ∗ for all A, B ∈ g . It follows that the map g ∋ X X ∗ ∈ U ( g )is a Lie algebra homomorphism, which by the defining property of the uni-versal enveloping algebra (Theorem 21 (i)) extends to a unique algebra ho-momorphism from U ( g ) to U ( g ). 53 .6 Dual, quotient, sum, and product spaces Dual spaces
The dual V ′ of a finite dimensional linear space V over K = R or = C is thespace of all linear forms l : V → K . Each element v ∈ V naturally definesa linear form L v on V ′ by L v ( l ) := l ( v ) and each linear form on V arises inthis way, so we can identify V ′′ ∼ = V . If { e (1) , . . . , e ( n ) } is a basis for V , thensetting f ( i )( e ( j )) := 1 { i = j } defines a basis { f (1) , . . . , f ( n ) } for V ′ called the dual basis . If V is equipped with an inner product, then setting h v | w := h v | w i defines a linear form on V and V ′ := {h v | : v ∈ V } . Through this identifica-tion, we also equip V ′ with an inner product. Then if { e (1) , . . . , e ( n ) } is anorthonormal basis for V , the dual basis is an orthonormal basis for V ′ . Eachlinear map A : V → W gives naturally rise to a dual map A ′ : W ′ → V ′ defined by A ′ ( l ) := l ◦ A, and indeed every linear map from W ′ to V ′ arises in this way, i.e., L ( W ′ , V ′ ) = { A ′ : A ∈ L ( V, W ) } . If V, W are equipped with inner products and A ∈L ( V, W ), then A ′ ( h φ | ) = h A ∗ φ | , where A ∗ denotes the adjoint of A , i.e., this is the linear map A ∗ ∈ L ( W, V )defined by h φ | Aψ i = h A ∗ φ | ψ i ( φ ∈ W, ψ ∈ V ) . If ( V, Π) is a representation of a Lie group G , then we can define grouphomomorphism Π ′ : G → L ( V ′ ) byΠ ′ ( A ) l := Π( A − ) ′ l = l ◦ Π( A − ) . In this way, the dual space V ′ naturally obtains the structure of a represen-tation of G . Note thatΠ ′ ( AB ) l = l ◦ Π(( AB ) − ) = l ◦ Π( A − )Π( B − ) = Π ′ ( A )(Π ′ ( B ) l ) , proving that Π ′ is indeed a group homomorphism. Similarly, if ( V, π ) is arepresentation of a Lie algebra g , then we can equip the dual space V ′ withthe structure of a representation of g by putting π ′ ( A ) l := − π ( A ) ′ ( l ) = − l ◦ π ( A ) , π ′ ([ A, B ]) l = − l ◦ π ([ A, B ]) = − l ◦ (cid:0) π ( A ) π ( B ) − π ( B ) π ( A ) (cid:1) = − (cid:0) π ′ ( B )( π ′ ( A ) l ) − π ′ ( A )( π ′ ( B ) l ) = π ′ ( A )( π ′ ( B ) l ) − π ′ ( B )( π ′ ( A ) l ) . This is called the dual representation or contragredient representation of G or g , respectively, associated with V , see [Hal03, Def. 4.21]. If two represen-tations of G and g are associated as in (A.6), then their dual representationsare also associated. Quotient spaces
By definition, a normal subgroup of a group G is a subgroup H such that A H := { AB : B ∈ H} = { BA : B ∈ H} =: H A ∀ A ∈ G , (A.9)or equivalently, if B ∈ H implies ABA − ∈ H for all A ∈ G . Sets of the form A H and H A are called left and right cosets , respectively. If H is a normalsubgroup, then left cosets are right cosets and vice versa, and we can equipthe set G / H := (cid:8) A H : A ∈ G} = (cid:8) H A : A ∈ G} of all cosets with a group structure such that( A H )( B H ) = ( AB ) H . We call G / H the quotient group of G and H . Note that as a set this isobtained from G by dividing out the equivalence relation A ∼ B ⇔ A = BC for some C ∈ H . If V is a linear space and W ⊂ V is a linear subspace, then we can definean equivalence relation on V by setting v ∼ v ⇔ v = v + w for some w ∈ W. The equivalence classes with respect to this equivalence relation are the setsof the form v + W := { v + w : w ∈ W } and we can equip the space V /W := { v + W : v ∈ V } a ( v + W ) + a ( v + W ) := (cid:0) a v + a v (cid:1) + W. An invariant subspace of a representation V of a Lie group G , Lie algebra g , or algebra a is a linear space W ⊂ V such that Aw ∈ W for all w ∈ W and A from G , g , or a , respectively. If W is an invariant subspace, then wecan equip the quotient space V /W with the structure of a representation bysetting A ( v + W ) := ( Av ) + W. Note that