The Exponential Map for Hopf Algebras
We give an analogue of the classical exponential map on Lie groups for Hopf ∗-algebras with differential calculus. The major difference with the classical case is the interpretation of the value of the exponential map, classically an element of the Lie group. We give interpretations as states on the...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2022 |
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Інститут математики НАН України
2022
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| Цитувати: | The Exponential Map for Hopf Algebras. Ghaliah Alhamzi and Edwin Beggs. SIGMA 18 (2022), 017, 17 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859566488892997632 |
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| author | Alhamzi, Ghaliah Beggs, Edwin |
| author_facet | Alhamzi, Ghaliah Beggs, Edwin |
| citation_txt | The Exponential Map for Hopf Algebras. Ghaliah Alhamzi and Edwin Beggs. SIGMA 18 (2022), 017, 17 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We give an analogue of the classical exponential map on Lie groups for Hopf ∗-algebras with differential calculus. The major difference with the classical case is the interpretation of the value of the exponential map, classically an element of the Lie group. We give interpretations as states on the Hopf algebra, elements of a Hilbert 𝐶∗-bimodule of 1/2 densities, and elements of the dual Hopf algebra. We give examples for complex-valued functions on the groups 𝑆₃ and ℤ, Woronowicz's matrix quantum group ℂq[𝑆𝑈₂], and the Sweedler-Taft algebra.
|
| first_indexed | 2026-03-13T16:53:20Z |
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| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 017, 17 pages
The Exponential Map for Hopf Algebras
Ghaliah ALHAMZI a and Edwin BEGGS b
a) Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia
E-mail: gyalhamzi@imamu.edu.sa
b) Department of Mathematics, Swansea University, Wales, UK
E-mail: e.j.beggs@swansea.ac.uk
URL: https://www.swansea.ac.uk/staff/science/maths/beggs-e-j/
Received June 15, 2021, in final form February 16, 2022; Published online March 09, 2022
https://doi.org/10.3842/SIGMA.2022.017
Abstract. We give an analogue of the classical exponential map on Lie groups for Hopf
∗-algebras with differential calculus. The major difference with the classical case is the
interpretation of the value of the exponential map, classically an element of the Lie group.
We give interpretations as states on the Hopf algebra, elements of a Hilbert C∗-bimodule
of 1
2 densities and elements of the dual Hopf algebra. We give examples for complex valued
functions on the groups S3 and Z, Woronowicz’s matrix quantum group Cq[SU2] and the
Sweedler–Taft algebra.
Key words: Hopf algebra; differential calculus; Lie algebra; exponential map
2020 Mathematics Subject Classification: 16T05; 46L87; 58B32
1 Introduction
For a Lie group G with Lie algebra g we have the exponential map exp: g → G [13]. We wish to
give a Hopf algebra generalisation of this map. The first thing is to decide what maps to what,
and the easy bit is what we map out of. Classically we have an algebra of smooth functions
C∞(G,C) with a ∗-structure which is just pointwise conjugation. This has a differential calculus
of 1-forms Ω1
G (again we take complex valued with a ∗-operation). Then the left invariant 1-
forms Λ1
G (equivalently the cotangent space at the identity) has a dual gC, the complexified
Lie algebra, and g is the real part of gC. We apologise to real differential geometers for the
seemingly unnecessary diversion through the complex numbers, but for Hopf algebras over C
this will become necessary. Algebraically, from the group G we take the algebra of complex
valued smooth functions C∞(G), and then the differential calculus Ω1
G for this algebra.
The target for the exponential map is more of a problem. To explain, we ignore analytic
complications (after all we will consider finite and discrete groups), and just take CG to be the
complex group algebra. We can view g ∈ G in two ways, first as g ∈ CG and secondly as the
state “evaluate at g” on the function algebra C(G). To complicate things further, the state can
be written using a GNS construction using a Hilbert space (12 densities) and we could consider g
to lie in this Hilbert space. All of these points of view will appear.
To motivate the exponential map for Hopf algebras we first look at the Lie group case.
Here the motivation is obvious, we know an enormous amount about Lie algebras and their
classification, and the exponential map allows us to use this to study Lie groups. Hopf algebras
are at a different stage of their history; we know little about the classification theory. We
know almost nothing about the non-bicovariant case, in fact the associative algebra U(g) in
this case was only worked out recently in terms of higher-order differential operators on Hopf
algebras (see [1] or [5, p. 515]). (To clarify, here U(g) denotes the algebra generated by invariant
vector fields, and does not refer to a deformation of a classical enveloping algebra for which
mailto:gyalhamzi@imamu.edu.sa
mailto:e.j.beggs@swansea.ac.uk
https://www.swansea.ac.uk/staff/science/maths/beggs-e-j/
https://doi.org/10.3842/SIGMA.2022.017
2 G. Alhamzi and E. Beggs
we use Uq(su2) below.) Of course, for a Hopf algebra H the dual of g is a quotient of H+ by
an ideal, but this just illustrates the problem in that we start with H to find g. Published
material on quantum Lie algebras concentrates on their properties (e.g., [12, 17]), especially
in the braided case, and not on a classification theory. We would hope that giving a very
general construction for the exponential map for Hopf algebras would motivate the study of the
corresponding quantum Lie algebras in their own right, including their classification theory. In
turn, this could be used in classification of Hopf algebras.
We begin by using the Kasparov, Stinespring, Gel’fand, Năımark and Segal (KSGNS) con-
struction (see [15]) in a case which (for C∗-algebras) gives a time varying state ψt on the function
algebra C(G). The KSGNS construction works by using a bimodule as half densities, and we
have m(t) an element of this bimodule. The construction for m(t) reduces to a differential equa-
tion, and solving this equation uses an actual exponential (Taylor series) in the group algebra CG.
In Section 3 we give a construction for the dynamics of states on an algebra from [2]. This
uses a vector field on an algebra, and for a Hopf algebra we can use the left invariant vector
fields in an exact correspondence with the classical case using Woronowicz’s calculi on Hopf
algebras [24]. We will do all this for four examples, the discrete group S3 in the simplest case,
and the group Z is not much more complicated (though it gives an interesting non-diffusion
evolution of states on Z). The quantum group Cq[SU2] has much more complicated formulae,
so we carry out calculations only in special cases. The Sweedler–Taft Hopf algebra is included
to stress that the method is more general than its initial motivation in C∗-algebras.
In solving the dynamics of the states we come on a much simpler and very direct interpretation
of the exponential as a power series in the dual Hopf algebra. If we had simply written this
in the beginning then questions would have been asked about its role and whether it was just
writing a power series to look like the classical case. But now we can be more clear about its
role, it plays a fundamental part in the dynamics of the states in the Hopf algebra using Hilbert
C∗-bimodules. Given an invariant vector field (i.e., an element of the “Lie algebra” of the Hopf
algebra) we get an exponential path in time t ∈ R lying in the dual Hopf algebra starting at ϵ
(i.e., evaluation at the identity). Of course, for many Hopf algebras the exponential will not lie in
the original dual Hopf algebra as it is an infinite series, but in a completion or formal extension.
