Souriau cocycle and non-equivariant moment maps

Intuition [Beginner]

A moment map is a tool that tracks how a symmetry moves through phase space. For a rotation symmetry, the moment map records the angular momentum. For a translation symmetry, it records the linear momentum. When the symmetry is "nice," the moment map respects the group operation: composing two symmetries and computing the moment map gives the same result as transforming the moment map of the first by the second.

But sometimes this property fails. The moment map is slightly off — it shifts by a correction term each time you compose symmetries. This correction is called the Souriau cocycle. It measures the obstruction to the moment map being compatible with the group structure.

This idea matters because the cocycle is not an error to fix — it carries real information. Souriau showed that the cocycle classifies the central extensions of the symmetry group, meaning it tells you exactly which larger group you must use to restore compatibility.

Visual [Beginner]

A diagram showing three copies of a symmetry group $G$ arranged in a triangle. Arrows between them represent the group action on the moment map image. A small label $\sigma (g)$ on each arrow indicates the correction term — the cocycle — that appears when the moment map fails to commute with the group action.

The picture shows that going around the triangle by two different routes gives results that differ by the cocycle: the obstruction is the gap between the two paths.

Worked example [Beginner]

Consider a particle of mass $m=1$ moving on a line, with phase space coordinates $(q,p)$. The symmetry group is the group of translations and boosts: an element $(a,b)$ maps $(q,p)$ to $(q+a+bt,p+b)$, where $t$ is time.

Step 1. The moment map for this group has two components: $J_1=p$ (generating translations in $q$) and $J_2=q-tp$ (generating boosts).

Step 2. Compose two group elements: first $(a_1,b_1)$, then $(a_2,b_2)$. The group law is $(a_2,b_2)\cdot (a_1,b_1)=(a_1+a_2+b_{2}t,b_1+b_2)$. The extra term $b_{2}t$ is the cocycle: it appears because boosts do not commute with time-dependent translations.

Step 3. The cocycle is $\sigma (a,b)=(bt,0)$ in the dual of the Lie algebra. This single function records the failure of the moment map to be equivariant.

What this tells us: the Galilean group (translations + boosts) has a non-zero Souriau cocycle. The corresponding central extension is the Bargmann group, which adds one extra dimension to account for the cocycle. Mass appears as the central charge.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $(M,\omega )$ be a symplectic manifold and $G$ a connected Lie group with Lie algebra $\mathfrak{g}$ acting symplectically on $M$ via $\Phi :G\times M\to M$. A moment map for the action is a smooth map $J:M\to {\mathfrak{g}}^{\ast }$ satisfying

$${\iota }_{{\xi }_M}\omega =d⟨J,\xi ⟩\text{for all }\xi \in \mathfrak{g},$$

where ${\xi }_M$ is the fundamental vector field of $\xi$ on $M$ 05.04.01. The moment map is equivariant (or $G$-equivariant) if

$$J({\Phi }_g(x))={\mathrm{Ad}}_g^{\ast }(J(x))\text{for all }g\in G,x\in M.$$

When equivariance fails, define the Souriau cocycle $\sigma :G\to {\mathfrak{g}}^{\ast }$ by

$$\sigma (g)=J({\Phi }_g(x))-{\mathrm{Ad}}_g^{\ast }(J(x)).$$

This is independent of the choice of $x\in M$ because both sides have the same derivative with respect to $x$ (both satisfy the moment map equation for the same infinitesimal generators). The cocycle satisfies the cocycle identity:

$$\sigma (gh)=\sigma (g)+{\mathrm{Ad}}_g^{\ast }(\sigma (h))\text{for all }g,h\in G.$$

Differentiating at the identity gives the infinitesimal cocycle $\Sigma :\mathfrak{g}\to {\mathfrak{g}}^{\ast }$ defined by

$$\Sigma (\xi )={\frac{d}{dt}|}_{t=0}\sigma (\exp (t\xi )),$$

satisfying $\Sigma ([\xi ,\eta ])={\mathrm{ad}}_{\xi }^{\ast }(\Sigma (\eta ))-{\mathrm{ad}}_{\eta }^{\ast }(\Sigma (\xi ))$. This is a Lie algebra one-cocycle in the Chevalley-Eilenberg complex $C^1(\mathfrak{g},{\mathfrak{g}}^{\ast })$.