this is a good definition since v = v + w for some w ∈ W implies Av = Av + Aw where Aw ∈ W by the assumption that W is invariant.A left ideal (resp. right ideal ) of an algebra a is a linear subspace i ⊂ a such that AB ∈ i (resp. BA ∈ i ) for all A ∈ a and B ∈ i . An ideal is a linearsubspace that is both a left and right ideal. If i is an ideal of a , then we canequip the quotient space a / i with the structure of an algebra by putting( A + i )( B + i ) := ( AB ) + i . To see that this is a good definition, write A ∼ A if A = A + B for some B ∈ i . Then A ∼ A and B ∼ B imply that A = A + C and B = B + D for some C, D ∈ i and hence A B = ( A + C )( B + D ) = A B + (cid:0) CB + A D + CD )with CB + A D + CD ∈ i , so A B ∼ A B . If a is a ∗ -algebra, then a ∗ -ideal of a is an ideal i such that A ∈ i implies A ∗ ∈ i . If i is a ∗ -ideal, thenwe can equip the quotient algebra a / i with an adjoint operation by putting( A + i ) ∗ := A ∗ + i . A linear subspace h of a Lie algebra g is said to be an ideal if [ A, B ] ∈ h for all A ∈ g and B ∈ h . Note that this automatically implies that also[ B, A ] = − [ A, B ] ∈ h . If h is an ideal of a Lie algebra, then we can equip thequotient space g / h with the structure of a Lie algebra by putting[ A + h , B + h ] := [ A, B ] + h . The proof that this is a good definition is the same as for algebras.56 he direct sum
The direct sum V ⊕ · · · ⊕ V n of linear spaces V , . . . , V n has already beendefined in Section 2.6. There is a natural isomorphism between V ⊕ · · · ⊕ V n and the Carthesian product V × · · · × V n = (cid:8)(cid:0) φ (1) , . . . , φ ( n ) (cid:1) : φ ( i ) ∈ V i ∀ i (cid:9) , which we equip with a linear structure by defining a (cid:0) φ (1) , . . . , φ ( n ) (cid:1) + b (cid:0) ψ (1) , . . . , ψ ( n ) (cid:1) := (cid:0) aφ (1) + bφ (1) , . . . , aφ ( n ) + bφ ( n ) (cid:1) . If V , . . . , V n are equipped with inner products, then we require that the innerproduct on V ⊕ · · · ⊕ V n is given by h φ (1) + · · · + φ ( n ) | ψ (1) + · · · + ψ ( n ) i := n X k =1 h φ ( k ) | ψ ( k ) i , (A.10)which has the effect that V , . . . , V n are (mutually) orthogonal. One has thenatural isomorphism ( V ⊕ V ) /V ∼ = V . In general, given a subspace V of some larger linear space W , there aremany possible ways to choose another subspace V such that W = V ⊕ V and hence W ∼ = ( W/V ) ⊕ V .If V is a linear subspace of some larger linear space W , and W is equippedwith an inner product, then we define the orthogonal complement of V as V ⊥ := { w ∈ W : h v | w i = 0 ∀ v ∈ V } . Then one has the natural isomorphisms
W/V ∼ = V ⊥ and W ∼ = V ⊕ V ⊥ , where the inner product V ⊕ V ⊥ is given in terms of the inner products on V and V ⊥ as in (A.10). Thus, given a linear subspace V of a linear space W that is equipped with an inner product, there is a canonical way to chooseanother subspace V such that W = V ⊕ V .If V , . . . , V n are representations of the same Lie group, Lie algebra, oralgebra, then we equip V ⊕ · · · ⊕ V n with the structure of a representationby putting A (cid:0) φ (1) + · · · + φ ( n ) (cid:1) := Aφ (1) + · · · + Aφ ( n ) . V, W are representations, then W is an invariant subspace of V ⊕ W andone has the natural isomorphism of representations ( V ⊕ W ) /W ∼ = V .If a , . . . , a n are algebras, then we equip their direct sum a ⊕ · · · ⊕ a n with the structure of an algebra by putting (cid:0) A (1)+ · · · + A ( n ) (cid:1)(cid:0) B (1)+ · · · + B ( n ) (cid:1) := A (1) B (1)+ · · · + A ( n ) B ( n ) . (A.11)If a , b are algebras, then b is an ideal of a ⊕ b and one has the naturalisomorphism ( a ⊕ b ) / b ∼ = a . Note that b is not a subalgebra of a ⊕ b since I b (unless a = { } ). For ∗ -algebras, we also put (cid:0) A (1) + · · · + A ( n ) (cid:1) ∗ := (cid:0) A (1) ∗ + · · · + A ( n ) ∗ (cid:1) . The direct sum of Lie algebras has already been defined in Section 2.6. Itis easy to see that this is consistent with the definition of the direct sum ofalgebras.
