Quantum gauge symmetries in Noncommutative Geometry

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by  Jyotishman Bhowmick

Quantum gauge symmetries in Noncommutative Geometry
Jyotishman Bhowmick1, Francesco D’Andrea2, Biswarup Das3, Ludwik Dabrowski4
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1 University of Oslo, Moltke Moes vei 35, 0316 Oslo, Norway
2 Universita` degli Studi di Napoli Federico II, P.le Tecchio 80, I-80125 Naples, Italy
3 Indian Statistical Institute, 203, B.T. Road, Kolkata, India
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1 4 Scuola Internazionale Superiore di Studi Avanzati (SISSA), via Bonomea 265, I-34136 Trieste, Italy
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2

c
e Abstract
D
Wediscussgeneralizationsofthe notionofi)thegroupofunitaryelementsofa(realorcom-
5 ∗
plex)finite dimensional C -algebra,ii) gaugetransformationsand iii) (real)automorphisms,
1
in the frameworkof compactquantum grouptheory and spectral triples. The quantumana-

] logue of these groups are defined as universal (initial) objects in some natural categories.
A
After proving the existence of the universal objects, we discuss several examples that are
Q
of interest to physics, as they appear in the noncommutative geometry approach to particle
.
h physics: inparticular,theC∗-algebrasMn(R), Mn(C)andMn(H),describingthe finite non-
at commutativespaceoftheEinstein-Yang-Millssystems,andthealgebrasAF =C⊕H⊕M3(C)
m and Aev =H⊕H⊕M4(C), that appear in Chamseddine-Connes derivation of the Standard
[ Model of particle physics minimally coupled to gravity. As a byproduct, we identify a“free”

1 versionof the symplectic group Sp(n) (quaternionic unitary group).
v
2
2
6
3 1 Introduction
.
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1
In the approach to particle physics from noncommutative geometry [13, 11], the dynamics of a
1
1 theory is obtained from the asymptotic expansion of the spectral action associated to an almost
v: commutativespectraltriple(A∞,H,D,J), i.e.aproductofthecanonical spectraltripleofaspin
i
X manifold and a finite-dimensional one (see e.g. [17] and references therein). A fundamental role
r is played by the group U(A∞) of unitary elements of the algebra, whose adjoint representation
a
u 7→ uJuJ−1 on H gives the group
G(A∞,J) = {uJuJ−1,u ∈ U(A∞)} (1.1)
of inner fluctuations of the real spectral triple [17, Sec. 10.8], also called “gauge group”of the
spectral triple, for its relation with thegauge group of physics[21]. For example, intheEinstein-
Yang-Mills system, thefinite-dimensionalspectral tripledescribingthe internalnoncommutative
space is built from the algebra A = M (C), with Hilbert space H = M (C) carrying the left
I n I n
regular representation and real structure J given by the hermitian conjugation; in this case
I
U(A ) = U(n) and G(A ,J ) is the classical gauge group SU(n), modulo a finite group given by
I I I
its center. In the more elaborated example of (the Euclidean version of) the Standard Model of
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Quantum gauge symmetries in NCG J. Bhowmick, F. D’Andrea, B. Das & L. Dabrowski
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elementary particles minimally coupled to gravity, the algebra is A = C⊕H⊕M (C) and the
F 3
group G(A ,J ) is U(1)×SU(2)×SU(3) modulo Z .
I I 6
More generally, consider a spectral triple based on an almost commutative algebra
A∞ := C∞(M)⊗A ≃ C∞(M → A ), dimA < ∞ ,
I I I
with M a closed Riemannian spin manifold. A basic idea is that every physical interaction
comes from a suitable“symmetry”of the above almost commutative space: particle interactions
from local gauge symmetries, and gravitational interactions from the symmetry under diffeo-
morphisms. It is natural to think that a first step in the unification of particle interactions with
gravity is the unification of this two types of symmetries. The key for this unification is the split
short exact sequence:
1 −→ Inn(A∞)−→ Aut(A∞) −→ Out(A∞)−→ 1 .