Since we are exponentiating a vector field X, the reader may be puzzled about why in various
places (e.g., Proposition 6.7) we get an exponential of minus X. The simple explanation is that
the “weight” defining the functional moves in the opposite direction to what the functional
applies to. Thus for a functional Tt : C0(R) → R and weight w(x)
Tt(f) =
∫
R
f(x+ vt)w(x)dx =
∫
R
f(x)w(x− vt)dt.
As pointed out by one of the referees, the reader should note the similarities in the construc-
tion here with Lévy processes on bialgebras [11].
The reader may ask why we continue to use an exponential with parameter in R in a noncom-
mutative setting. The differential setting of the KSGNS construction is very general, and could
be used with other Hopf algebras replacing C∞(R). However in [4] it is shown that using C∞(R)
is sufficient to describe quantum mechanics (the Schrödinger and Klein–Gordon equations) as
auto parallel paths using the proper time as parameter. This shows that the C∞(R) parameter
case is of interest, thought not the most general case. The importance of paths on C∗-algebras
parametrised by the reals is illustrated by the definition of suspension of an algebra.
2 Preliminaries
A first-order differential calculus Ω1
B on an algebra B is a B-bimodule with a derivation
d: B → Ω1
B, and so that Ω1
B is spanned by cdb where b, c ∈ B. For a ∗-algebra B, this will
The Exponential Map for Hopf Algebras 3
be a ∗-differential calculus if there is an antilinear map ∗ : Ω1
B → Ω1
B so that
(c.db)∗ = d(b∗).c∗.
The right vector fields χR
B consist of right module maps from Ω1
B to B, with evaluation
ev : χR
B ⊗Ω1
B → B.
For a left B-module M a left connection is a linear map ∇M : M → Ω1
B ⊗BM with the left
Leibniz rule for b ∈ B and m ∈M
∇M (b.m) = db⊗m+ b.∇M (m). (2.1)
In the case where M is a B-A-bimodule we have a left bimodule connection (M,∇M , δM ) when
there is a bimodule map
σM : M ⊗
A
Ω1
A → Ω1
B ⊗
B
M
for which we have the modified right Leibniz rule for a ∈ A
∇M (m.a) = ∇M (m).a+ σM (m⊗da).
Bimodule connections were introduced in [8, 9, 19] and extensively used in [10, 16].
For a Hopf algebra H we use the Sweedler notation ∆h = h(1)⊗h(2). A differential calculus
is called left covariant if there is a left H-coaction ∆L : Ω
1
H → H ⊗Ω1
H where ∆L(h.dk) =
h(1)k(1)⊗h(2)dk(2) for h, k ∈ H [23]. Similarly to the Sweedler notation, for a left coaction write
∆L(ξ) = ξ[−1]⊗ ξ[0] for ξ ∈ Ω1
H . We call Λ1
H the vector space of left invariant forms (i.e., ξ
such that ∆Lξ = 1⊗ ξ ). We now suppose that H has an invertible antipode, required by our
choice of right vector fields and left coactions. The left coaction on χR
H is defined to make the
evaluation ev : χR
H ⊗H Ω1
H → H a left comodule map, and is given by, for X ∈ χR
H and η ∈ Ω1
H
X[−1]⊗X[0](η) = X(η[0])(1)S
−1
(
η[−1]
)
⊗X
(
η[0]
)
(2)
.
Definition 2.1 ([18]). Two Hopf algebras H and H ′ are dually paired if there is a map
ev : H ′⊗H → C which obey, for all α, β ∈ H ′ and h, k ∈ H
ev⟨α, hk⟩ = ev⟨α(1), h, ⟩ev⟨α(2), k⟩, ev⟨αβ, h⟩ = ev⟨α, h(1)⟩ev⟨β, h(2)⟩,
ev⟨1H′ , h⟩ = ϵH(h), ev⟨α, 1H⟩ = ϵH′(h), ev⟨Sα, h⟩ = ev⟨α, Sh⟩.
They are a strictly dual pair if this pairing is nondegenerate.
If H is finite-dimensional the idea of dual is quite simply the linear dual. However for infinite-
dimensional Hopf algebras we must take more care. Notably the Hopf algebras Cq[SU2] and the
deformed enveloping algebra Uq(su2) are dually paired, but Uq(su2) is much smaller than the
continuous dual vector space of the C∗-algebra Cq[SU2].
Definition 2.2. A right integral ϕ : H → C is a linear map such that ϕ
(
h(1)
)
h(2) = 1H .ϕ(h). It
is said to be normalised if ϕ(1H) = 1.
Definition 2.3 ([18]). A Hopf algebra H which is also a ∗-algebra is called a Hopf ∗-algebra if
∆(h∗) = h(1)
∗⊗h(2)
∗, ϵ(h∗) = ϵ(h)∗, (S ◦ ∗)2 = id.
For a Hopf ∗-algebra we call a Haar right integral ϕ Hermitian if ϕ(h∗) = ϕ(h)∗.
4 G. Alhamzi and E. Beggs
3 The KSGNS construction and paths
The KSGNS construction [15] for a completely positive map from C∗-algebras A to B is given
by an B-A bimodule M and a Hermitian inner product
⟨ , ⟩ : M ⊗
A
M −→ B. (3.1)
Recall that the conjugate A-B-bimodule M is the conjugate C-vector space, with elements
m ∈ M for m ∈ M and m+ n = m + n and λm = λm for m,n ∈ M and λ ∈ C. The
actions of the algebras are a.m = ma∗ and m.b = b∗m for a ∈ A and b ∈ B [3]. If we forget
about completeness under a norm and positivity we can restate this in terms of more general
∗-algebras. We shall take B = C∞(R), and then in this case we just assume that ⟨ , ⟩ in (3.1) is
Hermitian, i.e., ⟨m,n⟩∗ = ⟨n,m⟩. We get ψ : A→ C∞(R) given by
ψt(a) = ⟨ma,m⟩, (3.2)
which is a time dependent linear functional, and in good cases a time dependent state.
We define the time evolution of ψt by imposing the condition ∇Mm = 0 on m in (3.2)
where∇M is a left B-connection as in (2.1). An obvious condition to place on the connection∇M
is that it preserves the inner product, i.e., that
d⟨m,n⟩ = (id⊗⟨ , ⟩)
(
∇Mm⊗n
)
+ (⟨ , ⟩⊗ id)
(
m⊗∇Mn
)
with∇Mn = p⊗ ξ∗ where∇Mn = ξ⊗ p. Note that this is just the usual preserving inner product
condition used in Riemannian geometry [5]. As special case we consider M = C∞(R)⊗A with
actions given by product making it into a C∞(R)-A bimodule. We define the inner product ⟨ , ⟩
on M
⟨f1⊗ a1, f2⊗ a2⟩ = f1f
∗
2ϕ(a1a
∗
2)
for fi ∈ C∞(R) and ai ∈ A, where ϕ : A → C is Hermitian map (i.e., ϕ(a∗) = ϕ(a)∗) and in
nice cases a positive map. In terms of the A valued function of time approach, this is just
⟨m,n⟩(t) = ϕ(m(t)n(t)∗) for m,n ∈M .
