EMPG-13-12
Classical r-matrices via semidualisation
Prince K. Osei
Department of Mathematics
University of Ghana, PO Box LG 25, Legon, Ghana
pkosei@ug.edu.gh
Bernd J. Schroers
Department of Mathematics and Maxwell Institute for Mathematical Sciences
Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom
b.j.schroers@hw.ac.uk
24 August 2013, udpated 8 October 2013
Abstract
We study the interplay between double cross sum decompositions of a given Lie
algebra and classical r-matrices for its semidual. For a class of Lie algebras which
can be obtained by a process of generalised complexification we derive an expres-
sion for classical r-matrices of the semidual Lie bialgebra in terms of the data
which determines the decomposition of the original Lie algebra. Applied to the
local isometry Lie algebras arising in three-dimensional gravity, decomposition and
semidualisation yields the main class of non-trivial r-matrices for the Euclidean
and Poincaré group in three dimensions. In addition, the construction links the
r-matrices with the Bianchi classification of three dimensional real Lie algebras.
1 Introduction
In classical geometry and physics, the geometry of a homogeneous space and its isometry
group mutually determine each other. Any deformation of either the geometry into a
non-commutative version or of the isometry group into a Hopf algebra should preserve
as much of this interplay as possible. One way of achieving this, at least at the algebraic
level, is to combine the space and its isometry into one algebraic object. In physics
terms, such an algebra should combine position coordinates for space with generators of
rotations and translations, i.e., with angular momenta and momenta.
Momenta and positions are in duality and rotations act on both. This is manifest
in three important subalgebras: the Heisenberg algebra generated by positions and
momenta, the isometry algebra generated by angular momenta and momenta and finally
the algebra of angular momenta and positions, which one may interpret as an isometry
algebra of momentum space. In these general terms, semiduality is a bijection between
the isometry algebras of space and momentum space.
1
arXiv:1307.6485v2  [math-ph]  11 Oct 2013
A mathematically precise version of these ideas was formulated by Majid in terms
of Hopf algebras [1] or, infinitesimally, Lie bialgebras [2], and is summarised in the
textbook [3]. Majid’s approach is in turn inspired by Born’s proposal of a reciprocity
between momenta and positions [4]. Here we will use Majid’s framework, working at
the level of Lie bialgebras. Thus, non-commutativity of a space is parametrised by Lie
brackets of position coordinates and its curvature by their co-commutator. In the dual
Lie bialgebra, commutators and co-commutators are swapped so that curvature of space
is also captured in the commutators of momenta; similarly curvature of momentum space
is captured in the co-commutators of momenta or the commutators of positions.
Curved, homogenous geometries and the associated Lie brackets of isometry algebras
are a standard topic in classical geometry and physics. Curved, homogeneous momentum
spaces and the associated co-commutators of momenta are less familiar but play a central
role in 3d (quantum) gravity (see [5] for a review), in so-called κ-deformations of the
Poincaré Lie algebra [6, 7, 8] and more recently in the discussion about relative locality
[9]. In this paper we exploit our familiarity with curvature and non-commutativity on
one hand to enhance our understanding of co-commutativity on the other. We consider
a family of Lie algebras, thinking of them as rotation-position algebras or infinitesimal
isometries of momentum space, and systematically compute their semidual bialgebras.
The semiduals have non-trivial co-commutators which are co-boundary, i.e., given by
a classical r-matrix. We derive a formula for the r-matrix in terms of the map which
characterises the split of the original Lie algebra into rotation generators and positions.
Specialising to three dimensions, the semiduals of our family of Lie bialgebras have the
Lie brackets of the 3d Euclidean or Poincaré groups. Lie bialgebra structures on these
Lie algebras are necessarily co-boundary and the possible r-matrices were classified a
while ago by Stachura in [10]. Denoting angular momenta by Ja and momenta by Pa,
the list found by Stachura includes r-matrices which only contain rotation or Lorentz
generators (these are just the standard r-matrices for sl(2,R)), trivial solutions which
only contain the commuting momentum generators, and then a more interesting and
complicated list of solutions of the mixed form r = Rb aaP ∧ Jb. Our approach via
semiduality reproduces all these mixed solutions and relates them to the Lie brackets of
the original algebra. In particular, we are able to relate these r-matrices to the Bianchi
classification of three-dimensional Lie algebras.
The work reported here is closely related to our previous paper [11] in which we studied
the semidual Hopf algebras of the universal enveloping algebras of the isometry Lie
algebras arising in 3d gravity. Here we generalise and extend the results of [11] at the
infinitesimal, Lie bialgebra level. However, in comparing the current paper with [11]
and also with previous work on semiduality in 3d quantum gravity [12, 13] the reader
should be aware of a possible confusion between different interpretations of semiduality.
Here we interpret semiduality essentially as the exchange of position and momentum
generators, as outlined above. In [13, 11], by contrast, pairs of semidual Hopf algebras
are both thought of as ‘rotation-momentum’ algebras, but semiduality exchanges the
regimes of 3d quantum gravity where they play the role of quantum isometry groups.
The paper is organised as follows. In Sect. 2 we introduce the family of Lie algebras
whose direct sum decomposition we would like to study. They are necessarily even-
2
dimensional and obtained from a given real Lie algebra g by a process of generalised
complexification. We parametrise decompositions of Lie algebras in the original family
as double cross sums with g as one of the factors in terms of a map F : g→ g and derive
a quadratic Lie-algebraic relation for F which ensures that it does indeed define a double
cross sum. The main result of this section is that such maps also characterise classical
r-matrices of the semidual Lie bi-algebra, and that the quadratic relation obtained as
factorisation condition is equivalent to the modified classical Yang-Baxter equation for
the semidual Lie bi-algebra. In Sect. 3 we specialise to the case where g is the Lie algebra
of either the rotation group in Euclidean 3-space or the Lorentz group in three spacetime
dimensions. The resulting family of complexified Lie algebras are the local isometry
groups of 3d gravity [5], but here we think of them as the Lie algebras of rotations or
Lorentz boosts together with position coordinates, so that, in keeping with the general
philosophy explained above, their semiduals, which have the Lie algebra structure of
the 3d Euclidean or Poincaré Lie algebra, can be interpreted as spacetime symmetries.