In the cases of physical interest, namely in the Einstein Yang Mills system and the the Standard
Model, Out(A ) is trivial, so that Out(A∞) = Out(C∞(M)) ≃ Diff(M) and the automorphism
I
group Aut(A∞) (the group of symmetries of the full“noncommutative space”) is a semidirect
product of the group of diffeomorphisms of M and the group Inn(A∞) = C∞(M → Inn(A ))
I
of smooth functions with values in Inn(A ) = U(A )/Z , where Z is the center of U(A ). The
I I I I I
group Inn(A∞) is what we call the local gauge group of the theory, while Inn(A ) is the global
I
gauge group, or gauge group ‘tout court’.
Ontheotherhand,thegroupG(A∞,J)in(1.1)isisomorphictothequotientU(A∞)/U(A ),
J
where A := {a ∈ A∞ : aJ = Ja∗} is a real ∗-subalgebra of the center of A∞ [21]. One has
J
G(A∞,J) ⊃ Inn(A∞), with equality iff A is exactly the center of A: this happens in both the
J
Einstein-Yang-Mills system and the Standard Model examples, and from G(A∞,J), one recovers
the local gauge transformations of physics, while G(A ,J ) gives the global ones.
I I
Given the importance of the group of gauge transformations in physics, it is very natural, in
the framework of noncommutative geometry, to look for compact quantum group analogues of
this notion. In fact, the idea of using quantum group symmetries to understand the conceptual
significance of the algebra A is mentioned in a final remark by Connes in [15]. In [9], an
F
approach along this line was made, where the quantum isometry group (in the sense of [23, 8])
of the finite part of the Standard Model was computed. It was shown that its coaction, once
extended to the whole spectral triple on C∞(M)⊗A , leaves invariant both the bosonic and
F
fermionic part of the spectral action, thus providingus with genuine quantum symmetries of the
Standard Model. In this article, we wish to continue the work in [9] by investigating the notion
of quantum gauge symmetries which might be helpful in having a better understanding of the
noncommutative geometry approach to particle physics. Since in many of the applications to
physics, the relevant algebra is the product of a commutative one with a finite noncommutative
one described by a finite-dimensional C∗-algebra, we restrict our attention to finite-dimensional
C∗-algebras. On the other hand, we need to consider both complex and real C∗-algebras, since
in one of the main applications of spectral triples to physics – the Standard Model of elementary
particles, the C∗-algebra involved is real.
It is evident that in order to have a correct quantum analogue of (1.1), we first need to make
sense of a compact quantum group version of the unitary group of a finite dimensional (possibly
real) C∗-algebra, and then use it to define the quantum gauge group. It is natural to wonder
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whether the free quantum groups A (n) or their twisted counterparts (denoted by A (n,R) in
u u
thisarticle), firstappearingintheseminalworksofWang andVan Daele[32,33,34]can playthe
role of quantum group of unitaries of M (C). The definition of these compact quantum groups
n
are recalled in Sec. 2.1. The structure and isomorphism classification of these quantum groups
were studied in [35]. Since then, a considerable amount of literature has been developed around
these quantum groups (see e.g. [3, 4, 10] for quantum symmetries of finite metric spaces and
graphs), which have also made contact with other branches of mathematics, like combinatorics
and free probability [5, 6]. We believe that the compact quantum group version of the unitary
group is also important from the point of view of compact quantum group theory. Indeed, we
will see that we obtain A (n,R) as the quantum unitary group of M (C) whose adjoint action
u n
preserves the state Tr(Rt.), where R is any positive invertible n×n matrix. The dependence
on R appears because unlike the classical case, a compact quantum group coaction on M (C)
n
does not need to preserve the usual trace. A byproduct of this construction for real C∗-algebras
shows that a“free”analogue of the symplectic group Sp(n) (quaternionic unitary group) can be
realized as the quantum unitary group of the real C∗-algebra M (H).
n
The plan of the paper is as follows. In Sec. 2 we recall some necessary background about
compact – and in particular free – quantum groups, spectral triples and real C∗-algebras. In
Sec. 3, inspired by the characterization of the group of unitaries of a C∗-algebra A as the
universal object in a certain category of groups having a trace preserving action on A, we define
the quantum analogue by passing to the category of quantum families (in the spirit of [37, 29])
and relaxing the condition of traciality of the state, that is necessary in order to accommodate
non-Kactypeexamples likeA (n,R). We provethat theuniversalobject– thatwecall quantum
u
unitary group – exists and has a compact quantum group structure, by explicitly computing it
for any finite-dimensional (complex and real) C∗-algebra.