The tensor product
The tensor product of two (or more) linear spaces has already been defined inSection 2.6. A proof similar to the proof of Lemma 19 shows that the tensorproduct is unique up to natural isomorphisms, i.e., if V ˜ ⊗ W and ( φ, ψ ) φ ˜ ⊗ ψ are another linear space and bilinear map which satisfy the definingproperty of the tensor product, then there exists a unique linear bijectionΨ : V ⊗ W → V ˜ ⊗ W such that Ψ( V ⊗ W ) = V ˜ ⊗ W .If V, W are representations of the same Lie group, then we equip V ⊗ W with the structure of a representation by putting A ( φ ⊗ ψ ) := Aφ ⊗ Aψ. (A.12)If
V, W are representations of the same Lie algebra or algebra, then we equip V ⊗ W with the structure of a representation by putting A ( φ ⊗ ψ ) := Aφ ⊗ ψ + φ ⊗ Aψ. (A.13)The reason why we define things in this way is that in view of (A.6), if g isthe Lie algebra of G , then the representation of g defined in (A.13) is therepresentation of g that is associated with the representation of G defined in(A.12). Note that (A.13) is bilinear in φ and ψ and hence by the definingproperty of the tensor product uniquely defines a linear operator on V ⊗ W .If a , b are algebras, then we equip their tensor product a ⊗ b with thestructure of an algebra by putting (cid:0) A (1) ⊗ B (1) (cid:1)(cid:0) A (2) ⊗ B (2) (cid:1) := (cid:0) A (1) A (2) ⊗ B (1) B (2) (cid:1) . a ⊗ b ) ∋ ( A, B ) AB ∈ a ⊗ b . We can identify a and b with the subalgebras of a ⊗ b given by a ∼ = { A ⊗ I : A ∈ a } and b ∼ = { I ⊗ B : B ∈ b } . Note that if we identify a and b with subalgebras of a ⊗ b , then every elementof a commutes with every element of b . If a , b are ∗ -algebras, then we equipthe algebra a ⊗ b with an adjoint operation by setting( A ⊗ B ) ∗ := ( A ∗ ⊗ B ∗ ) . If g and h are Lie algebras, then the universal enveloping algebra of theirdirect sum is naturally isomorphic to the tensor product of their universalenveloping algebras: U ( g ⊕ h ) ∼ = U ( g ) ⊗ U ( h ) . (A.14)Indeed, if { X , . . . , X n } is a basis for g and { Y , . . . , Y m } is a basis for h , thenwe can define a bilinear map ( A, B ) A ⊗ B from U ( g ) × U ( h ) into U ( g ⊕ h )by (cid:0) X k · · · X k n n , Y l · · · Y l m m (cid:1) X k · · · X k n n ⊗ Y l · · · Y l m m := X k · · · X k n n Y l · · · Y l m m . where we view g and h as sub-Lie-algebras of g ⊕ h such that [ X, Y ] = 0 foreach X ∈ g and Y ∈ h . In view of Theorem 21, the space U ( g ⊕ h ) togetherwith this bilinear map is a realization of the tensor product U ( g ) ⊗ U ( h ).On a philosophical note, recall that elements of a Lie algebra are relatedto elements of a matrix Lie group via an exponential map. We can view(A.14) as a reflection of the property of the exponential map that convertssums into products.If V and W are representations of algebras a and b , respectively, then wecan make V ⊗ W into a representation of a ⊗ b by setting( A ⊗ B )( φ ⊗ ψ ) := ( Aφ ) ⊗ ( Bψ ) . (A.15)Again, by bilinearity and the defining property of the tensor product, this is agood definition. Note that this is consistent with (A.14) and our definition in(2.29) where we showed that if V and W are representations of Lie algebras g and h , then V ⊗ W is naturally a representation of g ⊕ h . On the otherhand, one should observe that in the special case that a = b , our presentconstruction differs from our earlier construction in (A.13).59 .7 Irreducible representations Let g be a Lie algebra on