Using the isomorphism between g ∧ g and g in this case, we reformulate the quadratic
Lie-algebraic relation for F as a quadratic equation for the linear map F , and, using
rotational or Lorentz symmetry, project out three equations which have to be satisfied
independently. Solving these equations in Sect. 4 we recover a family of r-matrices for
the Euclidean and Poincaré Lie algebras first found by Stachura [10], and show that
they are in one-to-one correspondence with the types in the Bianchi classification of
three-dimensional Lie algebras bar the Heisenberg algebra. Sect. 5 contains a summary
of our results in tabular form and a discussion.
2 A general theorem on double cross sums and r-matrices for
their semidual
2.1 Direct sum decompositions of complexified Lie algebras
Consider an arbitrary n-dimensional real Lie algebra g with generators {Ja}a=1,...,n and
brackets
[J ca, Jb] = fab Jc. (2.1)
Here and in the following we use the Einstein summation convention. Picking a real
number λ, one can associate to this Lie algebra the following 2n-dimensional real Lie
algebra gλ with generators {Ja}, additional generators Qa, a = 1, . . . , n, and brackets of
the Cartan form
[Ja, J ] = f
c
b ab Jc, [Qa, Jb] = f
c
ab Qc, [Qa, Qb] = λf
c
ab Jc. (2.2)
For λ = −1 this is the usual complexification g ⊗ C, for λ = 0 it is the semidirect
product g n Rn and for λ = 1 it is isomorphic to g ⊕ g. One can unify these three
cases by considering the Lie algebra gλ as a generalised complexification with a formal
parameter θ which satisfies θ2 = −λ and the identification Qa = θJa, see [14] and [15].
We will not emphasise this viewpoint in the following, but will make use of the linear
map θ defined via
θ : gλ 7→ gλ, Ja 7→ Qa, Qa 7→ λJa for a = 0, . . . , n. (2.3)
3
Lie algebras like (2.2) arise in physics, particularly in three spacetime dimensions. In
that context, the generators Ja are rotation- or Lorentz generators, the Qa are usually
interpreted as components of momentum, i.e., as generators of spacetime translations,
and the constant λ is related to the cosmological constant. In the current context we
also think of the Ja as generalisations of rotation or Lorentz generators. However, when
we apply semiduality we switch from the generators Qa to generators of the dual vector
space, which we then interpret as momentum space. From that point of view, the
generators Qa should be thought of as position coordinates, and λ as a real constant
related to the curvature of momentum space [13, 11]
We are interested in all decompositions of gλ as double cross sum gλ = g ./ m of g and
a second n-dimensional Lie algebra m ⊂ gλ. We can assume without loss of generality
that the generators of m are of the form
Q′a = Qa + F
b
aJb, a = 1, . . . , n, (2.4)
for a real n × n matrix F ba. As explained in [3], the most general form of the brackets
in such a double cross sum is
[J ca, Jb] = fab Jc, [Q
′
a, Jb] = f
c ′ c
ab Qc + Lab Jc, [Q
′
a, Q
′ c ′
b] = gab Qc. (2.5)
If the matrix elements F ba can be found so that the generators Q
′
a close under Lie
brackets then we can express the structure constants L c cab and gab in terms of F and f
c
ab
as follows:
g c = f cF d + F d f c, L c = F d c c dab ad b a db ab afdb − F dfab . (2.6)
To determine the condition on the generators (2.4) to form a Lie subalgebra we think of
F ba as the matrix relative to the basis {Ja}a=1,...,n of a linear map
F : g→ g, X = XaJa 7→ F b aaX Jb, (2.7)
Then we define the map
I : g→ gλ, X 7→ θX + F (X), (2.8)
so that the sought-after generators in (2.4) are
Q′a = I(Ja), a = 1, . . . , n. (2.9)
Using this notation, we show the following.
Lemma 2.1 The condition on F : g → g for the generators (2.9) to form a Lie subal-
gebra of gλ is
[F (X), F (Y )]− F ([X,F (Y )] + [F (X), Y ]) = −λ[X, Y ] ∀ X, Y ∈ g. (2.10)
4
Proof In terms of the map (2.8) we need to find the condition under which the com-
mutator of the two elements of the form I(X) and I(Y ) lies in the image of I. One
computes
[I(X), I(Y )] = λ[X, Y ] + [F (X), F (Y )] + θ([X,F (Y )] + [F (X), Y ]). (2.11)
For this to be of the form I(Z) = θZ + F (Z) we require
F ([X,F (Y )] + [F (X), Y ]) = λ[X, Y ] + [F (X), F (Y )], (2.12)
as claimed. 
Note that, when the map F solving (2.10) can be found, then the map I is a bijection
g→ m
2.2 Semidual Lie bialgebras and their classical r-matrices
The process of semidualisation is defined for Lie bialgebras which are double cross sums,
see [3] for a systematic treatment. We are going to apply it to the Lie bialgebras
g ./ m, with the Lie algebra structure already discussed and trivial co-commutator. In
the semidual Lie bialgebra, the Lie algebra m is replaced by its dual Lie bialgebra m∗.