In Sec. 4, we generalize the construction (1.1) and define the quantum gauge group of a
finite dimensional spectral triple, and compute it for three examples, namely the Einstein Yang
Mills system, the spectral triple over the algebra Aev = H ⊕H ⊕M (C) and for the spectral
4
triple for the finite part of the Standard Model. Finally, in Sec. 5, we discuss some aspects of
quantum symmetries of finite-dimensional real C∗-algebras which were not dealt with, in [9]. In
particular, we prove the existence of quantum automorphism group for any finite-dimensional
real C∗-algebra and prove that for matrix algebras M (k), with k = R,C or H, the quantum
n
automorphism group coincides with the classical one.
Throughout the paper, by the symbol ⊗ we will always mean the algebraic tensor product
alg
over C, by ⊗ minimal tensor product of complex C∗-algebras or the completed tensor product
of Hilbert modules over complex C∗-algebras. The symbol ⊗R will denote the tensor product
over the real numbers. Unless otherwise stated, all algebras are assumed to be unital complex
associative involutive algebras. We denote by M(A) the multiplier algebra of the complex
C∗-algebra A, by L(H) the adjointable operators on the Hilbert module H and by K(H) the
compact operators on the Hilbert spaceH. With the symbol{e } weindicate the canonical
i 1≤i≤n
orthonormal basis of Cn, with {e } the standard basis of M (C) (e is the matrix with
ij 1≤i,j≤n n ij
1 in position (i,j) and zero everywhere else), and with I the n×n identity matrix.
n
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2 Compact quantum groups and spectral triples
2.1 Some generalities on compact quantum groups
Webeginbyrecallingthedefinitionofcompactquantumgroupsandtheircoactionsfrom[38,40].
Weshallusemostoftheterminologyof[33],forexampleWoronowiczC∗-subalgebra,Woronowicz
C∗-ideal, etc., however with the exception that Woronowicz C∗-algebras will be called compact
quantum groups, and we will not use the term compact quantum groups for the dual objects as
done in [33].
Definition 2.1. A compact quantum group (to be denoted by CQG from now on) is a pair
(Q,∆) given by a complex unital C∗-algebra Q and a unital C∗-homomorphism ∆ :Q → Q⊗Q
such that
i) ∆ is coassociative, i.e.
(∆⊗id)◦∆ = (id⊗∆)◦∆
as equality of maps Q → Q⊗Q⊗Q;
ii) Span (a⊗1 )∆(b) a,b∈Q and Span (1 ⊗a)∆(b) a,b∈Q are norm-dense in Q⊗Q.
Q Q
(cid:8) (cid:12) (cid:9) (cid:8) (cid:12) (cid:9)
For Q = C(G), where G is a compact topological group, conditions i) and ii) correspond to the
(cid:12) (cid:12)
associativity and the cancellation property of the product in G, respectively.
Definition 2.2. A unitary corepresentation of a compact quantum group (Q,∆) on a Hilbert
space H is a unitary element U ∈ M(K(H)⊗Q) satisfying
(id⊗∆)U = U U ,
(12) (13)
where we use the standard leg numbering notation (see e.g. [25]). The corepresentation U is
faithful if there is no proper C∗-subalgebra Q′ of Q such that U ∈ M(K(H)⊗Q′).
If Q = C(G), U corresponds to a strongly continuous unitary representation of G.
For any compact quantum group Q (see [38, 40]), there always exists a canonical dense
∗-subalgebra Q ⊂ Q which is spanned by the matrix coefficients of the finite dimensional
0
unitary corepresentations of Q and two maps ǫ : Q → C (counit) and κ : Q → Q (antipode)
0 0 0
which make Q a Hopf ∗-algebra.
0
Definition 2.3. A Woronowicz C∗-ideal of a CQG (Q,∆) is a C∗-ideal I of Q such that ∆(I) ⊂
ker(π ⊗π ), where π : Q → Q/I is the quotient map. The quotient Q/I is a CQG with the
I I I
induced coproduct.
If Q = C(G) are continuous functions on a compact topological group G, closed subgroups
of G correspond to the quotients of Q by its Woronowicz C∗-ideals. While quotients Q/I give
“compact quantum subgroups”, C∗-subalgebras Q′ ⊂ Q such that ∆(Q′) ⊂ Q′ ⊗ Q′ describe
“quotient quantum groups”.