which an adjoint operation is defined, and let h := { a ∈ g : a ∗ = − a } denote the real sub-Lie-algebra consisting of all skew-symmetric elements of g . It is not hard to see that g is the complexificationof h , i.e., each a ∈ g can uniquely be written as a = a + i a with a , a ∈ h . Let { x , . . . , x n } be a basis for g . The Lie bracket on g is uniquelycharacterized by the commutation relations[ x i , x j ] = n X j =1 c ijk x k , (A.16)where c ijk are the structure constants (see (A.16)). Likewise, the adjointoperation on g is uniquely characterized by its action on basis elements x ∗ i = X j d ij x j , (A.17)where d ij is another set of constants.By Theorem 20, the real Lie algebra h is the Lie algebra of some Liegroup G . By going to the universal cover, we can take G to be simplyconnected, in which case it is uniquely determined by h . Conversely, if G isa simply connected Lie group, h is its real Lie algebra, and g := h C is thecomplexification of h , then we can equip g with an adjoint operation suchthat the set of skew symmetric elements is exactly h , by putting ( a + i a ) ∗ := − a + i a for each a , a ∈ h .If V is a linear space and X , . . . , X n ∈ L ( V ) satisfy (A.16), then there ex-ists a unique Lie algebra homomorphism π : g → L ( V ) such that π ( x i ) = X i ( i = 1 , . . . , n ). If V is equipped with an inner product and the operators X , . . . , X n moreover satisfy (A.17), then π is a unitary representation. ByTheorem 21 (i) and Lemma 24, π can in a unique way be extended to a ∗ -algebra homomorphism π : U ( g ) → L ( V ). Moreover, if G is the simplyconnected Lie group associated with h , then by Theorem 18, there existsa unique Lie group homomorphism Π : G → L ( V ) such that (A.5) holds,so ( V, Π) is a representation of G . Since every element of h is skew sym-metric, ( V, π ) and hence also ( V, Π) are unitary representations of h and G ,respectively. To see that this is a sub-Lie-algebra, note that a , b ∈ h imply [ a , b ] ∗ = − [ a ∗ , b ∗ ] andhence [ a , b ] ∈ h . Equivalently, we may show that each a ∈ g can uniquely be written as a = Re( a ) + i Im( a ) with Re( a ) , Im( a ) self-adjoint. This follows easily by putting Re( a ) := ( a + a ∗ )and Im( a ) := i ( a ∗ − a ). W ⊂ V be a linear subspace. It is not hard to see that W is an invariant subspace of ( V, Π) ⇔ W is an invariant subspace of ( V, π ) ⇔ W is an invariant subspace of ( V, π ) . We say that V is irreducible if its only invariant subspaces are { } and V .Let V, W be two representations of the same Lie group G , Lie algebra g , oralgebra a . Generalizing our earlier definition for ie algebras, a homomorphism of representations (of any kind) is a linear map φ : V → W such that φ ( a v ) = a φ ( v ) (A.18)for all a ∈ G , a ∈ g , or a ∈ a , respectively. Homomorphisms of representa-tions are called intertwiners of representations. If φ is a bijection, then itsinverse is also an intertwining map. In this case we call φ an isomorphism and say that the representations are equivalent (or isomorphic ). If G is asimply connected Lie group, g its associated complexified Lie algebra, and U ( g ) its universal enveloping algebra, then it is not hard to see that(A.18) holds ∀ a ∈ G ⇔ (A.18) holds ∀ a ∈ g ⇔ (A.18) holds ∀ a ∈ U ( g ) . The following result can be found in, e.g., [Hal03, Thm 4.29]. In thespecial case of complex Lie algebras, we have already stated this in Proposi-tion 4.
