 simply connected Lie group admitting the algebra of $J_k$ as its own  Lie algebra.
As another  application, consider $X$ and $P$ defined on ${\cal S}(\mathbb R)$ as usual. The three symmetric operators $I,X,P$ enjoy the Lie algebra of Weyl-Heisenberg Lie group. Moreover $\Delta = X^2+P^2 +I^2$ is essentially self adjoint on ${\cal S}(\mathbb R)$, because it admits a dense set of analytic vectors (the finite linear combinations of eigenstates of the standard harmonic oscillator). Thus these operators admit unique self-adjoint extensions and are generators of a unitary representation of the (simply connected)  Weyl-Heisenberg Lie group. This example holds also replacing $L^2$ with another generic Hilbert space $\cal H$ and $X,P$ with operators verifying CCR on an dense invariant domain where $X^2+P^2$ (and thus also $X^2+P^2 +I^2$) is essentially self adjoint. It is possible to prove that the existence of the unitary rep of the Weyl-Heisenberg Lie group, if the space is irreducible, establishes the existence of a unitary operator from ${\cal H}$ to $L^2$ transforming  $X$ and $P$ into the standard operators. Following this way one builds up an alternate  proof of Stone-von Neumann's theorem.
As a last comment, I stress that usually ${\cal G}_\ell$ is not the group acting in the physical space and this fact may create some problem: Think of $SO(3)$ that is the group of rotations  one would like to represent at quantum level, while he/she ends up with a unitary representation of $SU(2) \neq SO(3)$. Usually nothing too terrible arises this way, since the only consequence is  the appearance of annoying phases as explained by Josh, and overall phases do not affect  states. Nevertheless sometimes some disaster takes place: For instance, a physical system cannot assume  quantum states that are coherent superpositions of both integer and semi-integer spin. Otherwise an internal phase would take place after a $2\pi$ rotation. What is done in these cases is just to forbid these unfortunate superpositions. This is one of the possible ways to realize superselection rules.      

A:

I'd like to add to Josh's answer, because he didn't really explain what a universal covering group is. Essentially
simply connected Lie group admitting the algebra of $ J_k$ as its own   Lie algebra. 
 As another   application, think $ X$ and $ P$ define on $ { \cal S}(\mathbb R)$ as usual. The three symmetric operators $ I, X, P$ delight the Lie algebra of Weyl - Heisenberg Lie group. Moreover $ \Delta = X^2+P^2 + I^2 $ is essentially self adjoint on $ { \cal S}(\mathbb R)$, because it accept a dense bent of analytic vectors (the finite linear combinations of eigenstates of the standard harmonic oscillator). therefore these operators admit unique self - adjoint extensions and are generator of a unitary representation of the (simply connected)   Weyl - Heisenberg Lie group. This example apply also replacing $ L^2 $ with another generic Hilbert space $ \cal H$ and $ adam, P$ with operators verifying CCR on an dense invariant domain where $ X^2+P^2 $ (and therefore also $ X^2+P^2 + I^2 $) is basically self adjoint. It is possible to prove that the existence of the unitary rep of the Weyl - Heisenberg Lie group, if the distance is irreducible, establishes the existence of a unitary operator from $ { \cal H}$ to $ L^2 $ transforming   $ X$ and $ P$ into the standard hustler. follow this way one builds up an alternate   proof of Stone - von Neumann's theorem. 
 As a last comment, I stress that usually $ { \cal G}_\ell$ is not the group acting in the forcible outer space and this fact may create some trouble: remember of $ SO(3)$ that is the group of rotation   one would like to represent at quantum grade, while he / she end up with a unitary representation of $ SU(2) \neq SO(3)$. normally nothing too terrible arises this way, since the only consequence is   the appearance of annoying phases as explained by Josh, and overall phases do not affect   states. Nevertheless sometimes some disaster take place: For case, a physical system cannot assume   quantum states that are coherent superpositions of both integer and semi - integer tailspin. Otherwise an internal phase would take place after a $ 2\pi$ rotation. What is done in these cases is just to forbid these unfortunate superposition. This is one of the possible way to realize superselection rule.      