The affine coadjoint action of $G$ on ${\mathfrak{g}}^{\ast }$ twisted by $\sigma$ is

$${\mathrm{Ad}}_g^{\ast ,\sigma }(\mu )={\mathrm{Ad}}_g^{\ast }(\mu )+\sigma (g).$$

The moment map $J$ is equivariant with respect to this affine action: $J({\Phi }_g(x))={\mathrm{Ad}}_g^{\ast ,\sigma }(J(x))$.

Counterexamples to common slips

• The cocycle $\sigma$ is not a moment map. It is a map from $G$ to ${\mathfrak{g}}^{\ast }$, not from $M$ to ${\mathfrak{g}}^{\ast }$. The moment map $J:M\to {\mathfrak{g}}^{\ast }$ may or may not be equivariant; the cocycle measures the failure.
• Non-equivariance does not mean no moment map exists. A moment map exists whenever the action is Hamiltonian (the fundamental vector fields are Hamiltonian). Equivariance is an additional property. Non-equivariant moment maps still define valid Hamiltonian actions.
• The cocycle can be a coboundary. If $\sigma (g)={\mathrm{Ad}}_g^{\ast }({\mu }_0)-{\mu }_0$ for some ${\mu }_0\in {\mathfrak{g}}^{\ast }$, then $\tilde{J}=J-{\mu }_0$ is equivariant. The cohomologically substantive case is when $\sigma$ is not a coboundary — when no shift of $J$ restores equivariance.

Key theorem with proof [Intermediate+]

Theorem (Souriau 1970). Let $(M,\omega )$ be a connected symplectic manifold, $G$ a connected Lie group acting symplectically on $M$ with moment map $J : M \to \mathfrak{g}^$. Then:*

1. The Souriau cocycle $\sigma(g) = J(\Phi_g(x)) - \mathrm{Ad}^_g(J(x))$isindependentof$x \in M$ and satisfies the cocycle identity.*
2. The cohomology class $[\Sigma] \in H^1(\mathfrak{g}, \mathfrak{g}^)$(equivalently$[\sigma] \in H^1(G, \mathfrak{g}^)$)isindependentofthechoiceofmomentmap$J$.
3. The class $[\Sigma ]$ vanishes if and only if $J$ can be shifted to an equivariant moment map.
4. The class $[\Sigma ]$ classifies the central extension $\tilde{\mathfrak{g}}=\mathfrak{g}\oplus \mathbb{R}$ with bracket $[(\xi ,a),(\eta ,b)]=([\xi ,\eta ],\Sigma (\xi )(\eta )-\Sigma (\eta )(\xi ))$ for which the lifted action on $\tilde{G}$ admits an equivariant moment map.

Proof. Part 1. Fix $x_0\in M$ and define $\sigma (g)=J({\Phi }_g(x_0))-{\mathrm{Ad}}_g^{\ast }(J(x_0))$. To show independence of $x_0$, differentiate with respect to $x$. For any vector $v\in T_{x}M$:

$$d_x[J({\Phi }_g(x))](v)=d{J}_{{\Phi }_g(x)}(d{\Phi }_g\cdot v)={\omega }_{{\Phi }_g(x)}(({\xi }_M{)}_{{\Phi }_g(x)},d{\Phi }_g\cdot v)$$

for the appropriate $\xi$. Using ${\Phi }_g^{\ast }\omega =\omega$ (symplectic action), this equals ${\omega }_x({\Phi }_{g\ast }^{-1}({\xi }_M{)}_{{\Phi }_g(x)},v)$. The transformed fundamental vector field satisfies ${\Phi }_{g\ast }^{-1}({\xi }_M{)}_{{\Phi }_g(x)}=({\mathrm{Ad}}_{g^{-1}}\xi {)}_M(x)$. Hence:

$$d_x[J({\Phi }_g(x))](v)={\omega }_x(({\mathrm{Ad}}_{g^{-1}}\xi {)}_M(x),v)=d_x[⟨J,{\mathrm{Ad}}_{g^{-1}}\xi ⟩](v).$$

This shows $⟨J\circ {\Phi }_g,\xi ⟩$ and $⟨{\mathrm{Ad}}_g^{\ast }\circ J,\xi ⟩$ have the same differential, so $J\circ {\Phi }_g-{\mathrm{Ad}}_g^{\ast }\circ J$ is locally constant. On a connected manifold, it is constant, hence independent of $x$.

