Describing certain Lie algebra orbits via polynomial equations
Let $\mathfrak{h}_3$ be the Heisenberg algebra and let $\mathfrak g$ be the 3-dimensional Lie algebra having $[e_1,e_2]=e_1$ $(=-[e_2,e_1])$ as its only non-zero commutation relations. We describe the closure of the orbit of a vector of structure constants corresponding to $\mathfrak{h}_3$ and $\mat...
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2019
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| author | Ivanova, N. Pallikaros, C. Іванова, Н. Паллікарос, К. |
| author_facet | Ivanova, N. Pallikaros, C. Іванова, Н. Паллікарос, К. |
| author_institution_txt_mv | [
{
"author": "N. Ivanova",
"institution": "European University of Cyprus"
},
{
"author": "C. Pallikaros",
"institution": "University of Cyprus"
}
] |
| author_sort | Ivanova, N. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2020-08-13T08:52:53Z |
| description | Let $\mathfrak{h}_3$ be the Heisenberg algebra and let $\mathfrak g$ be the 3-dimensional Lie algebra having $[e_1,e_2]=e_1$ $(=-[e_2,e_1])$ as its only non-zero commutation relations. We describe the closure of the orbit of a vector of structure constants corresponding to $\mathfrak{h}_3$ and $\mathfrak g$ respectively as an algebraic set giving in each case a set of polynomials for which the orbit closure is the set of common zeros.Working over an arbitrary infinite field, this description enables us to give an alternative way, using the definition of an irreducible algebraic set, of obtaining all degenerations of $\mathfrak{h}_3$ and $\mathfrak g$ (the degeneration from $\mathfrak g$ to $\mathfrak{h}_3$ being one of them. |
| first_indexed | 2026-08-04T01:06:15Z |
| format | Article |
| fulltext |
Çáiðíèê ïðàöü Iíñòèòóòó ìàòåìàòèêè ÍÀÍ Óêðà¨íè 2019, ò. 16, � 1, 84�99
ÓÄÊ 517.912:512.816
Describing certain Lie algebra orbits
via polynomial equations
N.M. Ivanova †, C.A. Pallikaros ‡
† European University of Cyprus, Nicosia
E-mail: ivanova.nataliya@gmail.com
‡ University of Cyprus, Nicosia
E-mail: pallikar@ucy.ac.cy
Ðîçãëÿäàþòüñÿ àëãåáðà Ãåéçåíáåðãà h3 òà òðèâèìiðíà àëãåáðà Ëi g ç íå-
íóëüîâèìè êîìóòàöiéíèìè ñïiââiäíîøåííÿìè [e1, e2] = e1 (= −[e2, e1]).
Îïèñàíî àëãåáðà¨÷íi ìíîæèíè, ùî ¹ çàìèêàííÿì îðáiò âåêòîðiâ ñòðóê-
òóðíèõ ñòàëèõ, ùî âiäïîâiäàþòü h3 i g, à ñàìå: ó êîæíîìó ç âèïàäêiâ
ïîáóäîâàíî íàáið ïîëiíîìiâ, òàêèõ, ùî ìíîæèíà ¨õ ñïiëüíèõ íóëiâ ¹
çàìèêàííÿì îðáiòè âåêòîðà ñòðóêòóðíèõ ñòàëèõ. Òàêèé îïèñ äîçâîëÿ¹
íàäàòè àëüòåðíàòèâíèé ïiäõiä äî çíàõîäæåííÿ âñiõ ìîæëèâèõ âèðîä-
æåíü h3 òà g ó äîâiëüíîìó íåñêií÷åííîìó ïîëi çà äîïîìîãîþ îçíà÷åííÿ
íåçâiäíî¨ àëãåáðà¨÷íî¨ ìíîæèíè.
Let h3 be the Heisenberg algebra and let g be the 3-dimensional Lie algebra
having [e1, e2] = e1 (= −[e2, e1]) as its only non-zero commutation rela-
tions. We describe the closure of the orbit of a vector of structure constants
corresponding to h3 and g respectively as an algebraic set giving in each
case a set of polynomials for which the orbit closure is the set of common
zeros. Working over an arbitrary in�nite �eld, this description enables us
to give an alternative way, using the de�nition of an irreducible algebraic
set, of obtaining all degenerations of h3 and g (the degeneration from g
to h3 being one of them).