Definition 2.4. We say that a CQG (Q,∆) coacts on a unital C∗-algebra A if there is a unital
C∗-homomorphism (called a coaction) α :A → A⊗Q such that:
i) (α⊗id)α = (id⊗∆)α,
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ii) Span α(a)(1 ⊗b) a ∈A, b ∈ Q is norm-dense in A⊗Q.
A
The coactio(cid:8)n is faithful if(cid:12) any CQG Q′(cid:9)⊂ Q coacting on A coincides with Q.
(cid:12)
It is well known (cf. [26, 34]) that condition (ii) in Def. 2.4 is equivalent to the existence of a
norm-dense unital ∗-subalgebra A of A such that the map α, restricted to A , gives a coaction
0 0
of the Hopf algebra Q , that is to say: α(A ) ⊂ A ⊗ Q and (id⊗ǫ)α = id on A .
0 0 0 alg 0 0
For later use, let us now recall the concept of certain universal CQGs defined in [32, 35] and
references therein.
Definition 2.5. For a fixed n×n positive invertible matrix R, A (n,R) is the universal C∗-
u
algebra generated by {u , i,j = 1,...,n} such that
ij
uu∗ = u∗u= I , ut(RuR−1) = (RuR−1)ut = I ,
n n
where u := ((u )), u∗ := ((u∗ )) and u := (u∗)t. It is equipped with the ’matrix’ coproduct ∆
ij ji
given on the generators by
∆(u )= u ⊗u .
ij ik kj
k
Note that u is a unitary corepresentation of AX(n,R) on Cn.
u
The A (n,R)’s are universal in the sense that every compact matrix quantum group (i.e. ev-
u
ery CQG generated by the matrix entries of a finite-dimensional unitary corepresentation) is
a quantum subgroup of A (n,R) for some R > 0, n > 0 [35]; in particular, the well-known
u
quantum unitary group SU (n) is a quantum subgroup of some A (n,R) (cf. Sec. 2.2). It may
q u
also be noted that A (n,R) is the universal object in the category of CQGs which admit a uni-
u
tary corepresentation on Cn such that the adjoint coaction on the finite-dimensional C∗-algebra
M (C) preserves the functional M (C) ∋m 7→ Tr(Rtm) (see [36]).
n n
More generally, for any invertible matrix F, an analogous construction can be done.
Definition 2.6 ([1, 2]). Let F ∈ GL (C). A CQG denoted A (n,F) is defined as the universal
n u
C∗-algebra generated by u= {u , i,j = 1,...,n} with the condition that both u and u′ = FuF−1
ij
are unitary; equipped with the standard ’matrix’ coproduct. A quantum subgroup of A (n,F),
u
denoted by A (n,F), is defined by the additional relation u= u′.
o
One immediately realizes that u′u′∗ = Fu(F∗F)−1utF∗ = I if and only if RuR−1ut = I
n n
and u′∗u′ = (F∗)−1utF∗FuF−1 = I if and only if utRuR−1 = I , where R = F∗F. Thus
n n
A (n,F) actually depend only on the modulus of F and is isomorphic to A (n,R) for R =F∗F.
u u
Thus, A (n,F) is also a quantum subgroup of A (n,R) for R = F∗F.
o u
Since we will need both the quantum groups mentioned above, for clarity, we will use the
symbolA (n,F)orA (n,F)whenF neednotbeapositivematrixanduseR whenitispositive.
u o
It is proved in [35] that A (n,F) can always be decomposed as A (n,F) ≃ A (n ,F )∗...∗
o o o 1 1
A (n ,F ), whereF ∈GL (C) satisfy F F ∈ CI (and n = n); thusup to a iterated free
o k k k nk k k k k k
product, in Definition 2.6 one can assume FF ∈ CI . Note also that concerning the notation
n P
for free quantum orthogonal groups, we follow here that of [1], which corresponds to B (Q) in
u
[35] for Q = F∗. We refer to [35] for a detailed discussion on the structure and classification of
such quantum groups.