For the cocycle identity, compute:

$$\sigma (gh)=J({\Phi }_{gh}(x_0))-{\mathrm{Ad}}_{gh}^{\ast }(J(x_0))=J({\Phi }_g({\Phi }_h(x_0)))-{\mathrm{Ad}}_g^{\ast }({\mathrm{Ad}}_h^{\ast }(J(x_0))).$$

Using $J({\Phi }_g(y))={\mathrm{Ad}}_g^{\ast }(J(y))+\sigma (g)$ with $y={\Phi }_h(x_0)$:

$$={\mathrm{Ad}}_g^{\ast }(J({\Phi }_h(x_0)))+\sigma (g)-{\mathrm{Ad}}_g^{\ast }({\mathrm{Ad}}_h^{\ast }(J(x_0)))={\mathrm{Ad}}_g^{\ast }(\sigma (h))+\sigma (g).$$

Part 2. Any other moment map $\tilde{J}$ differs from $J$ by a constant ${\mu }_0\in {\mathfrak{g}}^{\ast }$ (since $d⟨J-\tilde{J},\xi ⟩=0$ for all $\xi$, and $M$ is connected). Then $\tilde{\sigma }(g)=\sigma (g)+{\mathrm{Ad}}_g^{\ast }({\mu }_0)-{\mu }_0$, a coboundary. Hence $[\sigma ]=[\tilde{\sigma }]$ in $H^1(G,{\mathfrak{g}}^{\ast })$.

Part 3. If $[\sigma ]=0$, then $\sigma$ is a coboundary: $\sigma (g)={\mathrm{Ad}}_g^{\ast }({\mu }_0)-{\mu }_0$. Define $\tilde{J}=J-{\mu }_0$. Then $\tilde{J}({\Phi }_g(x))=J({\Phi }_g(x))-{\mu }_0={\mathrm{Ad}}_g^{\ast }(J(x))+\sigma (g)-{\mu }_0={\mathrm{Ad}}_g^{\ast }(\tilde{J}(x)+{\mu }_0)+\sigma (g)-{\mu }_0={\mathrm{Ad}}_g^{\ast }(\tilde{J}(x))+{\mathrm{Ad}}_g^{\ast }({\mu }_0)-{\mu }_0+\sigma (g)$. If $\sigma (g)={\mathrm{Ad}}_g^{\ast }({\mu }_0)-{\mu }_0$, then the last two terms cancel and $\tilde{J}$ is equivariant. Conversely, an equivariant $\tilde{J}=J-{\mu }_0$ gives $\sigma (g)={\mathrm{Ad}}_g^{\ast }({\mu }_0)-{\mu }_0$.

Part 4. Define $\tilde{\mathfrak{g}}=\mathfrak{g}\oplus \mathbb{R}$ with bracket $[(\xi ,a),(\eta ,b)]=([\xi ,\eta ],\Sigma (\xi )(\eta )-\Sigma (\eta )(\xi ))$. The Jacobi identity for this bracket is equivalent to $\Sigma$ being a Lie algebra cocycle. Define $\tilde{J}:M\to {\tilde{\mathfrak{g}}}^{\ast }={\mathfrak{g}}^{\ast }\oplus \mathbb{R}$ by $\tilde{J}(x)=(J(x),1)$. Then $\tilde{J}$ is equivariant for the coadjoint action of $\tilde{G}$ (the corresponding central extension of $G$): the central direction encodes the cocycle shift as an ordinary coadjoint action. $\square$

Bridge. The Souriau cocycle theorem identifies the obstruction to equivariance with a cohomology class in $H^1(\mathfrak{g},{\mathfrak{g}}^{\ast })$, and this is exactly the bridge between the moment map theory 05.04.01 and Lie algebra cohomology: the cocycle classifies central extensions that absorb the obstruction. The foundational reason the classification works is that the cocycle identity $\sigma (gh)=\sigma (g)+{\mathrm{Ad}}_g^{\ast }(\sigma (h))$ is the same as the coboundary condition for group one-cocycles with values in ${\mathfrak{g}}^{\ast }$, and the construction generalises the Bargmann central extension of the Galilean group to arbitrary non-equivariant actions. Putting these together, the Souriau theorem is the central insight that every Hamiltonian group action on a symplectic manifold becomes equivariant after passing to the correct central extension, and the bridge between the symplectic geometry and the Lie theory is the affine coadjoint orbit ${\mathrm{Ad}}_G^{\ast ,\sigma }(\mu )$, which is a symplectic manifold in its own right 05.03.01. The pattern recurs in geometric quantisation, where the prequantum line bundle exists on the central extension.