We consider the special case where ∇M is a bimodule connection. In [2] this is used to recover
classical geodesics, but we use this assumption as it gives us a role for vector fields. It also would
allow us to define a velocity for the paths, but we do not go into this.
We now take the C∞(R)-A bimodule C∞(R)⊗A in the previous theory. However, we quickly
find out that this bimodule will not in general contain the solution of the differential equations,
and so pass to a larger bimodule C∞(R, A), the infinitely differentiable functions from R to A.
Outside the case where A is finite-dimensional (and the two definitions are the same) we would
require some topology to define differentiable, but our infinite-dimensional examples are C∗-
algebras.
Proposition 3.1 ([2]). For a unital algebra A with calculus ΩA and C∞(R) with its usual
calculus Ω(R) we set M = C∞(R, A). Then a general left bimodule connection on M is of the
form, for m ∈ C∞(R)⊗A and ξ ∈ Ω1
A
∇M (m) = dt⊗
(
pm+ ∂m
∂t +X(dm)
)
, σM (1⊗ ξ) = dt⊗X(ξ)
for some p ∈ C∞(R, A) and X ∈ C∞(R, χR
)
where d is the derivation d: A → Ω1
A.
(
Note that
explicitly including time evaluation we have X(η)(t) = X(t)(η(t)) for η ∈ C∞(R,Ω1
A
)
.
)
Further
the connection preserves the inner product on M if for all a ∈ A and ξ ∈ Ω1
A.
⟨
(
pa+X(da) + ap∗
)
, 1⟩ = 0 = ⟨X(ξ∗)−X(ξ)∗, 1⟩. (3.3)
The Exponential Map for Hopf Algebras 5
Following from the classical theory, we shall call the first equality of the equation (3.3) the
divergence condition for p and the second the reality condition for X. In this case the divergence
div(X) ∈ A of X ∈ χR for all a ∈ A is given by
ϕ(div(X).a+X(da)) = 0.
In [2] it is shown that we can set p = 1
2 div(X) in Proposition 3.1.
In this paper we only consider the case of a Hopf algebra H and a left invariant right vector
field X. Now if both X ∈ χR and ξ ∈ Ω1
H are left invariant we find that X(ξ) ∈ H is left
invariant, so it is a multiple of the identity. We use the invariant derivative ω : H → Λ1
H which
is defined so that dh = ω
(
h(2)
)
.h(1),
ω(a) = da(2)S
−1
(
a(1)
)
∈ Λ1
H . (3.4)
Proposition 3.2. If ϕ is a Hermitian right Haar integral on H and X ∈ χR is left invariant,
then div(X) = 0.
Proof. As X
(
ω(a(2))
)
is just a number in the following expression
X(da) = X
(
da(3)S
−1
(
a(2)
)
a(1)
)
= X
(
ω
(
a(2)
))
a(1),
so ϕ(X(da)) = X
(
ω
(
a(2)
))
ϕ
(
a(1)
)
. For the Hermitian right Haar integral ϕ
(
a(1)
)
a(2) = ϕ(a).1
so ϕ(X(da)) = X(ω(1))ϕ(a) = 0. ■
Proposition 3.3. For a Hopf ∗-algebra H with a left invariant ∗-calculus and ϕ is a Hermitian
right Haar integral, to show that a left invariant right vector field X ∈ χR is real, it is sufficient
to check that X(η∗) = X(η)∗ for all η ∈ Λ1
H .
Proof. Recalling the property ϕ(a(1))a(2) = ϕ(a).1 for a right Haar integral, we have for ξ ∈ Ω1
1.ϕ(X(ξ∗)) = ϕ
(
X(ξ∗)(1)
)
X(ξ∗)(2),
1.ϕ(X(ξ)∗) = ϕ
(
X(ξ)∗(1)
)
X(ξ)∗(2) = ϕ
(
X(ξ)(1)
∗)X(ξ)(2)
∗,
and as X is left invariant and ev : χR ⊗Ω1 → H is a left comodule map we have
1.ϕ(X(ξ∗)) = ϕ
(
ξ∗[−1]
)
X
(
ξ∗[0]
)
= ϕ
(
ξ[−1]
∗)X(ξ[0]∗),
1.ϕ(X(ξ)∗) = ϕ
(
ξ[−1]
∗)X(ξ[0])∗.
For h ∈ H, ϕ(h) is in the field and ϕ(h∗) = ϕ(h)∗ and so
1.ϕ(X(ξ∗)) = X
(
ϕ
(
ξ[−1]
)∗
ξ[0]
∗) = X
((
ϕ
(
ξ[−1]
)
ξ[0]
)∗)
,
1.ϕ(X(ξ)∗) = X
(
ϕ
(
ξ[−1]
)
ξ[0]
)∗
.
Finally note that η = ϕ
(
ξ[−1]
)
ξ[0] is left invariant. ■
Theorem 3.4. The connection ∇Mm = dt⊗(ṁ + X(dm)) for left invariant X ∈ χR has
solutions of ∇Mm = 0 given by
m(t) = m(0)(1) exp(−t(X ◦ ω))
(
m(0)2
)
,
where we take the exponential as a power series in elements of H ′.
6 G. Alhamzi and E. Beggs
Proof. We solve ṁ = −X(dm) by using dm = ω(m(2))m(1), so
ṁ = −(X ◦ ω)
(
m(2)
)
m(1), m̈ = −(X ◦ ω)
(
ṁ(2)
)
m(1) − (X ◦ ω)
(
m(2)
)
ṁ(1). (3.5)
As ∆ and d
dt on M commute
d
dt
(
m(1)⊗m(2)
)
= ṁ(1)⊗m(2) +m(1)⊗ ṁ(2) = −m(1)⊗m(2)(X ◦ ω)
(
m(3)
)
,
and substituting this back into (3.5) gives
m̈ = (X ◦ ω)
(
m(3)
)
(X ◦ ω)
(
m(2)
)
m(1) = (X ◦ ω)2
(
m(2)
)
m(1)
using the product in H ′. Continuing with higher derivatives and using Taylor’s theorem to get
the answer, recalling that the first term in the exponential, the identity in H ′, is ϵ. ■
We can use this formula for m(t) in ψt(a) = ϕ(m(t)am(t)∗) to give
ψt(a) = exp(−tX ◦ ω)
(
m(0)(2)
)
exp(−tX ◦ ω)
(
n(0)(2)
)∗
ϕ
(
m(0)(1)an(0)
∗
(1)
)
, (3.6)
where n is an independent copy of m. Note that for a classical geodesic on a group starting
at the identity element we would have H = C∞(G) and m(0) would be a δ-function (or more
accurately 1
2 density) at the identity e ∈ G, giving
ψ0 = ϵ : C∞(G) → R.