We remark that the CQGs A (n) := A (n,I ) and A (n) := A (n,I ) are called the free
u u n o o n
quantum unitary group and free quantum orthogonal group, respectively, as their quotient by the
commutator ideal is respectively C(U(n)) and C(O(n)).
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Remark 2.7. Let n = 2m be even and F = σ ⊗I , where we identify M (C) with M (C)⊗
2 m 2m 2
M (C) and
m
0 −i
σ = (2.1)
2
i 0
(cid:18) (cid:19)
is the second Pauli matrix. In this case, the CQG A (2m,F) will be denoted A (m) and it is a
o sp
free version of the symplectic group Sp(m) (the group of unitary elements of M (H)), that can
m
be obtained as the quotient of A (m) by the commutator subalgebra (cf. Sec. 3.2). We will see
sp
in Sec. 3.2 that A (m) is the quantum unitary group of M (H). The identification of A (m)
sp m sp
with A (2m,F) for a special F was pointed out to us by T. Banica.
o
A matrix B (with entries in a unital ∗-algebra B) such that both B and Bt are unitary is
called a biunitary [7]. We will also need the following class of CQGs:
Definition 2.8. For a fixed n, we call A∗(n) the universal unital C∗-algebra generated by an
u
n×n biunitary u= ((u )) with relations
ij
ab∗c = cb∗a , ∀ a,b,c ∈ {u , i,j = 1,...,n} . (2.2)
ij
A∗(n) is a CQG with coproduct given by ∆(u )= u ⊗u .
u ij k ik kj
We will call A∗(n) the N-dimensional half-libePrated unitary group. This is similar to the
u
half-liberated orthogonal group A∗(n), that can be obtained by imposing the further relation
o
a = a∗ for all a ∈ {u , i,j,= 1,...,n} (cf. [7]).
ij
The analogue of projective unitary groups was introduced in [2] (see also Sec. 3 of [7]). Let
us recall the definition.
Definition 2.9. Let Q be a CQG which is generated by the matrix elements of a unitary corep-
resentation U. The projective version PQ of Q is the Woronowicz C∗-subalgebra of Q generated
by the entries of U⊗U(cf. section 3 of [7]). In particular, PA (n) is the C∗-subalgebra of A (n)
u u
generated by {u (u )∗ : i,j,k,l = 1,...,n}.
ij kl
In [34], Wang defines the quantum automorphism group of M (C), denoted by A (M (C))
n aut n
tobetheuniversalobjectinthecategory ofCQGswithacoaction onM (C)preservingthetrace
n
(and with morphisms given by CQGs homomorphisms intertwining the coactions). The explicit
definition is inTheorem 4.1 of [34]. We concludethis section bythefollowingproposition stating
Theorem 1(iv) from [2] (cf. also Prop. 3.1(3) of [7]) and a very special case (namely, q = 1) of
Theorem 1.1 from [28] together.
Proposition 2.10. We have PA (n) ≃ PA (n) ≃ A (M (C)) and A (M (C)) ≃ C(SO(3)).
u o aut n aut 2
Thus, PA (2) ≃ PA (2) ≃ C(SO(3)).
u o
2.2 Relation between free unitary groups and SU (n)
q
In this section, we discuss the relation between the quantum unitary groups A (n,R) and the
u
quantum groups SU (n) of [22, 31, 39].
q
For 0 < q ≤ 1, we recall the definition of SU (n) following the notations of [24, Sec. 9.2],
q
exceptthefactthatwewilluseu insteadofui todenotethematrixelementofuontherowiand
ij j
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Quantum gauge symmetries in NCG J. Bhowmick, F. D’Andrea, B. Das & L. Dabrowski
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column j. The CQG is generated by the matrix elements of an n-dimensional corepresentation
u = (u ), i,j = 1,...,n, with commutation relations
ij
u u = qu u u u = qu u ∀ i< j ,
ik jk jk ik ki kj kj ki
[u ,u ] = 0 [u ,u ] = (q−q−1)u u ∀ i< j, k < l ,
il jk ik jl il jk
and with determinant relation
D = (−q)||p||u u ...u = 1 ,
q 1,p(1) 2,p(2) n,p(n)
p∈Sn
X
where the sum is over all permutations p of the set {1,2,...,n} and ||p|| is the number of
inversions in p. The ∗-structure is given by
(u )∗ =(−q)j−i (−q)||p||u u ...u
ij p∈Sn−1 k1,p(l1) k2,p(l2) kn−1,p(ln−1)
X
with {k ,...,k } = {1,...,n}r{i}, {l ,...,l } = {1,...,n}r{j} (as ordered sets) and
1 n−1 1 n−1
the sum is over all permutations p of the set {l ,...,l }.