1. Introduction. In the second half of the twentieth century a
lot of works appeared on the study of different types of limit processes
between various physical or geometrical theories. Such limit processes
naturally lead to the notion of contraction (or degeneration). Possibly
the first work in this direction was Segal [11] who studied a limit process
of a family of some physically important isomorphic Lie groups. The
claim is that if two physical theories are related by a limit process, then
the associated invariance groups (and invariance algebras) should also
Describing certain Lie algebra orbits via polynomial equations 85
be related by some limit process. This led to a wide investigation of
contractions of Lie algebras from the physical point of view. Possibly, the
three most famous physical examples of contractions are the following.
• Contraction of relativistic mechanics to classical mechanics was
studied in works by Inönü and Wigner [6, 7]. Considering the phys-
ical limit process c → ∞ in special relativity theory they showed
how the symmetry group of relativistic mechanics (the Poincaré
group) contracts to the Galilean group which is the symmetry
group of classical mechanics.
• The relation between classical and quantum mechanics can also be
expressed in terms of a limit process or, in other words, a contrac-
tion [5]. Thus, one can consider classical mechanics as the limit of
quantum mechanics under the contraction h → a, where h is the
Weyl–Heisenberg algebra and a is the abelian Lie algebra of the
same dimension. Under this contraction the quantum mechanical
commutator [x, p] = i} (corresponding to the Heisenberg uncer-
tainty principle) maps to the Abelian case (that is, the classical
mechanics limit) under }→ 0.
• The porous medium equation ut = m−1∆(um− 1) can be contrac-
ted [13] (as m→ 0) to the equation ut = ∆ lnu, which is equivalent
to the equation defining the Ricci flow on R2.
In these (and many other publications) it is shown, in particular,
how some basic properties of the “contracted theories” can be recon-
structed from the corresponding properties of the “original” theories.
In an attempt to unify such observations, Zaitsev [14], independently
of Inönü and Wigner, suggested constructing “the theory of physical
theories” based on group limits of physical theories. This amounts to
including in a uniform system several physical theories being connected
together via certain relations. Recently, different types of contractions
have been widely used in elementary particle theory, analysis of differen-
tial equations and other areas of mathematical and theoretical physics.
Working over C or R, the statement “Lie algebra h1 is a contraction
of Lie algebra h2” can be rephrased as “h1 lies in the closure, in the
metric topology, of the orbit of h2 under the ‘change of basis’ action of
the group of invertible linear transformations”. In [4] the authors show
that over C the orbit closure in the metric topology coincides with the
orbit closure in the Zariski topology. Orbit closures with respect to the
86 N.M. Ivanova, C.A. Pallikaros
Zariski topology are called degenerations. The notion of degeneration is
well-defined not only over the fields C and R but also over an arbitrary
ground field. In fact, this concept of orbit closure under the action of
various groups arises naturally in many areas of mathematics (see, for
example, [10]).
In [8] we explored the possibility of investigating degenerations over
an arbitrary field using elementary algebraic techniques. For this we
needed to extend or modify techniques already used over the fields C, R
(for example contractions obtained as limit points resulting from the
action of diagonal matrices, also known as generalized Inönü–Wigner
contractions) in a way so that they can be applied to the case of dege-
nerations over an arbitrary field. In this paper, although we continue
our study of degenerations via an elementary algebraic approach, we
take a slightly different path and consider the possibility of obtaining all
degenerations (for certain examples of Lie algebras) ‘from first principles’
by direct application of the definition of an algebraic (Zariski-closed) set.
This involves obtaining explicit descriptions of the orbit closures under
consideration using polynomial equations.
The paper is organized as follows. In Section 2 we give some neces-
sary background, the setup being over an arbitrary infinite field F. In
particular, in Section 2.1 we recall some basic definitions and results on
irreducible algebraic sets and regular maps while in Section 2.2 we recall
the definition of degeneration together with some basic facts on Lie alge-
bra structure vectors and their orbits under the ‘change of basis’ action
of the general linear group. In Section 3 we perform some explicit com-
putations concerning the orbits (and their closure in the Zariski topol-
ogy) of certain given Lie algebra structure vectors corresponding to h3
and g2 ⊕ a1 respectively, where h3 denotes the Heisenberg algebra, g2
denotes the 2-dimensional non-Abelian Lie algebra and a1 denotes the
1-dimensional Abelian Lie algebra. This enables us to give a descrip-
tion of the orbit closures of these structure vectors as algebraic sets via
polynomial equations and, as a consequence, determine in an alterna-
tive way all degenerations of h3 and g2 ⊕ a1 over F. We also obtain
descriptions of the particular orbits described above as the intersection
of a Zariski-closed set with a Zariski-open set.