 A: 

 I'd wish to add to Josh's answer, because he didn't really explain what a cosmopolitan covering group is. Essentially
 silply connected Lie group admitting the cogebra of $J_k$ as its uwn  Lie algebra.
As another  epplucatiin, consider $X$ and $P$ dewined on ${\bal S}(\mathvb R)$ qs usual. Tis three symmefvic o'ecators $I,X,P$ enjoi the Lie alcebra of Weyl-Haireuberg Lie group. Moreover $\Delta = X^2+P^2 +I^2$ is essrnhially self adtoinu og ${\cam S}(\mathbb R)$, because it admits a dehse set of analytic fectors (the finite linear fomblnations of eigenshates of thg stwbdard harmonkc oscillauox). Thus thesg operators admit unique self-adjuint zxtensions qne agg generators of a unitary repvvsentathon of yhe (simply conkectev)  Wwyl-Heisenberg Lie gronp. This example holdf also re[lccing $L^2$ with another teberic Hiltert wpaze $\dak G$ and $D,P$ xith operatkrs verifyibg CCR on an dense ondqriant domain where $X^2+[^2$ (and thus also $X^2+P^2 +I^2$) is essentially selx asjoint. It is possible ti prove that the exisjence of tre unitary rep of the Weyl-Heisenberg Lie group, if the aoact lf urgeducible, establishes the existence of a unitwdy oierator from ${\cal M}$ to $L^2$ transforminb  $X$ wnd $P$ into thg standcdd operators. Followijg this way ine buildf up an alternate  proof of Stobe-von Neumanu's rheorem.
As a last clmment, I stxess tnat uxually ${\cal G}_\ell$ is not che grkup acting ln the phgrical space and ghix xact may create some problqm: Think if $SP(3)$ that ks tne grotp of rotahions  one would like to reprgsent dt quantum level, while he/she ends up with a unitary reptesantdtion of $SU(2) \ncq SO(3)$. Usually njthing too tertible arifes tfis way, sihce the only consezuence is  tha appearance mf annoyyng phasws as ebolained by Josn, and ovegajl pyases do not affecb  stxfes. Neverthelesw simetimes some dosartew nakxs plwwe: For instatce, x pfusicau system caknog asxume  quantum states thaf are coherent suprriositions of both integer and xemi-integer spin. Ouherwive en intrrnwl phase would take place afted a $2\pi$ rohatlon. What is djne ln trese cases is just to forbid these unfortunate su'erpositions. This is onw of the possible wcyx to realizx supewselectiot rules.      

A:

I'd likw to add to Josh's answer, because he disn't redlly fxplain what a universal covering group is. Essentially
simply connected Lie group admitting the algebra as own Lie As another application, on S}(\mathbb R)$ as The three symmetric $I,X,P$ enjoy the Lie algebra of Lie group. Moreover $\Delta = X^2+P^2 +I^2$ is essentially self adjoint on ${\cal R)$, because it admits a dense set of analytic vectors (the finite linear of of standard oscillator). Thus these operators admit unique self-adjoint extensions and are generators of a unitary representation of (simply connected) Weyl-Heisenberg Lie group. This example holds replacing $L^2$ with another Hilbert space $\cal H$ and with verifying CCR an invariant where $X^2+P^2$ (and also $X^2+P^2 +I^2$) is essentially self adjoint. It is possible to prove that the existence of the rep of Lie group, the is establishes the existence unitary operator from ${\cal H}$ to and $P$ into the standard operators. Following this one builds an alternate proof of Stone-von Neumann's As a last comment, I stress that usually G}_\ell$ is not the group acting in the physical space and this fact may create Think of $SO(3)$ that the group of one like represent quantum level, he/she ends up with a unitary representation of $SU(2) \neq SO(3)$. nothing too terrible arises this way, since the only consequence appearance annoying phases as by Josh, and overall do affect states. Nevertheless sometimes takes For system assume states that are coherent of both integer and semi-integer Otherwise an internal phase $2\pi$ rotation. What is done in these cases just to forbid these unfortunate superpositions. This one of the possible ways to realize superselection rules. A: I'd like add to because he didn't really explain what a universal group is. Essentially
 simply connected Lie group adMitting the AlgebRa oF $J_k$ As Its oWn  LiE algebra.
As anotHEr  apPlication, consider $X$ and $P$ DefinEd ON ${\cal s}(\MaThbb R)$ As usual. tHe THRee SyMmEtrIc OPeRatorS $I,X,p$ enjoy tHe Lie algebRa oF WEyl-HeisenberG liE group. MoreOveR $\Delta = X^2+P^2 +I^2$ is eSseNtiallY sElf ADjoinT on ${\Cal S}(\mAthbb R)$, BEcause It admits a DeNSe set oF AnalytiC VEcTors (The finite linear coMBiNAtions of eigensTates oF tHE sTANdaRd hArmonic oscIlLator). tHus thesE OpERATorS Admit unique seLf-adjoint exTEnsIons anD aRe gENeratoRs of a UnITarY representaTion Of the (simpLy connECted)  WeyL-heisenbErg Lie GroUp. THis eXAmPlE hoLdS AlsO RePlaCIng $l^2$ with anoThEr GenerIc HiLBERT spaCe $\cAl H$ aNd $X,P$ wIth operators vEriFyinG cCR On an dEnse iNvarIaNt domAin wheRe $X^2+P^2$ (aNd Thus also $X^2+P^2 +I^2$) is esSentIally self AdjOiNt. IT iS possIBle to pRovE thAt the exIstence OF thE uNITArY rep of the Weyl-HeiseNbERG LIe group, iF the spACe Is IRreducibLe, EstAbliSHEs the ExisTEnCe of a uniTary opERaToR from ${\caL H}$ To $L^2$ traNsForMinG  $X$ and $p$ Into The staNdard opeRatorS. following this wAY one builds up aN AlTERnATe  prOof Of Stone-von NEumaNN's thEoreM.
as A laST commEnt, I sTrESs THat usually ${\cal G}_\ell$ is NoT the grOup acTing in the physIcal space aND THis fact mAy crEAtE Some problem: ThiNk of $So(3)$ that is the GRoup of roTatioNs  one wouLd like to rEPResent at QuaNtuM leVel, WHIlE he/she ends up wITH a unItAry reprEseNtation Of $Su(2) \neQ SO(3)$. usuAlLy nothing Too terriBlE aRiSeS thIs way, SInce the oNlY coNsEquEnce iS  The appEaranCe of AnNoYIng Phases aS ExPLAineD bY JOsh, aNd oVeRall pHaseS Do nOt affecT  states. NeVerTHeleSs SoMetimes Some disaster tAkEs place: For InStaNce, a phYSIcal systEm cannot assume  quantum stATes that Are CoherEnt sUperpositIonS of botH inTEger anD semi-iNtegeR sPin. oTHerwiSE An IntErNal phase woULD taKe plaCe AfteR a $2\pi$ rotAtion. What is done in tHEse Cases is just to ForBid tHESe UnfORtUNatE sUPerPOSitions. This is onE of the possIbLE wAys to realiZE suPeRselectIon ruleS.      

A:

I'd lIKe to add To Josh's anSwer, becauSe He diDN'T reAlly explaiN what a unIversal coVEring GRoUp is. ESseNtiallY
 simply connected Lie grou p admittin g the  al geb ra  of$J_k $ as its own L ie a lgebra.
As another  ap plica ti o n, c o ns ider$X$ and $P $ def in ed  on  $ { \c al S} (\m athbb R )$ as usua l.Th e three symm e tr ic operato rs$I,X,P$ enjo y t he Lie  a lge b ra of  We yl-He isenbe r g Liegroup. Mo re o ver $\ D elta =X ^ 2+ P^2+I^2$ is essentia l ly self adjoint o n ${\c al S} ( \ mat hbb  R)$, beca us e ita dmits a de n s e  se t  of analyticvectors (th e  fi nite l in ear combin ation so f e igenstatesof t he standa rd har m onic os c illator ). Thu s t hes e op e ra to rsad m itu ni que sel f-adjoin tex tensi onsa n d aregen erat ors o f a unitary r epr esen t ati on of  the(sim pl y con nected )  We yl -Heisenberg Lie  gro up. Thisexa mp leho lds a l so rep lac ing  $L^2$with an o the rg e n er ic Hilbert space $ \c a l  H $ and $X ,P$ wi t hop e rators v er ify ingC C R onan d e ns e invari ant do m ai nwhere $ X^ 2+P^2$  ( and  th us al s o $X ^2+P^2  +I^2$)is es s entially selfa djoint. It is po s s ib l e to  pr ove that th e ex i sten ce o f  t heu nitar y rep  o f  t h e Weyl-Heisenberg L ie  group , ifthe space isirreducibl e , establis hest he existence of a  unit ary operat o r from $ {\cal  H}$ to$L^2$ tra n s forming $X $ a nd$P$ i nt o the standar d oper at ors. Fo llo wing th isway  on e b ui lds up an  alterna te  pr oo f o f Sto n e-von Ne um ann 's  th eorem . 
As alastcomm en t, I s tress t h at u sual ly  $ {\ca l G }_ \ell$  isn otthe gro up acting  in theph ys ical sp ace and thisfa ct may cre at e s ome pr o b lem: Thi nk of $SO(3)$ that is t h e group  of  rota tion s  one wo uld  liketor eprese nt atquant um  le v e l, wh i l ehe/ sh e ends upw i tha uni ta ry r epresen tation of $SU(2) \ n eqSO(3)$. Usual lynoth i n gtoo te r rib le ari s e s this way, sin ce the onl yc on sequence i s   t he  appear ance of  anno y ing pha ses as ex plained b yJosh , and  overall p hases do  not affe c t  st a te s. Ne ver theles ssom etime s some dis aster  takes  p lace:For i ns tance, a  physical system cannot  assum e  qu ant um states  th a t a re cohere nt s uperpositi ons  of  both  in t egerands em i-i n teger  spi n . Otherwi s eani n te rnal phasew o u ldtakepla c e afte r a$2\pi$ rotation.W hat is done in  the s e  ca ses is j us t to forbid th ese  u n f ortunate  s uperpositio ns. This  i s  oneof the  possi ble way s to realiz e su per selection  ru le s .     
A: 
I'd l iketo  add t o Josh ' s an s w er, because he d idn't r eally exp lainwh at a un i vers al coverin g group is.  Essen tial ly
 simply_connected Lie_group admitting the algebra_of $J_k$_as_its own__Lie algebra.
As another_ application, consider_$X$ and $P$ defined_on ${\cal S}(\mathbb_R)$_as usual. The three symmetric operators $I,X,P$ enjoy the Lie algebra of Weyl-Heisenberg Lie_group._Moreover $\Delta_=_X^2+P^2_+I^2$ is essentially self adjoint_on ${\cal S}(\mathbb R)$, because_it admits_a dense set of analytic vectors (the finite_linear_combinations of eigenstates_of the standard harmonic oscillator). Thus these operators admit_unique self-adjoint extensions and are generators_of a unitary_representation_of_the (simply connected) _Weyl-Heisenberg Lie group. This example holds_also replacing $L^2$ with another generic_Hilbert space $\cal H$ and $X,P$ with_operators verifying CCR on an dense_invariant domain where $X^2+P^2$ (and_thus also_$X^2+P^2 +I^2$) is essentially self_adjoint. It is_possible to_prove that the_existence of the unitary rep of_the Weyl-Heisenberg Lie_group, if the space is irreducible,_establishes_the existence of_a_unitary_operator from_${\cal H}$ to_$L^2$_transforming _$X$_and $P$ into the standard operators._Following_this way one builds up an alternate_ proof of Stone-von_Neumann's_theorem.
As a last comment,_I stress that usually ${\cal_G}_\ell$ is not the group acting_in the_physical space_and this fact may create some problem: Think of $SO(3)$ that_is the group of rotations _one would like to_represent at_quantum_level, while he/she_ends_up with_a unitary representation of $SU(2) \neq SO(3)$._Usually nothing_too terrible arises this way, since_the only consequence is__the appearance of annoying phases as_explained by Josh, and overall phases_do not affect  states._Nevertheless_sometimes_some disaster takes place: For_instance, a physical system cannot assume_ quantum states_that are coherent superpositions of both integer_and_semi-integer spin. Otherwise an internal phase_would_take place after a $2\pi$ rotation._What_is_done in these cases is_just to forbid these unfortunate superpositions._This is one of the possible ways to realize_superselection rules. _    

A:

I'd_like_to_add to Josh's answer, because he didn't really explain what_a universal_covering group is._Essentially