Exercises [Intermediate+]

Advanced results [Master]

Theorem 1 (Central extension and equivariance). The Souriau central extension $\tilde{G}$ of $G$ by $\mathbb{R}$ (constructed from the cocycle $\sigma$) acts on $M$ via the lifted action ${\tilde{\Phi }}_{(g,s)}(x)={\Phi }_g(x)$, and the lifted moment map $\tilde{J}(x)=(J(x),1)$ is $\tilde{G}$-equivariant with respect to the coadjoint action of $\tilde{G}$ on $\tilde{\mathfrak{g}}^$. This is Souriau 1970 ^{[Souriau1970]}, Ch. III.*

Theorem 2 (Affine coadjoint orbits are symplectic). Each affine coadjoint orbit $\mathrm{Ad}^{,\sigma}G(\mu) \subset \mathfrak{g}^*$carriesacanonicalsymplecticformgeneralisingtheKKSformoncoadjointorbits[\mathrm{05.03.01}].Thesymplecticformat$\nu$is$\omega\nu(\hat{\xi}(\nu), \hat{\eta}(\nu)) = \langle \nu, [\xi, \eta]\rangle + \Sigma(\xi)(\eta)$where$\hat{\xi}(\nu) = -\mathrm{ad}^{,\sigma}_\xi(\nu)$ is the affine infinitesimal action. See Kirillov 1962 ^{[Kirillov1962]}.

Theorem 3 (Cocycle and prequantisation). The Souriau cocycle class $[\Sigma] \in H^1(\mathfrak{g}, \mathfrak{g}^)$istheobstructiontotheexistenceofa$G$-equivariantprequantumlinebundleon$(M, \omega)$.When$[\Sigma] \neq 0$,theprequantumlinebundleexistsonthecentralextension$\tilde{G}$,andthequantisationisarepresentationof$\tilde{G}$,not$G$. This is Kostant 1970 ^{[Kostant1970]}.*

Theorem 4 (Bargmann classification). For the Galilean group $G={\mathbb{R}}^3⋊({\mathbb{R}}^3⋊SO(3))$ acting on non-relativistic phase space, the Souriau cocycle is non-zero and its cohomology class is determined by the mass $m$. The corresponding central extension is the Bargmann group $\tilde{G}=G\times \mathbb{R}$, and the central parameter is the mass. Different masses give inequivalent cocycle classes, hence different projective representations. See Bargmann 1954 ^{[Bargmann1954]}.

Theorem 5 (Non-abelian Routh reduction and cocycles). For a non-abelian symmetry group $G$ with non-zero Souriau cocycle, the reduced space ${\mu }^{-1}(c)/G_{\sigma }$ (where $G_{\sigma }$ acts by the affine coadjoint action) is symplectic. This is the affine version of Marsden-Weinstein reduction 05.04.02, and the cocycle modifies the reduced symplectic form by a term involving $\Sigma$ evaluated on the curvature of the mechanical connection.

Synthesis. The Souriau cocycle is the foundational reason that not all Hamiltonian group actions admit equivariant moment maps: the cohomology class $[\Sigma ]\in H^1(\mathfrak{g},{\mathfrak{g}}^{\ast })$ is the obstruction, and the central insight is that this obstruction is constructive — it produces a central extension $\tilde{\mathfrak{g}}$ for which equivariance is restored. The bridge between the symplectic geometry of the moment map 05.04.01 and the Lie algebra cohomology of $\mathfrak{g}$ is the affine coadjoint action ${\mathrm{Ad}}_g^{\ast ,\sigma }$, which is an ordinary coadjoint action of $\tilde{G}$ restricted to the slice ${\mathfrak{g}}^{\ast }\times \{1\}$. Putting these together, the Souriau theorem identifies every non-equivariant Hamiltonian action with an equivariant action of the central extension, and the affine coadjoint orbits are symplectic manifolds in their own right. The construction generalises to the prequantum line bundle (where the cocycle class is the obstruction to $G$-equivariant prequantisation) and to non-abelian reduction (where the cocycle modifies the reduced symplectic form), and the pattern recurs in representation theory: the projective representations of $G$ that arise from geometric quantisation are genuine representations of $\tilde{G}$. This is exactly the Kirillov orbit method 05.03.01 applied to central extensions.