4 Functions on a finite group G
We take H = C[G], the functions on a finite group G. A basis is δg for g ∈ G, the function
taking value 1 at g and zero elsewhere. This is a Hopf algebra with
ϵ(δg) = δg,e, ∆δg =
∑
x,y∈G : xy=g
δx⊗ δy, S(δg) = δg−1 .
The first-order left covariant differential calculi onH=C[G], correspond to subsets C⊆G\{e} [6].
The basis as a left module for the left invariant 1-forms is ea for a ∈ C, with relations and exterior
derivative for f ∈ C[G] being
ea.f = Ra(f)e
a, df =
∑
a∈C
(Ra(f)− f)ea,
where Ra(f)(g) = f(ga) denotes right-translation. We take eb for b ∈ C to be the dual basis to
ea ∈ Λ1, i.e., ev(eb⊗ ea) = eb(e
a) = δa,b. Now from (3.4)
ω(δg) =
∑
ea if g = e,
−ea if g−1 = a ∈ C,
0 otherwise,
so if we set X =
∑
Xaea ∈ h = (Λ1
H)′, for some Xa ∈ H, then
(X ◦ ω)(δg) =
∑
Xa if g = e,
−Xa if g−1 = a ∈ C,
0 otherwise.
(4.1)
We set ϕ to be the normalised Haar measure ϕ(f) = 1
|G|
∑
g∈G f(g).
The Exponential Map for Hopf Algebras 7
Now (X ◦ ω) is an element of the dual of H = C(G), which is the group algebra H ′ = CG.
To write elements of the dual Hopf algebra we first list the elements of G as g1, g2, . . . , gn and
then for β ∈ H ′ = CG we use a column vector notation
β =
β(δg1)
β(δg2)
...
β(δgn)
. (4.2)
It will be convenient to turn the calculation of the exponential on CG into a matrix exponential
using a differential equation. We set αt = exp(tX ◦ ω) so dαt
dt = αt.(X ◦ ω) and
dαt(δg)
dt
=
∑
xy=g
αt(δx)(X ◦ ω(δy)). (4.3)
Now we can write (4.3) as matrix equation α̇t = Tαt
α̇t(δgi) =
∑
k
αt(δgk).(X ◦ ω)
(
δg−1
k gi
)
, (4.4)
where
Tik = (X ◦ ω)
(
δg−1
k gi
)
=
∑
Xa if k = i,
−Xa if g−1
i gk = a ∈ C,
0 otherwise.
(4.5)
Now we have αt = exp(tT )β where β is the column vector corresponding to the identity in CG
and we use the matrix exponential. The calculus on H has a ∗-structure given by ea
∗
= −ea
−1
,
so by Proposition 3.3 the left invariant vector field X =
∑
Xaea is real if X(a)∗ = −Xa−1
.
Example 4.1. Let G = S3, set a ∈ C = {u, v, w} where u = (1, 2), v = (2, 3) and w = (1, 3),
and write X = Xueu +Xvev +Xwew where Xu, Xv, Xw ∈ C and the elements of S3 are listed
as
g1 = e, g2 = (1, 2, 3), g3 = (1, 3, 2), g4 = u, g5 = v, g6 = w. (4.6)
Now the matrix T in (4.5) becomes
T =
T11 0 0 −Xu −Xv −Xw
0 T22 0 −Xv −Xw −Xu
0 0 T33 −Xw −Xu −Xv
−Xu −Xv −Xw T44 0 0
−Xv −Xw −Xu 0 T55 0
−Xw −Xu −Xv 0 0 T66
,
where the diagonal elements of T are Tii = Xu+Xv+Xw. Now the solution to α̇t = Tαt is αt =
exp(tT )α0. Set α0 = e, the identity in S3, which is the column vector
(
1 0 0 0 0 0
)T
,
and time t = 1. We set Xu = ip, Xv = iq and Xw = ir and γ =
√
p2 + q2 + r2 − pq − pr − qr
for p, q, r ∈ C to get
exp(i(peu + qev + rew) ◦ ω)
8 G. Alhamzi and E. Beggs
=
1
3
ei(p+q+r)
2 cos(γ) + cos(p+ q + r)
cos(p+ q + r)− cos(γ)
cos(p+ q + r)− cos(γ)
−i
(
sin(γ)(2p− q − r)
γ
+ sin(p+ q + r)
)
−i
(
i sin(γ)(2q − p− r)
γ
+ sin(p+ q + r)
)
−i
(
i sin(γ)(2r − p− q)
γ
+ sin(p+ q + r)
)
∈ CS3. (4.7)
In our case C = {u, v, w} so a−1 = a for a ∈ C, and the reality condition is that p, q, r ∈ R and
as a result γ ∈ R. Note that the vector does not depend on the sign of the square root and that
the L2 norm of the vector in (4.7) is equal to 1. Now we look at time dependence of the state ψt
given by equations ψt(a) = ϕ(m(t)am(t)∗) and (3.6). We start with m(0) = δe at t = 0 where
everything is concentrated at the identity. We have ∆m(0) = ∆δe =
∑
g δg−1 ⊗ δg, so (3.6) gives
ψt(a) =
∑
g,h
ϕ(δg−1aδh−1)⟨exp(−X ◦ ω), δg⟩⟨exp(−X ◦ ω), δh⟩∗
=
∑
g
ϕ(δg−1a) |⟨exp(−X ◦ ω), δg⟩|2 , (4.8)
so ϕ is given by a probability density
∑
g δg−1 |⟨exp(−X ◦ω), δg⟩|2. In terms of the group algebra,
which is dual to the functions,
9ψt = e (2 cos(γ) + cos(p+ q + r))2 +
(
(132) + (123)
)
(cos(p+ q + r)− cos(γ))2
+ u
(
(2p− q − r) sin(γ)
γ
+ sin(p+ q + r)
)2
+ v
(
(2q − p− r) sin(γ)
γ
+ sin(q + p+ r)
)2
+ w
(
(2r − q − p) sin(γ)
γ
+ sin(r + q + p)
)2
.
To plot some example exponential of states we refer back to the ordering of group elements
in (4.6), and plot the weight of each element against time for 0 ≤ t ≤ 7. We display some
cases in Figure 1. This illustrates the conversion of the solution in (p, q, r) in particular cases to
a function of the parameter t. Note in general the exponential map will not be periodic as the
ratio between γ and p+ q + r is likely not to be rational.
Figure 1. The states ψt given by exponentials for vector fields X = iteu + 1
3 itev +
1
2 itew and
X = iteu + itev and X = iteu respectively.