1 n−1
From the defining relations, one derives the following ‘orthogonality’ relations between rows
resp. columns of u. For all a,b =1,...,n we have:
u (u )∗ = δ , (u )∗u = δ , (2.3)
ai bi a,b ia ib a,b
i i
q2X(i−b)u (u )∗ = δ , q2(Xa−i)(u )∗u = δ . (2.4)
ia ib a,b ai bi a,b
i i
X X
This is simply Prop. 8 of [24, Sec. 9.2.2], with quantum determinant D = 1 for SU (n), and
q q
cofactor matrix(−q)k−jAj = u˜k = S(uj),definedinpage 313 of [24]related totherealstructure
k j k
of SU (n) by the formula u∗ = S(u) = u˜t (cf. Sec. 9.2.4 of [24], case 2).
q
Now, equation (2.3) in matrix form is simply the unitarity condition uu∗ = u∗u = I . On
n
the other hand, if we call
1 qn−q−n
R = diag(1,q2,q4,...,q2(n−1)), [n] := , (2.5)
qn−1[n] q q−q−1
q
then
(RuR−1) = q2(i−j)(u )∗
ij ij
and(2.4)isequivalenttotheconditionsut(RuR−1)= (RuR−1)ut = I . ThisprovesthatSU (n)
n q
is a quantum subgroup of the free unitary group A (n,R), for R as in (2.5). Clearly R is not
u
unique, for example one can multiply R for a constant, or replace R with R−1 (SU (n) and
q
SUq−1(n) are isomorphic, for any q ∈ R+).
For n = 2, ϕ is the well known Powers state of M (C) [28, eq. (1)]. In fact, this case was
R 2
already dealt with in Rem 1.31 [30], where it was proved that SU (2) is isomorphic to A (2,F)
q o
for
1
0 −q2
F = .
1
q−2 0
(cid:18) (cid:19)
Clearly, A (2,F) is a quantum subgroup of A (2,R′) for R′ =F∗F = diag(q−1,q).
o u
Notice that [n] R = π(K ), where K is the element of the dual Hopf ∗-algebra U (su(n))
q 2ρ 2ρ q
implementing the modular automorphism (cf. eq. (3.2) of [18]) and π is the fundamental repre-
sentation described in [18, eq. (4.1)].
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2.3 Generalities on real C∗-algebras
WeneedtorecallsomebasicfactsaboutrealC∗-algebras, whichwearegoingtoneedthroughout
the article. For more details on real C∗-algebras, we refer the reader to [27] and [20].
Definition 2.11. A real Banach algebra is a real algebra A, equipped with a norm k·k, which
makes it a real Banach space, and satisfying the condition kx·yk ≤ kxkkyk. If the algebra is
unital, with unit 1 ∈ A, one also requires k1k = 1. A unital real C∗-algebra A is an unital real
Banach ∗-algebra (i.e. a real Banach algebra, which is simultaneously a real ∗-algebra), such that
for any x ∈A: i) kx∗xk= kxk2 and ii) 1+x∗x is invertible in A.
The following result characterizes all finite dimensional real C∗-algebras.
Proposition 2.12. Let A be a finite dimensional real C∗-algebra. Then A ∼= M (D ) ⊕
n1 1
M (D )⊕M (D )⊕...⊕M (D ) (asreal C∗-algebras) for some positive integersn ,n ,...n ,
n2 2 n3 3 nk k 1 2 k
where for each i= 1,2,...k, D is either R, C or H.
i
For a real C∗-algebra A, the ∗-algebra AC = A⊗R C is a complex C∗-algebra, known as the
complexification of A. Moreover, A is the fixed point algebra of the antilinear automorphism σ
on AC = A⊗RC, given byσ(a⊗Rz) = a⊗Rz. Note that σ commutes with the involution on AC,
given by (a⊗Rz)∗ = a∗⊗Rz. Throughout this article, the symbol σ will stand for this antilinear
automorphism.