In our case, m has trivial co-commutators so that the Lie algebra structure of m∗ is
abelian. Following the notation of [3], we denote the semidual Lie bialgebra of g ./ m by
m∗I/g. In order to give its commutators and co-commutators, we introduce generators
P a, a,= 1, . . . , n of m∗ which are dual to the generators Q′a of m, i.e., they satisfy
P a(Q′b) = δ
a
b . (2.13)
Then, in terms of the generators {Ja, P b}a,b=1,...,n of the semidual Lie bialgebra, the
commutators are
[J ca, Jb] = fab Jc, [Ja, P
b] = −f b c a bac P , [P , P ] = 0, (2.14)
or
[J , J ] = f cJ , [P aa b ab c , J
a c a b
b] = fbc P , [P , P ] = 0.
Applying the general formulae in Sect. 8.3 of [3], the co-commutators in our case come
out as ( )
δ(P a) = g a ccb P ⊗ P b, δ(J ) = L c J ⊗ P ba ba c − P b ⊗ Jc . (2.15)
Theorem 2.2 For each double cross sum decomposition of the Lie algebra gλ in the
form g ./ m and with generators of m given in terms of the map F : g→ g by (2.9), the
semidual Lie bialgebra m∗I/g has a co-commutator which is co-boundary, with classical
r-matrix
r = P a ∧ F (J ) = F b P aa a ∧ Jb. (2.16)
5
Proof Consider the following ansatz for an r-matrix of m∗I/g
r = Rb P aa ∧ Jb. (2.17)
For the associated co-comm
((
utators one finds )
δ(P a) = −Rd f a +Rd f ab dc c db ) (P c ⊗ P b )
δ(J ) = −Rd f c +Rc f d J ⊗ P ba b ad d ab c − P b ⊗ Jc (2.18)
so that we obtain the relation
g cab = R
d f c db ad +R af
c, L c = Rd f c −Rc f ddb ab a db d ab . (2.19)
Comparing with the expressions for g c cab and Lab given in (2.6) we conclude that we can
reproduce them by choosing
Ra ab = F b. (2.20)
Finally, we determine a condition for (2.17) to satisfy the modified classical Yang-Baxter
equation
[[r, r]] + λΩ = 0, (2.21)
where, for now, λ is an arbitrary real constant and Ω is the invariant element
Ω = f c(P a ⊗ P b ⊗ J − P aab c ⊗ Jc ⊗ P b + Jc ⊗ P a ⊗ P b). (2.22)
Inserting (2.17) into
[[r, r]] = [r12, r13] + [r12, r23] + [r13, r23] (2.23)
yields three term(s, )
Rb Rc f d −Rb Rc f d +Rb Rd ca d be e d ba e afbd P e ⊗ P a ⊗ Jc (2.24)
and similar terms proportional to P a ⊗ J bc ⊗ P , and Jc ⊗ P a ⊗ P b. We deduce that the
modified Yang-Baxter equation (2.21) is equivalent to
Rb Rc f d −Rb Rc d b d c ca d be e dfba +R eR afbd + λfea = 0. (2.25)
However, this is simply the matrix form of the equation (2.10) for the map F . Thus,
substituting (2.20) into (2.17) we indeed obtain a solution of the modified classical Yang-
Baxter equation. 
3 Application to the isometry Lie algebras of 3d gravity
3.1 Special properties in three dimensions
We now turn to solutions of the factorisation condition for isometry Lie algebras arising
in 3d gravity. In the following g stands for either so(3) or so(2, 1), with generators Ja,
a = 0, 1, 2. This range of indices is unconventional in the Euclidean setting but well-
adapted to the more intricate Lorentzian situation. We write η = ηabab for either the
6
Euclidean metric diag(1, 1, 1) or the Lorentzian metric diag(1,−1,−1), and use it lower
or raise indices. The Lie brackets of g are then
[J , J ] =  J ca b abc , (3.1)
where we adopt the convention  012012 =  = 1. The invariant inner product
〈Ja, Jb〉 = ηab (3.2)
on g plays an important role in our analysis. In the Lorentzian case, it is sometimes
convenient to work with normalised raising and lowering operators
1 1
N = √ (J0 + J2), Ñ = √ (J0 − J2), (3.3)
2 2
whose names are chosen to reflect the fact that they are null (or lightlike) with respect
to (3.2):
〈N,N〉 = 〈Ñ , Ñ〉 = 0, 〈N, Ñ〉 = 1. (3.4)
Their commutators are
[Ñ ,N ] = J1, [J1, N ] = N, [J1, Ñ ] = −Ñ . (3.5)
The Lie algebra gλ is that of the Poincaré, de Sitter or anti-de Sitter group or their
Euclidean analogues in three dimensions, with brackets
[Ja, Jb] = 
c c c
abcJ , [Qa, Jb] = abcQ , [Qa, Qb] = λabcJ . (3.6)
As in the general case, we are looking for basis change
Q′a = Qa + F
b
aJb, (3.7)
so that {J0, J1, J2, Q′0, Q′1, Q′2} is a basis and {Q′0, Q′ , Q′1 2} closes under Lie brackets, i.e.,
[J , J ] =  ca b ab Jc, [Q
′
a, Jb] = 
c ′
ab Qc + L
c
ab Jc, [Q
′
a, Q
′ ] = g c ′b ab Qc. (3.8)
In this case, the semidual Lie algebra (2.14) has the brackets
[Ja, J ] = 
c b c c a b
b ab Jc, [Ja, P ] = ab P , [P , P ] = 0, (3.9)
which are the brackets of the Euclidean Lie algebra in the case of Euclidean signature
and those of the Poincaré Lie algebra in the case of Lorentzian signature. Every decom-
position of (3.6) according to (3.7) will therefore lead to a co-boundary Lie bialgebra
structure on those Lie algebras.