Proposition 25 (Schur’s lemma)(a)
Let V and W be irreducible representations of a Lie group, Lie algebra,or algebra, and let φ : V → W be an intertwiner. Then either φ = 0 or φ is an isomorphism. (b) Let V be an irreducible complex representation of a Lie group, Lie al-gebra, or algebra, and let φ : V → V be an intertwiner. Then φ = λI for some λ ∈ C . By definition, the center of an algebra is the subalgebra C ( a ) := { C ∈ a : [ A, C ] = 0 ∀ A ∈ a } . The center is trivial if C ( a ) = { λI : λ ∈ K } . Thefollowing is adapted from [Hal03, Cor. 4.30]. Corollary 26 (Center)
Let ( V, π ) be an irreducible complex representationof an algebra a and let C ∈ C ( a ) . Then π ( C ) = λI for some λ ∈ C . Proof
Define φ : V → V by φv := π ( C ) v . Then φ ( Av ) = π ( C ) π ( A ) v = π ( CA ) v = π ( AC ) v = π ( A ) π ( C ) v = A ( φv ) for all A ∈ a , so φ : V → V is anintertwiner. By part (b) of Schur’s lemma, φ = λI for some λ ∈ C .61 .8 Semisimple Lie algebras A Lie algebra g is called irreducible (see [Hal03, Def. 3.11]) if its only idealsare { } and g , and simple if it is irreducible and has dimension dim( g ) ≥ semisimple if it can be written as the direct sumof simple Lie algebras. Recall the definition of the center of a Lie algebrain (A.1). Lemma 27 (Trivial center)
The center of a semisimple Lie algebra istrivial.
Proof If g is simple and A is an element of its center, then the linear spacespanned by A is an ideal. Since dim( g ) ≥ { } and g , this implies that A = 0. If g = g ⊕ · · · ⊕ g n is the direct sum ofsimple Lie algebras, then we can write any element A of the center of g as A = A (1) + · · · + A ( n ) with A ( k ) ∈ g . By the definition of the Lie bracketon g (see (2.24)), A ( k ) lies in the center of g for each k , and hence A = 0 bywhat we have already proved.The following proposition is similar to [Hal03, Prop. 7.4]. Proposition 28 (Inner product on Lie algebra)
Let g be a Lie algebraon which an adjoint operation is defined, let h := { a ∈ g : a ∗ = − a } , and let G be the simply connected Lie group with Lie algebra h . Assume that G iscompact. Then the Lie algebra g , equipped with the map g ∋ x ad x ∈ L ( g ) , is a faithful representation of itself. It is possible to equip g with an innerproduct such that this is a unitary representation, i.e., ad x ∗ = (ad x ) ∗ ( x ∈ g ) . Proof
By [Hal03, Prop. 7.7], the center of g is trivial. By Lemma 15 and theremarks below it, it follows that g , equipped with the map g ∋ ad X ∈ L ( g ),is a faithful representation of itself. This representation naturally gives riseto a representation of G . By assumption, G is compact, so by Theorem 22,we can equip g with an inner product so that this representation is unitary.It follows that the representation of h on g is also unitary and hence therepresentation of g on itself is a unitary representation.The following theorem follows from [Hal03, Thm 7.8]. Theorem 29 (Semisimple algebras)
Let G be a compact simply connectedLie group and let g be the complexification of its Lie algebra. Then g issemisimple. roof (main idea) If g is not simple, then it has some ideal i that is neither { } nor g . Let i ⊥ denote the orthogonal complement of i with respect to theinner product on g defined in Proposition 28. It is shown in [Hal03, Prop. 7.5]that i ⊥ is an ideal of g and one has g ∼ = i ⊕ i ⊥ , where ⊕ denotes the directsum of Lie algebras. Continuing this process, one arrives at a decompositionof g as a direct sum of simple Lie algebras.In fact, the converse statement to Theorem 29 also holds: if g is a semisim-ple complex Lie algebra, then it is the complexification of the Lie algebra ofa compact simply connected Lie group. This is stated (with references for aproof) in [Hal03, Sect. 10.3].Let G be a compact simply connected Lie group, let h be its real Liealgebra, let g := h C be the complexification of h , and let U ( g ) denote theuniversal enveloping algebra of g . The Casimir element is the element C ∈ U ( g ) defined as c := − X j x j , where { x , . . . , x n } is a basis for h that is orthonormal with respect to theinner product from Proposition 28. We cite the following result from [Hal03,Prop. 10.5].