The following result recalls the complexifications and the formulas of σ for the finite dimen-
sional C∗-algebras M (R), M (C) and M (H).
n n n
Proposition 2.13. Let A:= Mn(k), and AC := A⊗RC. Then:
1. if k= R, then AC = Mn(C) and σ(m) = m;
2. if k= C, then AC = Mn(C)⊕Mn(C) and σ(m1⊕m2)= m2⊕m1;
3. if k = H, then AC = M2(C)⊗Mn(C) ∼= M2n(C) and σ(m) = (σ2 ⊗1n)m(σ2 ⊗1n), where
σ is the matrix (2.1).
2
2.4 Real spectral triples
In noncommutative geometry, compact Riemannian spin manifolds are replaced by real spectral
triples. Recallthataunitalspectraltriple(A∞,H,D)isthedatumof: aHilbertspaceH,aunital
associative involutive algebra A∞ with a faithful unital ∗-representation π : A → B(H) (the
representationsymbolisusuallyomitted)anda(not-necessarilybounded)self-adjointoperatorD
onHwithcompactresolventandhavingboundedcommutatorswithalla ∈ A∞,seee.g.[14,16].
A spectral triple is even if thereis a Z -grading γ on H commutingwith A∞ and anticommuting
2
with D; we will set γ = 1 when the spectral triple is odd. A spectral triple is real if there is an
antilinear isometry J :H → H, called the real structure, such that
J2 = ǫ1 , JD = ǫ′DJ , Jγ = ǫ′′γJ , (2.6)
and
[a,JbJ−1] = 0, [[D,a],JbJ−1] = 0 , (2.7)
for all a,b ∈ A∞ 1. ǫ, ǫ′ and ǫ′′ are signs and determine the KO-dimension of the space [14].
1In some examples (not in thepresent case) conditions (2.7) haveto beslightly relaxed, see e.g. [19].
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Quantum gauge symmetries in NCG J. Bhowmick, F. D’Andrea, B. Das & L. Dabrowski
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A canonical commutative example is given by (C∞(M),L2(M,S),D/) – where C∞(M) are
complex-valued smoothfunctionsonaclosed Riemannianspinmanifold,L2(M,S) istheHilbert
spaceofsquareintegrablespinorsandD/ istheDiracoperator. ThisspectraltripleisevenifMis
even-dimensional. In fact, from any commutative real spectral triple it is possible to reconstruct
a closed Riemannian spin manifold. We refer to [16] for the exact statement.
While we always tacitly assume that H is a complex Hilbert space, we allow the possibility
that A∞ is a real ∗-algebra. Note that to any real spectral triple (A∞,H,D,γ,J) over a real
∗-algebra A∞, we can associate a real spectral triple (B∞,H,D,γ,J) over a complex ∗-algebra
B∞,asshowninLemma3.1of[9]. WeletB∞ bethequotientA∞C /kerπC,whereA∞C ≃ A∞⊗RC
is the complexification of A∞, with conjugation defined by (a⊗Rz)∗ = a∗⊗Rz for a ∈ A∞ and
z ∈ C, and πC :A∞C → B(H) is the ∗-representation
πC(a⊗Rz) = zπ(a) , a ∈ A∞, z ∈ C. (2.8)
It was observed in [9] that kerπC may be nontrivial since the representation πC is not always
faithful. For example, if A∞ is itself a complex ∗-algebra (every complex ∗-algebra is also a real
∗-algebra) and π is complex linear, then for any a ∈ A∞ the element a⊗R1+ia⊗Ri of A∞C is
in the kernel of πC. In fact, if A∞ is a complex algebra,
B∞ ≃ A∞ .
We close this section with a remark. While usually A∞ is only a pre-C∗-algebra for the
operator norm, in the finite-dimensional case it is a C∗-algebra, and to make this fact more
evident it will be denoted by A, without the ∞ supscript.