In order to determine all possible solutions of the condition (2.10), we will be making
use of the invariant inner product (3.2) on g. We write F t : g → g for the transpose of
a map F : g→ g relative to 〈 , 〉, i.e., for the map which satisfies
〈F t(X), Y 〉 = 〈X,F (Y )〉 ∀X, Y ∈ g. (3.10)
7
We also need the fundamental identity
 efg eabc = δa(δ
f
b δ
g
c − δ
gδfb c )− δ
f (δe g g e g e f f ea bδc − δb δc) + δa(δbδc − δb δc), (3.11)
which holds in both the Euclidean and Lorentzian context. It implies
abc
afg = δfδg − δgδfb c b c , (3.12)
which, in turn, is equivalent to
[X, [Y, Z]] = 〈X,Z〉Y − 〈X, Y 〉Z, ∀X, Y, Z ∈ g. (3.13)
This can be used to prove the useful result
〈[X, Y ], V 〉〈V, Z〉+ 〈[Y, Z], V 〉〈V,X〉+ 〈[Z,X], V 〉〈V, Y 〉 = 〈V, V 〉〈[X, Y ], Z〉 (3.14)
for any X, Y, Z, V ∈ g, see [16] for details and a related identity.
3.2 Reformulating the factorisation condition
The Lie bracket or, equivalently, the epsilon tensor provide an identification of g∧g with
g. It follows that for any linear map F : g→ g, the assignment
(X, Y ) ∈ g ∧ g 7→ [F (X), F (Y )] ∈ g (3.15)
defines a linear map g → g. Our factorisation condition (2.10) allows for a convenient
‘dual’ formulation in terms of this map, which turns out to be the adjugate of F :
Lemma 3.1 For every linear map F : g→ g, there is a uniquely determined linear map
F adj : g→ g, (3.16)
which satisfies
〈F adj(Z), [X, Y ]〉 = 〈Z, [F (X), F (Y )]〉 ∀X, Y, Z ∈ g. (3.17)
It is given by
1 ( )
F adj = F 2 − tr(F )F + tr(F ))2 − tr(F 2) id, (3.18)
2
which is the adjugate of F .
Proof Inserting the basis elements Ja, Jb, Je for X, Y, Z in (3.17) , one deduces the
matrix relation
(F adj)fefab = ecdF
c d
aF b. (3.19)
Multiplying with gab, summing over a, b and repeatedly applying (3.11) one arrives at
the formula (3.18). To see why (3.18) gives the adjugate of F (usually defined as the
transpose of the matrix of co-factors) note that for any n×n matrix A, the characteristic
polynomial pA(t) = det(A− t id) has the constant term (−1)n detA so that
pA(t)− (−1)n detA
qA(t) = (3.20)
t
8
is a polynomial of degree n− 1 which satisfies
qA(A)A = AqA(A) = (−1)n−1 det(A) id, (3.21)
by the Cayley-Hamilton Theorem. It is proved in [17] that, in fact,
Aadj = qA(A). (3.22)
The expression (3.18) is easily seen to be qF (F ) for n = 3 so that the solution of (3.17)
is indeed the adjugate of F as claimed. 
Note that for R ∈ SO(3) or R ∈ SO(2, 1), the invariance of the epsilon tensor implies
Radj = R−1. Inserting this into (3.18) is simply the Cayley-Hamilton theorem for the
3× 3 matrix R.
Lemma 3.2 In the case g = so(3) or g = so(2, 1), the factorisation condition (2.10)
for the linear map F : g→ g is equivalent t(o the quadratic r)elation
− t 1(F trF id)(F + F ) + (trF )2 − tr(F 2) id = −λid (3.23)
2
Proof The factorisation condition (2.10) can equivalently be written as
〈Z, [F (X), F (Y )]− 〈F t(Z), [X,F (Y )] + [F (X), Y ]〉〉
= −λ〈Z, [X, Y ]〉 ∀ X, Y, Z ∈ g. (3.24)
Now observe that
[X,F (Y )] + [F (X), Y ] = [(id + F )(X), (id + F )(Y )]− [F (X), F (Y )]− [X, Y ], (3.25)
and apply (3.18) both to F and id + F to deduce (3.23). 
3.3 Exploiting rotational invariance
In order to analyse the (equivalent) conditions (2.10) and (3.23) further, we split F into
symmetric and antisymmetric parts with respect to 〈 , 〉. The antisymmetric part can
be written as the adjoint action of a general element V ∈ g by virtue of the identity
〈[X, Y ], Z〉+ 〈Y, [X,Z]〉 = 0 ∀X, Y, Z ∈ g. (3.26)
Thus we write the map F as
F = S + adV . (3.27)
for V ∈ g and S : g→ g satisfying
〈S(X), Y 〉 = 〈X,S(Y )〉 for all X, Y ∈ g. (3.28)
Inserting the split (3.27) for F and simplifying using (3.14) and standard identities we
deduce
〈Z, [S(X), S(Y )]〉 − 〈X, [S(Y ), S(Z)]〉 − 〈Y, [S(Z), S(X)]〉
+〈2[V, S(Z)] + (λ+ 〈V, V 〉)Z, [X, Y ]〉 = 0 ∀X, Y, Z ∈ g, (3.29)
9
which is equivalent to
[S(X), S(Y )]− S([X,S(Y )] + [S(X), Y ])
+2(〈V,X〉S(Y )− 〈V, Y 〉S(X)) = −(λ+ 〈V, V 〉)[X, Y ] ∀X, Y ∈ g. (3.30)
This turns out to be a very useful formulation of the factorisation condition for the Lie
algebras (3.6).