2. Preliminaries and generalities. We begin this section by recal-
ling some basic facts on irreducible algebraic sets. We refer the reader
to Geck [2] for more details and for proofs of the main results from the
theory we will be using.
Describing certain Lie algebra orbits via polynomial equations 87
2.1. Algebraic sets. Fix F to be an arbitrary infinite field and
let m be a positive integer. We consider the ring F [X] = F[X1, . . . , Xm]
of polynomials in the indeterminates X1, . . . , Xm over F. For each α =
(α1, . . . , αm) ∈ Fm there exists a unique F-algebra homomorphism evα :
F[X1, . . . , Xm] → F such that evα(Xi) = αi for all i. Given α =
(α1, . . . , αm) ∈ Fm and f ∈ F[X1, . . . , Xm] we will be writing more
simply f(α) = f(α1, . . . , αm) = evα(f).
Definition. Let S be any subset of F[X1, . . . , Xm]. The algebraic set
V(S) determined by S is defined by
V(S) = {α ∈ Fm : f(α) = 0 for all f ∈ S}.
A subset of Fm is called algebraic if it is of the form V(S) for some
subset S ⊆ F[X1, . . . , Xm]. For any subset V ⊆ Fm, the vanishing ideal
I(V ) of V is defined by
I(V ) = {f ∈ F[X1, . . . , Xm] : f(α) = 0 for all α ∈ V }.
It is immediate from the above definition that if S1, S2 are sub-
sets of F[X1, . . . , Xm] with S1 ⊆ S2, then V(S2) ⊆ V(S1) (see [2, Re-
mark 1.1.4]).
It can be shown (see, for example, [2, Remark 1.1.4 and Lemma 1.1.5])
that arbitrary intersections and finite unions of algebraic sets in Fm are
again algebraic. The empty set ∅ and Fm itself are clearly algebraic.
Thus, the algebraic sets in Fm form the closed sets of a topology in Fm,
which is called the Zariski topology. A subset X ⊆ Fm is open if its
complement Fm \X is algebraic (closed).
We will denote by V the closure of a subset V of Fm in the Zariski
topology.
An essential role in our investigation is played by the notion of irre-
ducibility of algebraic sets.
Definition. Let Z ⊆ Fm be a nonempty algebraic set. We say that Z is
reducible if we can write Z = Z1 ∪ Z2, where Z1, Z2 ⊆ Z are nonempty
algebraic subsets with Z1 6= Z and Z2 6= Z. Otherwise, we say that Z is
irreducible.
Remark 1 (see [2, Example 1.1.13]). Our assumption that F is infinite
ensures that Fm is irreducible.
88 N.M. Ivanova, C.A. Pallikaros
Definition. Let s, r be positive integers and let V ⊆ Fs and W ⊆
Fr be nonempty algebraic sets. We say that Φ: V → W is a reg-
ular map if there exist f1, . . . , fr ∈ F[X1, . . . , Xs] such that Φ(α) =
(f1(α), . . . , fr(α)) for all α ∈ V .
One can then observe (see [2, p. 23]) that regular maps are continuous
in the Zariski topology.
Remark 2 (see [2, Remark 1.3.2]). Let V , W be as in the definition
above and let Φ: V →W be a regular map. Assume that V is irreducible.
Then the Zariski closure Φ(V ) ⊆W is also irreducible.
2.2. Degenerations of Lie algebras. We keep the setup of the
previous subsection. In particular F denotes an arbitrary infinite field
but now we assume further that m = n3 for some integer n ≥ 2 we have
fixed. Also let G be the general linear group GL(n,F).
Now let α = (α1, . . . , αm) ∈ Fm be given. For the rest of our dis-
cussion, it will be convenient to relabel the components of α as fol-
lows. For 1 ≤ r ≤ m relabel αr as αi(r),j(r),k(r) where i(r), j(r),
k(r) are the unique integers with 1 ≤ i(r), j(r), k(r) ≤ n satisfying
r− 1 = (i(r)− 1)n2 + (j(r)− 1)n+ (k(r)− 1). We will write α = (αi,j,k)
or α = (αijk) for short. For example, in the case n = 2 (m = 8) we have
for α ∈ Fm,
α = (α1, α2, α3, α4, α5, α6, α7, α8)
= (α111, α112, α121, α122, α211, α212, α221, α222).