3 Quantum unitary group of a finite-dimensional C∗-algebra
3.1 The case of complex C∗-algebras
Let A be a finite-dimensional complex C∗-algebra, that is,
m
A = M (C) (3.1)
ni
i=1
M
for some positive integers m and n . For a = a ⊕...⊕a ∈ A, we denote by Tr the trace map:
i 1 m
m ni
Tr(a) := (a ) .
i kk
i=1k=1
XX
AnyfaithfulstateofAisoftheformTr(R·)forsomepositiveinvertibleoperatorR := ⊕ R ∈ A
i i
with normalization Tr(R)= 1, called the density matrix of the state. Since in the following, the
normalization of R is irrelevant, in the particular case when R = 1 I is a scalar multiple of the
Tr(I)
identity, one can equivalently use the map Tr(·). Let π : A → B(L2(A,Tr(R·)) be the GNS
R
representation of a finite dimensional complex C∗-algebra A with respect to the faithful state
Tr(R·) as above. We define the functional
ϕ (π (a)) = Tr(Ra) , (3.2)
R R
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Quantum gauge symmetries in NCG J. Bhowmick, F. D’Andrea, B. Das & L. Dabrowski
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TheabovefunctionaliswelldefinedsincetheGNSrepresentationofaC∗-algebrawithrespectto
a faithful state is faithful. Throughout this article, the symbol ϕ will stand for this functional.
R
We startbystating thefollowing Lemma, whichgives acharacterization of the unitarygroup
of a finite dimensional complex C∗-algebra.
Lemma 3.1. Let A be a finite dimensional complex C∗-algebra, viewed as a subalgebra of
B(L2(A,Tr)) via the GNS representation π, and denote by π = π| its restriction to the
U U(A)
group U(A) of unitary elements of A. Then (U(A),π ) is the universal (final) object in the
U
category whose objects are pairs (G,π˜), with G a compact group and π˜ a unitary representation
of G on L2(A,Tr) satisfying π˜(g) ∈ A for all a ∈ A, and whose morphisms are continuous group
homomorphisms intertwining the representations.
Proof. Clearly (U(A),π ) is an object in the category (as a linear space L2(A,Tr)≃ A since the
U
normalized trace is a faithful state, and then π is a faithful representation). Moreover, if (G,π˜)
is any object in the category, since π is faithful there exists a unique morphism φ :G → U(A)
U
intertwining the representations, which is defined by φ(g) = (π )−1π(g) for all g ∈ G. This
U
shows the universality of (U(A),π ).
U
e
We defineanotionof quantumfamily ofunitariesbytakingasuitablenoncommutativeanalogue
of thischaracterization. Notice thatwhileU(A)isafinal objectinthecategory describedabove,
since the functor C is contravariant, the C∗-algebra C(U(A)) is a initial object in the dual
category.
Definition 3.2. Let A be a finite-dimensional complex C∗-algebra, R ∈ A a positive invertible
operator, ϕ as in (3.2), and let π : A → B L2(A,ϕ ) be the associated GNS representation.
R R R
We denote by Cu(A,R) the category whose objects are pairs (Q,U), with Q a unital C∗-algebra
(cid:0) (cid:1)
and U a unitary element in π (A)⊗Q such that:
R
(i) Ad =U(· ⊗1 )U∗ preserves the state ϕ on π (A),
U Q R R
(ii) AdU∗ = U∗(· ⊗1Q)U preserves the state ϕR−1 on πR(A),
A morphism φ: (Q,U) → (Q′,U′) is a C∗-homomorphisms such that (id⊗φ)(U) = U′.
We call Cu(A,R) the category of quantum families of R-unitaries of A.
Remark 3.3. Notice that condition (i) is equivalent to the condition that U not only preserves
the inner product ha,bi = ϕ (a∗b) of the GNS representation (that follows from U∗U =1), but
R R
also the sesquilinear form (a,b) = ϕ (ab∗). If we consider the subcategory whose objects (Q,U)
R R
are compact matrix quantum groups, condition (ii) can be derived from (i) using the properties
of the antipode.
To start with, we will prove that the universal (initial) object in the category Cu(Mn(C),R)
exists and is in fact A (n,Rt). Using this result we will prove that for any finite-dimensional
u
complex C∗-algebra A, Cu(A,R) has a universalobject which is in fact a CQG. We will call this
CQG the quantum R-unitary group of A and denote it by the symbol Q (A,R).
u
Proposition 3.4. The universal object in the category Cu(Mn(C),R) exists and it is isomorphic
to (A (n,Rt),U ), where U is the faithful unitary corepresentation defined by
u n n
n
U = π (e )⊗u , (3.3)
n R ij ij
i,j=1
X
10