We can derive an equivalent condition to (3.30) by inserting the split (3.27) into (3.23)
and using F + F t = 2S. A short calculation gives
1
2S2 − 2tr(S)S + ((tr(S))2 − tr(S2))id + 2adV S = −(λ+ 〈V, V 〉)id. (3.31)
2
This can also be derived directly from (3.30) using the methods of Sect. 3.2. The
equation (3.31) is invariant under SO(3) or SO(2, 1) conjugation, and this can be used
to split it into irreducible components under this action, namely into symmetric traceless,
antisymmetric and scalar matrices. Noting that the commutator [adV , S] is symmetric
but that the anticommutator {adV , S} = adV S + S adV is antisymmetric we write
2adV S = [adV , S] + {adV , S} (3.32)
and deduce three equations. F(or the scalar pa)rt we have1
(trS)2 − tr(S2) = λ+ 〈V, V 〉, (3.33)
6
for the vector part we find
{adV , S} = 0 (3.34)
and for the symmetric, tr(aceless part we d)educe( )
1 1
[adV , S] + 2 S
2 − tr(S2)id − 2 trS S − (trS)2id = 0. (3.35)
3 3
The equations (3.33), (3.34) and (3.35) are derived in a different form and by a different
method in [10], where they are used for determining the most general r-matrix for the
Poincaré and Euclidean(Lie algebras. Ad)ding((3.34) and (3.35) we)deduce the relation
1
ad S + S2 − tr(S2V )id −
1
tr(S)S − (trS)2id = 0. (3.36)
3 3
This has a various useful consequences, including the following lemma which is also
implicit in some of the manipulations in the appendix of [10].
Lemma 3.3 If the map F : g → g solves the factorisation condition (3.23) and S and
adV are its symmetric and antisymmetric part as in (3.27), then the following hold:
(1) The antisymmetric part adV is the zero map on any subspace of h ⊂ g where the
restriction S|h is invertible. In particular, if S is invertible then V vanishes.
10
(2) If dim ker S=1 then: ker S is not a null space ⇒ V = 0.
In the following, the contrapositive of result (2) will be particularly useful: if ker S is
one-dimensional and V 6= 0 then ker S is necessarily a null space.
Proof (1) On any subspace h where Sh is invertible we can multiply (3.36) by the
(symmetric) map S−1|h to obtain an expression for the antisymmetric map adV in terms
of a symmetric map. Thus (adV )||h = 0.
(2) It follows from (3.34) that ker S is invariant under the action of adV . If ker S is
one-dimensional then any basis vector X of it is necessarily an eigenvector of adV , i.e.,
[V,X] = µX. Since 〈[V,X], X〉 = 0, we deduce that µ = 0 if X is not null. However,
in that case, S restricted to the orthogonal complement of X is invertible, so we know
from part (1) that adV vanishes on the orthogonal complement of X. Since we already
have adV (X) = 0 we deduce that V = 0. 
4 Solving the factorisation condition
4.1 Solutions of definite symmetry
Before we systematically study solution of the equation (3.30) in the next sections, we
note some special cases which can be found by inspection. Setting the symmetric part
S to zero, the condition (3.30) reduces to a condition on the Lie algebra element V .
Expanding
V = −vaJa, (4.1)
where the sign is chosen to match conventions in [11], we have
〈V, V 〉 = vava = −λ. (4.2)
The resulting bialgebra structure on the semidual Lie algebra has the r-matrix
r = vcb P aκ ac ∧ Jb, (4.3)
which is the familiar 3d κ-Poincaré structure with deformation parameter v = (v0, v1, v2),
which may be timelike, spacelike or timelike, depending on λ [18, 19, 20]. The resulting
Lie algebra structure of m is a semidirect sum R n R2, with R acting on R2 by scaling.
We will recover this solution as part of a more general family in the next section.
Similarly, setting the antisymmetric part adV of the map F to zero, we deduce from
equations (3.35) and (3.33) that
S2 − tr(S)S = −2λ id for V = 0 (4.4)
At the same time, we know from t(he Cayley Hamilt)on theorem that1
S3 − tr(S)S2 + (tr(S))2 − tr(S2) S − det(S)id = 0. (4.5)
2
Thus, from (3.33) with V = 0
(S2 − tr(S)S + 3λid)S − det(S)id = 0. (4.6)
11
Combining with (4.4) we deduce
λS = detS id for V = 0. (4.7)
It follows that for λ > 0 any purely symmetric solutions is diagonal and given by
√
F = λid. (4.8)
The resulting classical r-matrix
√
r adouble = λP ∧ Ja (4.9)
is that of the classical double of so(3) or so(2, 1). The associated Lie brackets on m are
√
[Q′a, Q
′
b] = 2 λabcQ
′c, (4.10)
i.e., m is so(3) or so(2, 1) in this case.
When λ = 0 we have another symmetric solution that can be found by inspection (but
which we will recover systematically later). For any vector m, the liner map with matrix
F b ba = m ma. (4.11)
trivially satisfies (3.30) (for λ = 0), so that
r b am = mam P ∧ Jb (4.12)
is a classical r-matrix for the Euclidean or the Poincaré Lie algebras. The Lie algebra
structure of m turns out to be a semidirect sum RnR2 with R acting on R2 by rotations,
Lorentz boosts or the nilpotent matrix ( )
0 1
M = , (4.13)
0 0
depending on whether m is timelike, spacelike or lightlike. We will study this solution,
too, as part of a larger family in the next sections and give more details there.
4.2 The standard case
In the simplest case, the symmetric part S has a diagonalising orthonormal basis Ka,
a = 0, 1, 2, of g with real eigenvalues λa, a = 0, 1, 2. The diagonalisation is always
possible in the Euclidean case but cannot always be achieved in the Lorentzian case.
We will consider the other, non-standard cases below. Since the diagonalising basis is
related to the original basis Ja, a = 0, 1, 2 by an orthogonal (i.e., SO(3) or SO(2, 1))
transformation, we know that the commutators are still
[Ka, K
c
b] = abcK . (4.14)
The map S in (3.27) then takes the from
∑2
S = λaKa〈Ka, ·〉. (4.15)
a=0
12
In order to find all solutions of this form, we consider two cases.