Proposition 30 (Casimir element)
The definition of the Casimir elementdoes not depend on the choice of the orthonormal basis { x , . . . , x n } of h .Moreover c lies in the center of U ( g ) . In irreducible representations, the Casimir element has a simple form.
Lemma 31 (Representations of Casimir element)
For each irreduciblerepresentation ( V, π ) of g , there exists a constant λ V ≥ such that π ( c ) = λ V I . Proof
Proposition 30 and Corollary 26 imply that for each irreducible rep-resentation (
V, π ) of U ( g ), there exists a constant λ ∈ C such that π ( c ) = λI .By Theorem 22, we can equip V with an inner product such that it is a uni-tary representation of h . This means that x j is skew symmetric and hence i x j is hermitian, so c = P i ( i x j ) is a positive operator. In particular, itseigenvalues are ≥ The inner product from Proposition 28 is not completely unique; at best it is onlydetermined up to a multiplicative constant. So the Casimir operator depends on thechoice of the inner product, but once this is fixed, it does not depend on the choice of theorthonormal basis. .9 Some basic matrix Lie groups For any finite-dimensional linear space V over V = R or = C , we let GL( V )denote the general linear group of all invertible linear maps A : V → V . Inparticular, we write GL( n ; R ) = GL( R n ) and GL( n ; C ) = GL( C n ).The special linear group SL( V ) is defined asSL( V ) := (cid:8) A ∈ GL( V ) : det( A ) = 1 (cid:9) . Again, we write SL( n ; R ) = SL( R n ) and SL( n ; C ) = SL( C n ). If V is a finite-dimensional linear space over C and V is equipped with an inner product h ·| · i , then we call U( V ) := { A ∈ L ( V ) : A is unitary } the unitary group andSU( V ) := { A ∈ U( V ) : det( A ) = 1 } the special unitary group , and write U( n ) := U( C n ) and SU( n ) := SU( C n ).If V is a finite-dimensional linear space over R and V is equipped withan inner product h ·| · i , then an operator O ∈ L ( V ) that preserves the innerproduct as in (A.8) is called orthogonal . (This is the equivalent of unitarityin the real setting.) We callO( V ) := { A ∈ L ( V ) : A is orthogonal } denote the orthogonal group andSO( V ) := { A ∈ O( V ) : det( A ) = 1 } the special orthogonal group , and write O( n ) := O( R n ) and SO( n ) := SO( R n ).There also exists a group O( n ; C ), which consists of all complex matrices thatpreserve the bilinear form ( v, w ) := P i v i w i . Not that this is not the innerproduct on C n ; as a result O( n ; C ) is not the same as U( n ).Unitary operators satisfy | det( A ) | = 1 and orthogonal operators satisfydet( A ) = ±
1. The group O(3) consists of rotations and reflections (andcombinations thereof) while SO(3) consists only of rotations.By [Hal03, Prop. 3.23], for K = R or = C , the Lie algebra of GL( n, K )is the space M n ( K ) of all K -valued n × n matrices, and the Lie algebra ofSL( n, K ) is given by sl ( n, K ) = { A ∈ M n ( K ) : tr( A ) = 0 } .