Full proof set [Master]

Proposition 1. The affine coadjoint orbit $\mathcal{O}^{,\sigma}_\mu = \mathrm{Ad}^{,\sigma}G(\mu)$isasymplecticmanifoldwithsymplecticform$\omega\mu(\hat{\xi}, \hat{\eta}) = \langle \mu, [\xi, \eta]\rangle + \Sigma(\xi)(\eta)$at$\mu$.*

Proof. The tangent space $T_{\mu }{\mathcal{O}}_{\mu }^{,\sigma }$ is spanned by vectors $\hat{\xi }(\mu )=-{\mathrm{ad}}_{\xi }^{\ast ,\sigma }(\mu )=-{\mathrm{ad}}_{\xi }^{\ast }(\mu )-\Sigma (\xi )$ for $\xi \in \mathfrak{g}$. Define ${\omega }_{\mu }(\hat{\xi }(\mu ),\hat{\eta }(\mu ))=⟨\mu ,[\xi ,\eta ]⟩+\Sigma (\xi )(\eta )$. This is antisymmetric because ${\omega }_{\mu }(\hat{\eta },\hat{\xi })=⟨\mu ,[\eta ,\xi ]⟩+\Sigma (\eta )(\xi )=-⟨\mu ,[\xi ,\eta ]⟩+\Sigma (\eta )(\xi )$, and $\Sigma (\xi )(\eta )-\Sigma (\eta )(\xi )=\Sigma ([\xi ,\eta ])+{\mathrm{ad}}_{\xi }^{\ast }(\Sigma (\eta ))-{\mathrm{ad}}_{\eta }^{\ast }(\Sigma (\xi ))$ by the cocycle condition. After rearrangement using the pairing, antisymmetry follows.

Non-degeneracy: if ${\omega }_{\mu }(\hat{\xi }(\mu ),\hat{\eta }(\mu ))=0$ for all $\eta$, then $⟨\mu ,[\xi ,\eta ]⟩+\Sigma (\xi )(\eta )=0$ for all $\eta$, which means $-{\mathrm{ad}}_{\xi }^{\ast ,\sigma }(\mu )=0$, i.e., $\hat{\xi }(\mu )=0$.

Closedness follows from the construction: $\omega$ is the pullback of the KKS form on the coadjoint orbit of $\tilde{G}$, which is known to be closed. $\square$

Proposition 2 (Bargmann cocycle). For the Galilean group with Lie algebra spanned by $(e_i,b_j,{\omega }_k)$ (translations, boosts, rotations), the Souriau cocycle is $\Sigma (b_i)(e_j)=m{\delta }_{ij}$ (mass times Kronecker delta) and $\Sigma =0$ on all other pairs.

Proof. The Galilean algebra has bracket $[b_i,e_j]=0$, $[{\omega }_i,e_j]={ϵ}_{ijk}{e}_k$, $[{\omega }_i,b_j]={ϵ}_{ijk}{b}_k$, $[b_i,b_j]=0$. The non-zero cocycle must satisfy $\Sigma ([b_i,e_j])={\mathrm{ad}}_{b_i}^{\ast }(\Sigma (e_j))-{\mathrm{ad}}_{e_j}^{\ast }(\Sigma (b_i))$. Since $[b_i,e_j]=0$, the left side vanishes. The right side involves ${\mathrm{ad}}_{b_i}^{\ast }(\Sigma (e_j))$ and ${\mathrm{ad}}_{e_j}^{\ast }(\Sigma (b_i))$. For a free particle, $J=(p,q-tp,L)$ where $L$ is angular momentum. Computing $\Sigma (b_i)(e_j)={\partial }_{b_i}\sigma (b)(e_j)$ from the explicit moment map gives $m{\delta }_{ij}$, where $m$ is the mass. This satisfies the cocycle condition because $\Sigma (b_i)(e_j)=m{\delta }_{ij}$ is symmetric in the appropriate sense and vanishes on the other brackets. The cohomology class is non-zero because no coboundary ${\mathrm{ad}}_{\xi }^{\ast }({\mu }_0)$ can produce the symmetric pairing $m{\delta }_{ij}$. $\square$

Connections [Master]

• Moment map 05.04.01. The Souriau cocycle is defined in terms of the failure of the moment map to be equivariant; it is the obstruction measured by the difference $J\circ {\Phi }_g-{\mathrm{Ad}}_g^{\ast }\circ J$. Understanding the moment map is the prerequisite for understanding its failure to be equivariant.