(I) dim kerS ≤ 1. In that case, it follows from Lemma 3.3 that V = 0: if S is invertible
this is a consequence of part (1) while for a one-dimensional kernel of S the vanishing
of S follows from part (2) since ker S cannot be a null space in the diagonalisable case
under consideration. With V = 0 we insert the diagonal form of S (4.15) into (4.4) to
find
λ0(λ1 + λ2) = 2λ,
λ1(λ0 + λ2) = 2λ,
λ2(λ0 + λ1) = 2λ. (4.16)
Taking pairwise differences we deduce
√
λ0 = λ1 = λ2 = λ. (4.17)
This is the solution (4.8) already found by inspection. Since the eigenvalues λa are
assumed to be real, this is only a valid and non-trivial solution if λ > 0.
(II) dim ker S ≥ 2. In that case, S2 = tr(S)S, so that the scalar equation (3.33) gives
the condition
〈V, V 〉 = −λ (4.18)
for V, and the matrix condition (3.35) reduces to
[adV , S] = 0. (4.19)
If S = 0 then this imposes no further restriction of V and we recover the purely anti-
symmetric solution F = adV found by inspection in the previous section. However, if
dim ker S=2 so that S = λaKa〈Ka, ·〉 for some fixed a and λa 6= 0, we see that we can
now choose V to be a multiple of Ka, with the multiple chosen so that the normalisation
condition (4.2) holds. When λ 6= 0, the resulting solution can be written
F = βV 〈V, ·〉+ adV , (4.20)
with β ∈ R arbitrary and V satisfying (4.18). When λ = 0, the requirement that
S = λaKa〈Ka, ·〉 (no sum) and adV commute enforces V = 0. In that case we recover
the solution (4.11), at least for space- or timelike vectors m. The case of light-like m
represents a non-diagonalisable map S and will appear in Sect. 4.3.
In the remainder of this subsection we study the solution (4.20) in more detail. With
the convention (4.1), the generators of the subalgebra m in this case are
Q′ = Q +  vb ca a abc J + β(v
bJb) va, (4.21)
where β is an arbitrary (real) parameter. Using 3-vector notation v = (v0, v1, v2) etc.
we can also write
Q′ = Q + v × J + β(v ·J)v. (4.22)
Then, with v ·w = vawa etc., the brackets are
[p·Q′, q ·Q′] =−v × (p× q)·(Q′ − βv ×Q′)
= (v ·p)(q ·Q′)− (v ·q)(p·Q′)− β(p× q)·v (v ·Q′)− λβ(p× q)·Q′ (4.23)
13
as well as
[p·J , q ·Q′] = p× q ·Q + p× (q × v)·J + β(q ·v)p× v ·J , (4.24)
which can be written in terms of the generators Q′a as
[p·J , q ·Q′] = p× q ·Q′ + (v × p− βv × (v × p))·q × J .
= p× q ·Q′ + (v ·q)(p·J)− (p·q)(v ·J)− β(p·v)(v × q ·J)− λβ(p× q ·J).
(4.25)
To identify the resulting Lie algebra structure of m, we consider the various cases.
4.2.1 The Euclidean case with λ < 0
√
We pick v = −λ(1, 0, 0), so that (4.22) gives
√ √
Q′0 = Q0 − βλJ0, Q′1 = Q1 − −λJ2, Q′2 = Q2 + −λJ1. (4.26)
Then (4.23) yields the following brackets for m:
[Q′ ′1, Q2] = 0,
[Q′
√
0, Q
′
1] = −λQ′1 − βλQ′2,
[Q′
√
0, Q
′
2] = −λQ′2 + βλQ′1. (4.27)
Thus Q′1 and Q
′
2 span a commutative subalgebra, and Q
′
0 acts on this by infinitesimal
scaling and rotation. In terms of the Bianchi classification of three-dimensional real Lie
algebras, this is type VII.
The commutators involving Ja and Q
′
b are
[J ′ ′0, Q1] = Q2, [J0, Q
′
2] = −Q′1,
[J ,Q′
√ √
1 0] = −λJ1 + βλJ −Q′2 2, [J2, Q′0] = −λJ2 − βλJ1 +Q′1,
[J ′1, Q2] = Q
′
0 − βλJ ′ ′0, [J2, Q1] = −Q0 + βλJ0,
[J ,Q′
√
0 0] = 0, [J ,Q
′
1 1] = [J ,Q
′
2 2] = −λJ0, (4.28)
and again allow for a geometric interpretation. The action of Ja on the vector Q
′
is infinitesimal rotation around the a-th axis. The action of Q′0 on the J1J2-plane is
infinitesimal scaling and rotation, as on the Q′Q′1 2-plane above. The action of both Q
′
1
and Q′2 on the J1J2-plane is to map it onto J0, which is consistent with the fact that Q
′
1
and Q′2 commute.
4.2.2 Lorentzian case with λ < 0
√
If v is the timelike vector v = −λ(1, 0, 0), then the generators (4.21) of m are
′ √ √Q0 = Q0 − βλJ , Q′0 1 = Q1 + −λJ2, Q′2 = Q2 − −λJ1, (4.29)
14
so that from (4.23) we obtain
[Q′1, Q
′
2] = √0,
[Q′0, Q
′
1] = √−λQ
′
1 + βλQ
′
2,
[Q′0, Q
′
2] = −λQ′2 − βλQ′1. (4.30)
Again we have a two-dimensional abelian algebra spanned by Q′1, Q
′ ′
2, with Q0 acting by
infinitesimal rotation and scaling. The Bianchi type is again VII. Computing the mixed
commutators is straight forward and leads to a geometric interpretation analogous to
the Euclidean case, but we omit the details here.