The Exponential Map for Hopf Algebras 9
5 Functions on the integers Z
We shall apply the finite group methods of Section 4 to the group Z, which needs to be treated
with care. We shall use rapidly decreasing functions and an un-normalised Haar measure ϕ(f) =∑
n∈Z f(n) and infinite matrices. The column vector notation of (4.2) becomes, truncating the
infinite vectors
β =
β(δ2)
β(δ1)
β(δ0)
β(δ−1)
β(δ−2)
, z−1 =
0
0
0
1
0
, z0 =
0
0
1
0
0
, z1 =
0
1
0
0
0
. (5.1)
We have used the pairing of the group algebra CZ with the functions to give the vectors corre-
sponding to Zn ∈ CZ, the basis elements corresponding to n ∈ Z. We look at equation (4.4) in
the case of the integers, which becomes
α̇t(δi) =
∑
k
αt(δk)(X ◦ ω)(δi−k). (5.2)
We use C = {+1,−1} for the calculus, giving two generators e+1, e−1 with dual invariant vector
fields e+1 and e−1. Now for the vector field X = X+1e+1 +X−1e−1 (4.1) becomes
(X ◦ ω)(δg) =
X+1 +X−1 if g = 0,
−X−1 if g = 1,
−X+1 if g = −1,
0 otherwise,
so (5.2) becomes
α̇t(δi) = −αt(δi+1)X
+1 − αt(δi−1)X
−1 +
(
X+1 +X−1
)
αt(δi). (5.3)
To describe this more easily we use matrices Nn (infinite in both directions, we only consider
a part centred on the 0, 0 entry)
N1 =
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
0 0 0 0 1
0 0 0 0 0
, N−1 =
0 0 0 0 0
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
, N2 =
0 0 1 0 0
0 0 0 1 0
0 0 0 0 1
0 0 0 0 0
0 0 0 0 0
,
etc., and NnNm = Nn+m. If we write αt as a column vector similarly to (5.1), then we can write
the differential equation
α̇t =
(
−X+1N−1 +
(
X+1 +X−1
)
N0 −X−1N1
)
αt.
To find the exponential we need to use a generalised hypergeometric function [22]
0F1(; a;x) = 1 +
x
a1!
+
x2
a(a+ 1)2!
+ · · · .
Proposition 5.1.
exp(−aN1 + (a+ b)N0 − bN−1)
= ea+b
(
0F1(; 1; ab)N0 +
∑
n>0
0F1(;n+ 1; ab)
n!
(
(−a)nNn + (−b)nN−n
))
,
10 G. Alhamzi and E. Beggs
so
αt = exp(tX ◦ ω) = exp
(
−tX−1z1 +
(
tX−1 + tX+1
)
z0 − tX+1z−1
)
= et(X
+1+X−1)
(
0F1
(
; 1; t2X−1X+1
)
z0
+
∑
n>0
0F1
(
;n+ 1; t2X−1X+1
)
n!
((
−tX−1
)n
zn + (−tX+1)nz−n
))
.
Proof. Using the trinomial theorem
(−aN1 + (a+ b)N0 − bN−1)
n =
∑
i,j,k≥0:i+j+k=n
n!
i!j!k!
(−aN1)
i((a+ b)N0)
j(−bN−1)
k
=
∑
i,j,k≥0:i+j+k=n
n!
i!j!k!
(−a)i(a+ b)j(−b)kNi−k,
so
exp(−aN1 + (a+ b)N0 − bN−1) =
∑
i,j,k≥0
(−a)i(a+ b)j(−b)k
i!j!k!
Ni−k
= ea+b
∑
i,k≥0
(−a)i(−b)k
i!k!
Ni−k. (5.4)
If we set i − k = n we get sums depending on the sign of n. For n ≥ 0 we have the coefficient
of Nn in (5.4) being
ea+b
∑
k≥0
(−a)k+n(−b)k
(k + n)!k!
= ea+b (−a)n
n!
∑
k≥0
n!
(k + n)!k!
(ab)k = ea+b (−a)n
n!
0F1(;n+ 1; ab).
For n < 0 we have (putting m = −n) k = i+m and the coefficient of N−m in (5.4) is
ea+b
∑
i≥0
(−a)i(−b)i+m
i!(i+m)!
= ea+b (−a)m
m!
∑
i≥0
m!
(i+m)!i!
(ab)i = ea+b (−b)m
n!
0F1(; 1 +m; ab).
To calculate exp(tX ◦ ω) we put a = tX−1 and b = tX+1 and apply the matrix exponential to
z0 ∈ CZ to find the answer. ■
Now we follow the equation (4.8) in finding the state corresponding to the initial state which
is the dual element z0, corresponding to m(0) = δ0,
ψt(a) = ϕ(m(t)am(t)∗) =
∑
r∈Z
ϕ(δra) |exp(−tX ◦ ω)(δ−r)|2 .
As ϕ(δra) = a(r) we see that the state is the element of the dual
∑
n zn| exp(−tX ◦ ω)(δ−n)|2.
Now we restrict to the real vector field case
(
X+1
)∗
= −X−1, where X+1 +X−1 is imaginary,
so
∣∣et(X++X−1)
∣∣ = 1, and
∣∣X+1
∣∣2 = −X+1X−1 > 0. To plot a graph we take the case
∣∣X+1
∣∣ = 1
ψt = z0
∣∣
0F1
(
; 1;−t2
)∣∣2 +∑
n≥1
(zn + z−n)
∣∣∣∣∣0F1
(
;n+ 1;−t2
)
n!
∣∣∣∣∣
2
|t|2n.
The Exponential Map for Hopf Algebras 11
Figure 2. The time evolution of states on the integers for the exponential and diffusion respec-
tively.
We plot this for integers −4 ≤ n ≤ 4 in the range 0 ≤ t ≤ 5 in the first graph in Figure 2, plotted
using standard functions in Mathematica, and there it can be seen that there is a damped oscil-
latory behaviour. We should compare this geodesic calculation with the usual diffusion equation
on Z which also gives a time dependent state. Diffusion is defined in terms of a Lagrangian ∆
and Lagrangians on graphs have been studied for some time (e.g., [7]). We use the special case
for diffusion of a density f : Z× [0,∞) → R given by, for λ > 0,
df(n)
dt
= −λ(∆f)(n) = −λ(2f(n)− f(n− 1)− f(n+ 1)).
This is just (5.3) with f(n) = αt(δn) but with X
+1 = X−1 = −λ < 0, i.e., with an “imaginary”
vector field satisfying X−1 = X+1. We start at t = 0 with f(n) = 0 for n ̸= 0 and f(0) = 1, and
this is just the same as the initial condition for αt previously. Now the Proposition 5.1 gives the
solution for f as a function of t in the case λ = 1 as
f = exp(tX ◦ ω) =
(
0F1
(
; 1; t2
)
z0 +
∑
n>0
0F1
(
;n+ 1; t2
)
n!
(
tnzn + tnz−n
))
e−2t ∈ CZ.
We plot this for −4 ≤ n ≤ 4 and 0 ≤ t ≤ 5 as before in the second graph in Figure 2. For both
the exponential and the diffusion we have one real parameter,
∣∣X+1
∣∣ and λ respectively. We
can see from the graphs that the behaviour of the states in the two cases is different, with the
exponential giving “damped oscillations” and the diffusion giving a monotonic decrease at z0.