64y [Hal03, Prop. 3.24], the Lie algebras of U( n ) and O( n ) are given by u ( n ) = { A ∈ M n ( C ) : A ∗ = − A } and o ( n ) = { A ∈ M n ( R ) : A ∗ = − A } . Moreover, again by [Hal03, Prop. 3.24], the Lie algebras of SU( n ) and SO( n )are given by su ( n ) = { A ∈ M n ( C ) : A ∗ = − A, tr( A ) = 0 } and so ( n ) = o ( n ) . By [Hal03, formula (3.17)], the complexifications of the real Lie algebrasintroduced above are given by gl ( n, R ) C ∼ = gl ( n, C ) , u ( n ) C ∼ = gl ( n, C ) , su ( n ) C ∼ = sl ( n, C ) , sl ( n, R ) C ∼ = sl ( n, C ) , so ( n, R ) C ∼ = so ( n, C ) . As mentioned in [Hal03, Sect. 1.3.1], the following Lie groups are compact:O( n ) , SO( n ) , U( n ) , and SU( n ) . By [Hal03, Prop 1.11, 1.12, and 1.13] and [Hal03, Exercise 1.13], the followingLie groups are connected:GL( n ; C ) SL( n ; C ) U( n ) SU( n ) , and SO( n ) . By [Hal03, Prop. 13.11], the group SU( n ) is simply connected. By [Hal03,Example 5.15], SU(2) is the universal cover of SO(3).Of further interest are the real and complex symplectic groups SP( n, R )and SP( n, C ), and the compact symplectic group SP( n ); for their definitionswe refer to [Hal03, Sect. 1.2.4]. A.10 The Lie group SU(1,1)
Let us define a Minkowski form { · , · } : C → C by { v, w } := v ∗ w − v ∗ w . Note that this is almost identical to the usual definition of the inner producton C (in particular, it is conjugate linear in its first argument and linear in65ts second argument), except for the minus sign in front of the second term.Letting M := (cid:18) − (cid:19) , we can write { v, w } = h v | M | w i , where h · , · i is the usual inner product. The Lie group SU(1 ,
1) is the matrixLie group consisting of all matrices Y ∈ L ( C ) with determinant 1 thatpreserve this Minkowski form, i.e.,det( Y ) = 1 and { Y v, Y w } = { v, w } ( v, w ∈ C ) . The second condition can be rewritten as h Y v | M | Y w i = h v | M | w i whichholds for all v, w if and only if Y ∗ M Y = M, (A.19)where Y ∗ denotes the usual adjoint of a matrix. Since( e tA ) ∗ M e tA = M + t ( A ∗ M + M A ) + O ( t ) , it is not hard to see that a matrix of the form Y = e tA satisfies (A.19) if andonly if A ∗ M + M A = 0 ⇔ M A ∗ M = − A, and the Lie algebra su (1 ,
1) associated with SU(1 ,
1) is given by su (1 ,
1) = (cid:8) A ∈ M ( C ) : M A ∗ M = − A, tr( A ) = 0 (cid:9) . It is easy to see that A = (cid:18) A A A A (cid:19) ⇒ M A ∗ M = (cid:18) A − ( A ) ∗ − ( A ) ∗ A (cid:19) and in fact the map A M A ∗ M satisfies the axioms of an adjoint operation.Let su (1 , C denote the Lie algebra su (1 , C := (cid:8) A ∈ M ( C ) : tr( A ) = 0 (cid:9) , equipped with the adjoint operation A M A ∗ M . Then su (1 ,
1) is the realsub-Lie algebra of su (1 , C consisting of all elements that are skew symmetricwith respect to the adjoint operation A M A ∗ M .A basis for su (1 , C is formed by the matrices in (2.8), which satisfy thecommutation relations (2.7). The adjoint operation A M A ∗ M leads to theadjoint relations (2.9). Some elementary facts about the Lie algebra su (1 , C are already stated in Section 2.4. Note that the definition of the “Casimiroperator” in (2.10) does not follow the general definition for compact Liegroups in Proposition 30, but is instead defined in an analogous way, replacingthe inner product by a Minkowski form.66 .11 The Heisenberg group Consider the matrices X := , Y := , Z := . We observe that XX = 0 , XY = Z, XZ = 0 ,Y X = 0 , Y Y = 0 , Y Z = 0 ,ZX = 0 , ZY = 0 , ZZ = 0 . The
Heisenberg group H [Hal03, Sect. 1.2.6] is the matrix Lie group consistingof all 3 × B = I + xX + yY + zZ ( x, y, z ∈ R ) . To see that this is really a group, we note that if B is as above, then itsinverse B − is given by B − = − xX − yY + ( xy − z ) Z. It is easy to see that { X, Y, Z } is a basis for the Lie algebra h of H . In fact,the expansion formula for e t ( xX + yY + zZ ) terminates and e t ( xX + yY + zZ ) = I + t ( xX + yY + zZ ) + t xyZ ( t ≥ . The basis elements
X, Y, Z satisfy the commutation relations[
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