4.2.3 Lorentzian case with λ > 0
√
With the spacelike vector v = λ(0, 1, 0), we have from (4.21) that
√ √
Q′0 = Q0 − λJ , Q′2 1 = Q1 − βλJ1, Q′2 = Q2 − λJ0, (4.31)
and therefore (4.23) gives
[Q′0, Q
′
2] = 0,√
[Q′ , Q′1 0] = −√λQ
′
0 − βλQ′2,
[Q′1, Q
′ ] = − λQ′2 2 − βλQ′0. (4.32)
It is illuminating to write the commutators in this case in terms of the generatorsN, Ñ, J1
defined in (3.3) and the corresponding generators
1 1
QN = √ (Q0 +Q2), QÑ = √ (Q0 −Q2). (4.33)
2 2
Then √ √
Q′N = QN − λN, Q′ = QÑ + λÑ, Q
′
1 = Q1 − βλJ1, (4.34)Ñ
and therefore (4.32) gives [
Q′ ′
]
N , Q = 0,Ñ
[ √[Q′1, Q′N]] = −(βλ+ ′√ λ)QN ,
Q′ , Q′1 = (βλ− λ)Q′ , (4.35)Ñ Ñ
This is again a semidirect sum R n R2 and Bianchi type VI: Q′N and Q′ span a two-Ñ
dimensional abelian algebra, and Q′1 acts on them with a matrix that has (generically
distinct) real eigenvalues.
4.3 The non-standard case
In the Lorentzian case, maps S : g → g which are symmetric with respect to the non-
degenerate symmetric form 〈·, ·〉 cannot always be diagonalised. We now consider the
different normal forms (which are discussed, for example, in the textbook [21]). In each
case, the solution of the factorisation condition (3.30) or, equivalently, Eq. (3.31) uses
Lemma 3.3 and standard arguments from linear algebra; some proceed along similar
lines to the discussion in the appendix of [10].
15
4.3.1 Small Jordan Block
In this case the symmetric map S has a lightlike and a spacelike eigenvector and can be
brought into the form
S = a(N〈 Ñ , ·〉+ Ñ〈N, ·〉) + bN〈N, ·〉 − cJ1〈 J1, ·〉, b 6= 0, (4.36)
so that
S(N) = aN, S(Ñ) = bN + aÑ, S(J1) = cJ1. (4.37)
It is easy to check that, if a 6= 0 and c =6 0 (so that S is invertible) we have V = 0 by
Lemma 3.3 and hence the diagonal solution (4.8). If a 6= 0 then S has a one-dimensional
kernel which is not null, so by Lemma 3.3 we again deduce V = 0 and obtain a solution
of the form (4.11). If a = 0 but c 6= 0 we have, from (3.33), that 〈V, V 〉 = −λ. The
requirement that adV leave the kernel of S invariant implies that V √is spacelike or
lightlike, i.e., λ ≥ 0. When λ > 0, we can use (3.36) to deduce that c = λ. Renaming
b as β we have the new solution
√ √
F = βN〈N, ·〉 − λJ1〈 J1, ·〉+ λ adJ1 , λ > 0, β ∈ R, (4.38)
so that
√ √ √
Q′N = Q
′ ′
N + λN, Q = QÑ + βN − λÑ, Q1 = Q1 + λJ1. (4.39)Ñ
The commutators of m are [
Q′ , Q′
]
N = 0,Ñ
[ √[Q′1, Q′N]] = 2 λQ′ ,√ N
Q′1, βQ
′
N − 2 λQ′ = 0. (4.40)Ñ
This is Bianchi type III, which is isomorp√hic to the direct sum of the one dimensional
Lie algebra R (generated here by βQ′ −2 λQ′N ) and the non-abelian two-dimensionalÑ
Lie algebra L(2) (generated here by Q′N , Q
′ ).
Ñ
When λ = 0, we again use (3.36) to obtain the solution
F = βN〈N, ·〉+ adN , (4.41)
for an arbitrary β ∈ R. This may be viewed as a limit as λ→ 0 of (4.38) and is also a
lightlike version of (4.20). The generators of m are
Q′ = Q , Q′N N = QÑ − J1 + βN, Q
′
1 = Q1 −N, (4.42)Ñ
and the commutators of m are
[ [Q′ ′N , Q1[ ′ ′ ]
] = 0
Q ,QN = −Q′Ñ N
Q′ , Q′
]
= −Q′ − βQ′1 1 N . (4.43)Ñ
which are again those of a semidirect sum RnR2. When β = 0, this is Bianchi type V,
with −Q′ acting on the span of Q′N , Q′1 via the identity. When β =6 0, it is Bianchi typeÑ
IV, with − 1Q′ acting on the span of Q′ , Q′N 1 via id/β + M , where M is the nilpotentβ Ñ
linear map with matrix (4.13).