6 The exponential map on quantum SU2
We use the matrix quantum group Cq[SU2] as given by Woronowicz [24] and quantum envelop-
ing algebra Uq(su2) as given in [14, 20]. There is a dual pairing between H = Cq[SU2] and
H ′ = Uq(su2). (We just say dual pairing as H is infinite-dimensional and we need to be careful
about duals.)
Definition 6.1. For q ∈ C∗ with q2 ̸= −1, we define the quantum group Cq[SU2] to have
generators a, b, c, d with relations
ba = qab, ca = qac, db = qbd, dc = qcd, cb = bc,
da− ad = q
(
1− q−2
)
bc, ad− q−1bc = 1.
12 G. Alhamzi and E. Beggs
This is a Hopf algebra with coproduct, antipode and counit
∆(a) = a⊗ a+ b⊗ c, ∆(b) = b⊗ d+ a⊗ b, ∆(c) = c⊗ a+ d⊗ c,
∆(d) = d⊗ d+ c⊗ b, S(a) = d, S(b) = −qb, S(c) = −q−1c, S(d) = a,
ϵ(a) = ϵ(d) = 1, ϵ(b) = ϵ(c) = 0.
This is a Hopf ∗-algebra with a∗ = d, d∗ = a, c∗ = −qb and b∗ = −q−1c for q real. We define
a grade on monomials in generations by |a| = |c| = 1 and |b| = |d| = −1.
Definition 6.2. Uq(su2) has generators X+, X−, q
±H
2 , where we have relations
q−
H
2 q
H
2 = q
H
2 q−
H
2 = 1, q
H
2 X±q
−H
2 = q±X±, [X+, X−] =
qH − q−H
q − q−1
,
and comultiplication, counit and antipode
∆q±
H
2 = q±
H
2 ⊗ q±
H
2 , ∆X± = X±⊗ q
H
2 + q−
H
2 ⊗X±,
ϵ
(
q±
H
2
)
= 1, ϵ(X±) = 0, S(X±) = −q±X±, S
(
q±
H
2
)
= q∓
H
2 .
As in Definition 2.1, these are dually paired by ⟨α, tij⟩ = ρ(α)ij ∈ C where α ∈ Uq(su2) and
t11 = a, t12 = b, t21 = c and t22 = d and ρ : Uq(su2) → M2(C) is the representation (where
r =
√
q)
ρ(q
H
2 ) =
(
r 0
0 1
r
)
, ρ(X+) =
(
0 1
0 0
)
, ρ(X−) =
(
0 0
1 0
)
.
Definition 6.3 ([24]). The left covariant 3D calculus for the quantum group Cq[SU2] has gen-
erators e0 and e±. The relations are
e±a = qae±, e±b = q−1be±, e±c = qce±, e±d = q−1de±,
e0a = q2ae0, e0b = q−2be0, e0c = q2ce0, e0d = q−2de0,
and exterior derivative and the ∗-operator
da = ae0 + qbe+, db = ae− − q−2be0, dc = ce0 + qde+, dd = ce− − q−2de0,
e0∗ = −e0, e+∗ = −q−1e−, e−∗ = −qe+.
Now using ω : H → Λ1
H from (3.4) we calculate
ω(a) = q−2e0, ω(b) = q−1e−, ω(c) = q2e+, ω(d) = −e0. (6.1)
We define e0, e+, e− to be the dual basis of e0, e+, e−, i.e., ⟨ei, ej⟩ = δij . Now every ei ◦ω gives
a map from Cq[SU2] to C. We shall identify ei ◦ ω as an element of Uq(su2). The first step is to
apply ei ◦ ω to a product.
Lemma 6.4. For all g, h ∈ Cq[SU2]
(e± ◦ ω)(gh) = (e± ◦ ω)(g)ϵ(h) + q−|g|ϵ(g)(e± ◦ ω)(h),
(e0 ◦ ω)(gh) = (e0 ◦ ω)(g)ϵ(h) + q−2|g|ϵ(g)(e0 ◦ ω)(h).
The Exponential Map for Hopf Algebras 13
Proof. By definition
ω(gh) = d
(
g(2)h(2)
)
S−1
(
g(1)h(1)
)
= ω(g)ϵ(h) + g(2)ω(h)S
−1
(
g(1)
)
.
Now ejg(2)e
iS−1
(
g(1)
)
= 0 unless i = j, so we need to show
g(2)e
±S−1
(
g(1)
)
= ϵ(g)q−|g|e±, g(2)e
0S−1
(
g(1)
)
= ϵ(g)q−2|g|e0.
It is enough to do this on the generators
a(2)e
±S−1(a(1)) = ae±S−1(a) + ce±S−1(b)
= ae±d− q−1ce±b = q−1
(
ad− q−1cb
)
e± = q−1e±,
and similarly for e0 and b, c, d. ■
We can use Lemma 6.4 to identify the coproduct of ei ◦ ω, where the linear map g 7→ ϵ(g) is
just 1 ∈ Uq(su2). To do this we need to identify the map g 7→ qs|g|ϵ(g).
Lemma 6.5. For s ∈ R and g ∈ Cq[SU2] we have ⟨qsH , g⟩ = qs|g|ϵ(g).
Proof. As ∆qsH = qsH ⊗ qsH where
〈
qsH , hg
〉
=
〈
qsH , h
〉〈
qsH , g
〉
we only have to check the
formula on the generators, and this is
〈
qsH , tij
〉
= ρ
(
qsH
)i
j using ρ
(
qsH
)
=
(
qs 0
0 q−s
)
. ■
Proposition 6.6. In Uq(su2) we have (ei ◦ ω)(h) = ev(νi⊗h) where νi ∈ Uq(su2) is given by,
where r =
√
q,
ν0 = e0 ◦ ω =
1− q−2H
q2 − 1
, ν+ = e+ ◦ ω = r3q
−H
2 X−, ν− = e− ◦ ω = r−1q
−H
2 X+.
Proof. First we check that νi = ei ◦ ω on the generators
ρ(ν0) =
(
q−2 0
0 −1
)
, ρ(ν+) =
(
0 0
q2 0
)
, ρ(ν−) =
(
0 q−1
0 0
)
.
Thus ν0 applied to b, c gives zero, and this is also true for e0 ◦ ω by (6.1). Also ν+ applied
to a, b, d gives zero, and this is also true for e+ ◦ ω by (6.1). Lastly ν− applied to a, c, d gives
zero, and this is also true for e− ◦ ω. We are left with the cases, for
(e0 ◦ ω)(a) = e0
(
q−2e0
)
= q−2 = ⟨ν0, a⟩, (e0 ◦ ω)(d) = e0
(
−e0
)
= −1 = ⟨ν0, d⟩,
(e+ ◦ ω)(c) = q2 = ⟨ν+, c⟩, (e− ◦ ω)(b) = q−1 = ⟨ν−, b⟩. (6.2)
Now we need to show that we get equality on products of generators, which we show by induction.