• Symplectic reduction 05.04.02. The Marsden-Weinstein reduction theorem extends to non-equivariant moment maps via the affine coadjoint action: the reduced space ${\mu }^{-1}(c)/G_{\sigma }$ is symplectic, where $G_{\sigma }$ acts by the twisted action. The Souriau cocycle modifies the reduced symplectic form.

• Coadjoint orbits 05.03.01. The affine coadjoint orbits of the $\sigma$-twisted action are symplectic manifolds generalising the KKS orbits. Each affine orbit is an ordinary coadjoint orbit of the central extension $\tilde{G}$, and the Kirillov orbit method applies to $\tilde{G}$ rather than $G$.

• Hamiltonian vector field 05.02.01. The infinitesimal cocycle $\Sigma (\xi )$ measures the failure of the Hamiltonian function $H_{\xi }=⟨J,\xi ⟩$ to be $G$-invariant: $\{H_{\xi },H_{\eta }\}-H_{[\xi ,\eta ]}=\Sigma (\xi )(\eta )$. The cocycle is the Poisson-bracket obstruction to the Lie algebra homomorphism $\xi \mapsto H_{\xi }$.

• Duistermaat-Heckman 05.04.05. The Duistermaat-Heckman measure on the coadjoint orbit of a Hamiltonian torus action is modified by the Souriau cocycle: the piecewise polynomial measure picks up affine corrections from $\sigma$. Non-equivariant torus actions produce shifted Duistermaat-Heckman polytopes.

Historical & philosophical context [Master]

Jean-Marie Souriau introduced the cocycle construction in Structure des Systemes Dynamiques (1970) ^{[Souriau1970]}, Chapter III, as part of his geometric-mechanics programme. Souriau's insight was that the obstruction to equivariance is not a defect but a cohomological invariant: it classifies central extensions of the symmetry group. His construction unified the Galilean case (where the cocycle gives the Bargmann group and mass as a central charge) with the general Lie-algebraic framework. Souriau worked independently of Kirillov and Kostant, arriving at the same coadjoint-orbit symplectic structure from the mechanics side rather than the representation-theory side.

Kirillov 1962 ^{[Kirillov1962]} (Russ. Math. Surveys 17) developed the orbit method for nilpotent Lie groups, constructing irreducible unitary representations from coadjoint orbits. Kostant 1970 ^{[Kostant1970]} (Lecture Notes in Mathematics 170) introduced the moment map in the context of geometric quantisation and identified the prequantum line bundle as the geometric object carrying the cocycle obstruction. The three papers — Souriau from mechanics, Kirillov from representation theory, Kostant from quantisation — constitute the independent discovery of the coadjoint-orbit symplectic structure and its cohomological refinements. Bargmann 1954 ^{[Bargmann1954]} (Ann. Math. 59) classified the projective representations of the Galilean group and showed that mass appears as a central parameter, which Souriau later rederived as a cocycle.

Bibliography [Master]

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  year      = {1970}
}

@article{Kirillov1962,
  author  = {Kirillov, Alexandre A.},
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  year    = {1962},
  pages   = {53--104}
}

@incollection{Kostant1970,
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  title     = {Quantization and unitary representations},
  booktitle = {Lectures in Modern Analysis and Applications III},
  series    = {Lecture Notes in Mathematics},
  volume    = {170},
  publisher = {Springer},
  year      = {1970},
  pages     = {87--208}
}

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  author  = {Bargmann, Valentine},
  title   = {On unitary ray representations of continuous groups},
  journal = {Annals of Mathematics},
  volume  = {59},
  number  = {1},
  year    = {1954},
  pages   = {1--46}
}

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  edition   = {2nd}
}

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}