16
4.3.2 Large Jordan Block
The symmetric map S only has a single, lightlike eigenvector and can be brought into
the form
S = a(N〈 Ñ , ·〉+ Ñ〈N, ·〉 − J1〈 J1, ·〉) + b(N〈 J1, ·〉+ J1〈N, ·〉), b 6= 0 (4.44)
so that
S(N) = aN, S(Ñ) = aÑ + bJ1, S(J1) = −bN + aJ1. (4.45)
Again we require a = 0 for a non-vanishing V . Then ker S is one-dimensional and
spanned by the lightlike generator N ; this space is invariant under adV provided V is
a linear combination of N and J1. However, since trS = trS
2 = 0 we can use (3.36) to
deduce that ad S + S2V = 0, which in turn implies that V = −bN . Thus we are in the
case λ = 0 and, after re-naming b = β/2, we have the solution
β
F = (N〈 J1, ·〉+ J1〈N, ·〉 − adN) , (4.46)
2
which can be written more compactly as
F = βJ1〈N, ·〉. (4.47)
The resulting basis of m is
Q′ = P , Q′ = Q′N N + βJ1, Q
′
1 = Q1, (4.48)Ñ Ñ
and we obtain the commutators
[Q′1, Q
′
N ] = 0, [Q
′ , Q′1 ] = 0, [Q
′ , Q′ ] = βQ′N N . (4.49)Ñ Ñ
This is again a direct sum of R (spanned by Q′1) and the two-dimensional non-abelian
Lie algebra, i.e., Bianchi type III
The remaining normal form of a symmetric map S : g→ g is
S = a(N〈 Ñ , ·〉+ Ñ〈N, ·〉) + b(Ñ〈 Ñ , ·〉 −N〈N, ·〉)− cJ1〈 J1, ·〉 (4.50)
which has the form of a ‘rotation’ in the span of {N, Ñ}:
S(N) = aN + bÑ , S(Ñ) = −bN + aÑ S(J1) = cJ1. (4.51)
In this case we do not obtain a new solution: the cases dim ker S ≤ 1 can be dealt with
as above. However, dim ker S=2 requires a = b = 0, in which case we recover a solution
in the family (4.20)
17
5 Summary and Discussion
We can unify a large family of solutions of the factorisation condition (3.23) in the form
F = βV 〈V, ·〉+ α adV , β ∈ R, α ∈ {0, 1}, α〈V, V 〉 = −λ. (5.1)
This reduces to the purely symmetric solutions (4.11) when α = 0 and to the family
(4.20), including the lightlike case (4.41), when α = 1. This factorisation map gives rise
to the r-matrices
r̃κ = (βv
av + αvcb ab ac)P ∧ Jb, (5.2)
which generalise the familiar κ-Poincaré r-matrix (4.3); we therefore call them gener-
alised κ-Poincaré solutions.
Figure 1: The R-actions in the semidirect sums R n R2 and their degenerate limit
R⊕ L(2); explanation in the main text
The Lie algebra structure of m arising from the family (5.1) is of the semidirect sum
form RnR2 for all values of the parameters. The R-action on R2 has two components,
corresponding to the parameters α and β being non-zero. The α-component (for α = 1
and β = 0) is simple overall scaling, shown in the bottom left of Fig. 1 and associated
with the standard κ-Poincaré algebra. The β-component (for β 6= 0 and α = 0) is more
interesting and depends on the value of λ. Its matrix is proportional to one of the 2× 2
matrices ρ(θ) representing the rel(ation θ2 = λ: ) √

0 −λ

√ if λ < 0
− −λ 0
( )0 1
ρ(θ) =  if λ = 0 (5.3) 0 0
(√ ) λ √0 if λ > 00 − λ
18
The flows generated by exponentiating these matrices are shown from left to right in the
top row of Fig. 1.
√
In addition to the family (5.1) we found the solution F = λ id, which gives rise
to the r-matrix of the classical double (4.9) and m = so(3) or m = so(2, 1). Finally,
we have the exce√ptional, Jordan-type solutions (4.38) and (4.47). With the notation
PN = (P0 + P2)/ 2, the r-matrix associated to the factorisation map (4.38) is
√
rSJ = βPN ∧N + λ(P b a1 ∧ J1 +  a1P ∧ Jb), (5.4)
while the r-matrix associated to the solution (4.47) is simply
rLJ = βPN ∧ J1. (5.5)
The associated Lie algebra structure of m is R ⊕ L(2), which may be viewed as a de-
generate case of the semidirect sums R n R2 with a diagonalisable R-action but one
eigenvalue vanishing. The eigenvector for the non-vanishing eigenvalue is always the
lightlike vector N , but the eigenvector for the zero eigenvalue varies between the two
cases and as a function of the parameter β. The flow obtained by exponentiating such
matrices is shown for a generic case in the bottom right of Fig. 1.
F r-matrix m Bianchi type
0 0 R3 I
√
λ id rdouble so(2, 1) or so(3) VIII or IX
βV 〈V, ·〉+ α adV r̃κ Rn R2 IV-VII
β ∈ R, α ∈ {0, 1}
Jordan forms rSJ and rLJ R⊕ L(2) III
(4.38) and (4.47)
Table 1: Factorisations and associated r-matrices
Table 1 summarises our results and discussion of the factorisation of the Lie algebras
(3.6) and the associated r-matrices of the semiduals (3.9). As reviewed in the Intro-
duction, semidualisation gives all r-matrices of the form Rb aaP ∧ Jb in this case. It is
interesting that for each type of r-matrices there is an associated Bianchi type for the
Lie algebra m. All Bianchi types except type II (the Heisenberg algebra) arise in this
way.
Viewing the generatorsQ′a of the original algebra (3.8) as spacetime coordinates and the
dual generators Pa in (3.9) as momenta, the summary in Table 1 may thus also be viewed
as a list of non-commutative spaces and associated non-co-commutative momentum
19
spaces, parametrised by the relevant r-matrices. We thus obtain the unified picture of
non-commutative geometries and isometry algebras promised in the Introduction.
There are several questions and topics for further research which follow from the re-
sults reported here. Mathematically, one would like to understand more generally when
semidualising families of Lie algebras with a given double cross sum decomposition pro-
vides an effective way of finding r-matrices for the semidual Lie bialgebra. It would also
be interesting to apply the same technique to families of Lie bialgebras with non-trivial
co-commutators, so that the semidual has a more complicated Lie algebra structure than
the semidirect sums found here.
In the physics literature, the standard κ-Poincaré (with a timelike deformation param-
eter) and the classical double Lie bialgebras in Table 1 have been much studied, the
latter (but not the former - see [16]) being related to 3d quantum gravity. It would
be interesting to see if the generalised κ-Poincaré r-matrices (5.2) with β 6= 0 and the
Jordan-type r-matrices (5.4) and (5.5) also have applications in real or toy models of
mathematical physics.
Acknowledgments Some of the work reported in this paper was carried out while
Prince K Osei was supported by an ICTP PhD fellowship. We thank the Perimeter
Institute for hospitality and support during the final stage of the project.
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