Suppose that is the dual ei ◦ ω = νi when applied to products of ≤ n generators. If g, h are
product of ≤ n generators, then
ν±(gh) = ν±(g)ϵ(h) +
〈
q−H , g
〉
ν±(h) = ν±(g)ϵ(h) + q−|g|ν±(h),
which is (e± ◦ ω)(gh) by Lemma 6.4. Also from Lemma 6.5
ν0(gh) =
1
q2 − 1
ϵ(g)ϵ(h)− 1
q2 − 1
〈
q−2H , gh
〉
=
ϵ(g)ϵ(h)
q2 − 1
(
1− q−2|g|q−2|h|),
whereas
(e0 ◦ ω)(gh) =
ϵ(h)
q2 − 1
(
ϵ(g)− q−2|g|ϵ(g)
)
+
q−2|g|
q2 − 1
(
ϵ(h)− q−2|h|ϵ(h)
)
ϵ(g)
=
ϵ(g)ϵ(h)
q2 − 1
(
1− q−2|g|q−2|h|) = ν0(gh)
as required. ■
14 G. Alhamzi and E. Beggs
The previously calculated exponentials in this case give a formal exponential of a linear
combination of the ν in Uq(su2). We consider that there is little reason to just write out such
a formal sum. However, we can calculate the time evolution of certain states on Cq[SU2]. We
use states of the form ψt(a) = ϕ(m(t)am(t)∗) where (m(t)) is defined in Theorem 3.4. First we
look at the real vector field X = ie0.
Proposition 6.7. For any monomial y in the generators a, b, c, d we have ν0(y) = −[−|y|]q2ϵ(y)
using the q2 integer [n]q2 =
(
1− q2n
)
/
(
1− q2
)
, and
exp(itν0)(y) = exp
(
−it[−|y|]q2
)
ϵ(y).
Proof. The first result is given by using the definition of ν0 in Proposition 6.6 with Lemma 6.5.
Then ν20(y) = ν0(y(1))ν0(y(2)) = [−|y(1)|]q2 [−|y(2)|]q2ϵ(y(1))ϵ(y(2)). As ϵ annihilates b and c we
must have y containing no b, c for this to be nonzero. As ∆a = a⊗ a+b⊗ c and ∆d = d⊗ d+c⊗ b
we have ν20(y) = ν0(y)
2. This continues with higher powers. ■
Proposition 6.8. For the real vector field X = ie0 we have (summing over monomials in m(0))
m(t) = m(0) exp
(
it[−|m(0)|]q2
)
.
Proof. We have from Theorem 3.4 and Proposition 6.7
m(t) = m(0)(1) exp(−ite0 ◦ ω)
(
m(0)(2)
)
= m(0)(1) exp
(
it[−|m(0)(2)|]q2
)
ϵ
(
m(0)(2)
)
.
Note that for a monomial m(0), we have |m(0)(2)| = |m(0)| giving the answer. ■
We now look at the X = γe+ + δe− case. In the previous examples we had ψ0(a) =
ϕ(m(0)am(0)∗) = ϵ(a). However, because of the complexity of the calculation we will simply
calculate m(t) for m(0) a generator.
Proposition 6.9. For X = γe+ + δe− we calculate m(t) for m(0) being generator to be
m(0) = a, m(t) = a cosh
(
t
√
qγδ
)
− b
q2γ√
qγδ
sinh
(
t
√
qγδ
)
,
m(0) = b, m(t) = b cosh
(
t
√
qγδ
)
− a
q−1δ√
qγδ
sinh
(
t
√
qγδ
)
,
m(0) = c, m(t) = c cosh
(
t
√
qγδ
)
− d
q2γ√
qγδ
sinh
(
t
√
qγδ
)
,
m(0) = d, m(t) = d cosh
(
t
√
qγδ
)
− c
q−1δ√
qγδ
sinh
(
t
√
qγδ
)
.
Proof. We get the corresponding element of the dual x = X ◦ ω = γν+ + δν−. Then xn(h) =
x
(
h(1)
)
xn−1
(
h(2)
)
, so from (6.2)
xn(a) = q−1δxn−1(c), xn(b) = q−1δxn−1(d),
xn(c) = q2γxn−1(a), xn(d) = q2γxn−1(b).
These give
xodd(a) = xodd(d) = 0, xeven(b) = xeven(c) = 0,
x2n(a) = (qγδ)n = x2n(d), x2n+1(b) = qn−1γnδn+1, x2n+1(c) = qn+2γn+1δn,
The Exponential Map for Hopf Algebras 15
so for the formal exponential e−tx(h) =
∑
n≥0
(−tx)n
n!
(h)
e−tx(a) =
∑
n≥0
t2n(−x)2n
2n!
(a) =
∑
n≥=0
t2n(qγδ)n
2n!
= cosh
(
t
√
qγδ
)
= e−tx(d),
e−tx(b) =
∑
n≥0
t2n+1(−x)2n+1
(2n+ 1)!
(b) = −
∑
n≥0
t2n+1qn−1γn+1δn
(2n+ 1)!
=
−q−1δ√
qγδ
sinh
(
t
√
qγδ
)
,
e−tx(c) =
∑
n≥0
t2n+1(−x)2n+1
(2n+ 1)!
(c) = −
∑
n≥0
t2n+1qn+2γn+1δn
(2n+ 1)!
=
−q2γ√
qγδ
sinh
(
t
√
qγδ
)
,
and we use Theorem 3.4 to get the answer. ■
Example 6.10. For X = γe+ + δe− to be real by Proposition 3.3 we have γ∗ = −q−1δ, so√
qγδ =
√
−q2|γ|2 = ±iq|γ|. Then a special case of Proposition 6.9 gives
m(0) = a, m(t) = a cosh(iq|γ|t)− b
q2γ
iq|γ|
sinh(iq|γ|t) = a cos(q|γ|t)− b
qγ
|γ|
sin(q|γ|t).
Using an exponential map on an initial state will give a time evolution preserving the normali-
sation as long as we use a real invariant vector field. We now check this in our case of γe++δe−.
We begin with the Haar integral on Cq[SU2] which is zero on all basis elements anbrcs and dnbrcs
except
ϕ((bc)r) =
(−1)rqr
[r + 1]q2
.
Starting from m(0) = a we find from ϕ(m(t)m(t)∗) that ψt(1) =
q2
1+q2
which is independent of t
as required.
7 Sweedler–Taft algebra
We will now look at the Sweedler–Taft algebra H of dimension 4. This Hopf algebra does not
have a normalised Haar integral, and since we use the integral in finding the “state” we shall get
an algebraic construction which is nothing like the C∗-algebra framework. The Sweedler–Taft
algebra [21] is a unital algebra with generators x, t and relations
t2 = 1, x2 = 0, xt = −tx,
so it is 4-dimensional with basis {1, t, x, xt}. The following operations make it into a Hopf
algebra
∆1 = 1⊗ 1, ∆t = t⊗ t, ∆x = x⊗ t+ 1⊗x, ∆(tx) = tx⊗ 1 + t⊗ tx,
ϵ(t) = ϵ(1) = 1, ϵ(x) = ϵ(tx) = 0, St = t, Sx = tx.
We make the Sweedler–Taft algebra into a Hopf ∗-algebra by t∗ = t and x∗ = x.
There is a unique 2D bicovariant calculus with right ideal I ⊂ H+ = ker ϵ generated by x−xt
and the ∗ operation above has S((x− xt)∗) = x− xt and so gives a ∗-calculus [5]. We take the
basis e1 = [x] ∈ H+/I = Λ1 and e2 = [t− 1] of Λ1 and relations
dt = te2, dx = xe2 + e1, d(xt) = te1,
e2t = −te2, e1t = te1, e1x = xe1, e2x = −xe2 − 2e1,
e1 ∧ e1 = e2 ∧ e2 = 0, e1 ∧ e2 = e2 ∧ e1, de2 = 0, de1 = −e1 ∧ e2.
16 G. Alhamzi and E. Beggs
Then dt∗ = −e2t gives e2∗ = −e2 and dx∗ = xe2 + e1 gives e1∗ = −e1. We introduce a basis of
left invariant right vector fields e1 and e2 dual to e1, e2 ∈ Λ1. From (3.4) we find ω(t) = −e2,
ω(x) = −e1 and ω(tx) = −e1. Now we find ei ◦ ω ∈ H ′. We get ei ◦ ω(1) = 0 and
e2 ◦ ω(t) = −1, e2 ◦ ω(x) = e2 ◦ ω(tx) = 0, e1 ◦ ω(t) = 0,
e1 ◦ ω(x) = e1 ◦ ω(tx) = −1.
The dual basis elements corresponding to 1, t, x, tx are δ1, δt, δx, δtx and their products are
δ1.δ1 = δ1, δt.δt = δt, δx.δt = δx = δ1.δx, δtx.δ1 = δtx = δt.δtx,
and any other product is zero. Then e1 ◦ ω = −δt and e2 ◦ ω = −δx − δtx. The vector field
X = ae1 + be2 for a, b ∈ C has
X ◦ ω = −aδt − b(δx + δtx),
(X ◦ ω)2 = a2δ2t + b2(δx + δtx)
2 + abδt(δx + δtx) + ab(δx + δtx)δt = −a(X ◦ ω),
so adding a time parameter s as usual (t being used already)
exp(−s(X ◦ ω)) = ϵ− s(X ◦ ω) + (−s(X ◦ ω))2
2!
+
(−s(X ◦ ω))3
3!
+ · · ·
= ϵ− eas − 1
a
(X ◦ ω),
so from Theorem 3.4
m(s) = m(0) +m(0)(1)
1− esa
a
(X ◦ ω)(m(0)(2)). (7.1)
We now turn to the inner product and definition of real vector fields. The Haar integral [5] is
given by ϕ(tx) = λ where λ is arbitrary and ϕ of all other basis elements, including 1, to be zero.
For Proposition 3.2 we require ϕ to be Hermitian, so λ is imaginary. Applying this ϕ in the
formula for the inner product gives all left invariant vector fields being real, as ϕ(1) = 0. Note
that this definition of reality corresponds to preserving the inner product in Proposition 3.1,
rather than what in this case is the stronger condition on duals in Proposition 3.3.
Example 7.1. Set m(0) = t+ x and X = ae1 + be2 as above. Then by (7.1)
m(s) = t+ x+
1− esa
a
(t(X ◦ ω)(t) + 1(X ◦ ω(x)) + x(X ◦ ω)(t))
= t+ x+
1− esa
a
(−tb− xb− a).
To calculate the value of the “state” ψs in ϕ(m(t)am(t)∗) we setm(s) = m11+mtt+mxx+mtxtx
for mi ∈ C then for the Haar measure ϕ the map a 7→ ϕ(mam∗) is the element of the dual
λδ1(−m1mtx +mtmx −mxmt +mtxm1) + λδt(m1mx −mtmtx −mxm1 +mtxmt)
+ λδx(−m1mt +mtm1) + λδtx
(
|m1|2 − |mt|2
)
.
Substituting the values for m(s) gives
ψs = λ(δt − δx)
(
esa − esa +
∣∣esa − 1
∣∣2( b
a
− b
a
))
+ λδtx
(∣∣esa − 1
∣∣2(1− ∣∣∣∣ ba
∣∣∣∣2
)
− 1− b
a
(
esa − 1
)
− b
a
(
esa − 1
))
.
The Exponential Map for Hopf Algebras 17
Acknowledgements
We would like to thank the editor and referees for many useful comments. Computer algebra
and graphs were done on Mathematica.
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1 Introduction
2 Preliminaries
3 The KSGNS construction and paths
4 Functions on a finite group G
5 Functions on the integers Z
6 The exponential map on quantum SU_2
7 Sweedler-Taft algebra
References
|
| id | nasplib_isofts_kiev_ua-123456789-211528 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-13T16:53:20Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Alhamzi, Ghaliah Beggs, Edwin 2026-01-05T12:26:18Z 2022 The Exponential Map for Hopf Algebras. Ghaliah Alhamzi and Edwin Beggs. SIGMA 18 (2022), 017, 17 pages 1815-0659 2020 Mathematics Subject Classification: 16T05; 46L87; 58B32 arXiv:2203.04549 https://nasplib.isofts.kiev.ua/handle/123456789/211528 https://doi.org/10.3842/SIGMA.2022.017 We give an analogue of the classical exponential map on Lie groups for Hopf ∗-algebras with differential calculus. The major difference with the classical case is the interpretation of the value of the exponential map, classically an element of the Lie group. We give interpretations as states on the Hopf algebra, elements of a Hilbert 𝐶∗-bimodule of 1/2 densities, and elements of the dual Hopf algebra. We give examples for complex-valued functions on the groups 𝑆₃ and ℤ, Woronowicz's matrix quantum group ℂq[𝑆𝑈₂], and the Sweedler-Taft algebra. We would like to thank the editor and referees for many useful comments. Computer algebra and graphs were done on Mathematica. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Exponential Map for Hopf Algebras Article published earlier |
| spellingShingle | The Exponential Map for Hopf Algebras Alhamzi, Ghaliah Beggs, Edwin |
| title | The Exponential Map for Hopf Algebras |
| title_full | The Exponential Map for Hopf Algebras |
| title_fullStr | The Exponential Map for Hopf Algebras |
| title_full_unstemmed | The Exponential Map for Hopf Algebras |
| title_short | The Exponential Map for Hopf Algebras |
| title_sort | exponential map for hopf algebras |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211528 |
| work_keys_str_mv | AT alhamzighaliah theexponentialmapforhopfalgebras AT beggsedwin theexponentialmapforhopfalgebras AT alhamzighaliah exponentialmapforhopfalgebras AT beggsedwin exponentialmapforhopfalgebras |