(The above ordering in fact amounts to writing α = (αijk) ∈ Fn3
where
the triples (i, j, k) are placed in lexicographic order.)
In a similar manner we relabel the indeterminates X1, . . . , Xm in
F[X1, . . . , Xm] and we write F[X] (= F[X1, . . . , Xm]) = F[Xijk : 1 ≤
i, j, k ≤ n].
Definition. An element λ = (λijk) ∈ Fm is called a Lie algebra struc-
ture vector if there exists an n-dimensional Lie algebra g over F and an
ordered F-basis b̂ = (b1, . . . , bn) of g such that [bi, bj ] =
∑n
k=1 λijkbk for
1 ≤ i, j ≤ n. In such a case we call λ = (λijk) the structure vector
of g relative to b̂. We denote by Ln(F) the subset of Fm consisting of
precisely those elements of Fm which are Lie algebra structure vectors.
We refer the reader to [9] for the basic definitions and properties of
Lie algebras.
Describing certain Lie algebra orbits via polynomial equations 89
The properties of the Lie bracket ensure that Ln(F) is an algebraic
subset of Fm. This is because Ln(F) = V(S) where S is the union of the
following three subsets of F[Xijk : 1 ≤ i, j, k ≤ n] (see, for example, [9,
pp. 4–5] for a proof of this fact):
{Xiik : 1 ≤ i, k ≤ n}, {Xijk +Xjik : 1 ≤ i, j, k ≤ n},{∑
k
(XijkXklr +XjlkXkir +XlikXkjr) : 1 ≤ i, j, l, r ≤ n
}
.
Remark 3. We have the following natural action of G = GL(n,F) on
Ln(F) by ‘change of basis’. Let g = (gij) ∈ G and let λ = (λijk) ∈ Ln(F).
Also let g be an n-dimensional Lie algebra over F and b̂ = (b1, . . . , bn)
be an ordered F-basis of g such that λ = (λijk) is the structure vector
of g relative to b̂. Now let b̂′ = (b′1, . . . , b
′
n) be the basis of g defined
by b′j =
∑n
i=1 gijbi for 1 ≤ j ≤ n. Also let λ′ = (λ′ijk) ∈ Fm be the
structure vector of g relative to b̂′ (so we have [b′i, b
′
j ] =
∑n
k=1 λ
′
ijkb
′
k for
1 ≤ i, j ≤ n). We will write λ′ = λg (clearly, λ′ ∈ Ln(F)). We call g the
transition matrix from basis b̂ to basis b̂′ of g.
It is well known and easy to check, that the above process describes
a well-defined (right) action of G on Ln(F). (See, for example, [8, Re-
mark 2.6] where some details of such a check are given.)
Observe that the orbits relative to the action defined in the preceding
remark correspond precisely to the isomorphism classes of n-dimensional
Lie algebras over F. We denote by O(µ) the orbit of the Lie algebra
structure vector µ ∈ Ln(F) under the action of GL(n,F) described above.
Example. It is immediate that the zero vector 0 = (0F, . . . , 0F) of Fn3
belongs to Ln(F) as it corresponds to the n-dimensional Abelian Lie
algebra over F (under any choice of basis). Its orbit consists of precisely
one point and hence it is Zariski-closed.
Remark 4. (i) For each g ∈ GL(n,F), making use of the action de-
scribed in Remark 3, we define a function Φg : Ln(F)→ Ln(F): µ 7→ µg,
(µ ∈ Ln(F)). Then Φg is a regular map and hence continuous in the
Zariski topology. (To see this we fix g ∈ GL(n,F). It follows from
the change of basis process that for each µ ∈ Ln(F) we get Φg(µ) =
(evµ(f1), . . . , evµ(fn3)) where, for 1 ≤ i ≤ n3, fi is polynomial in F[X]
which only depends on g.)
90 N.M. Ivanova, C.A. Pallikaros
(ii) In view of item (i), one can give an elementary proof of the
fact that the closure of an orbit in Ln(F) is a union of orbits (see, for
example, [8, Lemma 3.1]).
Definition. Let g1, g2 be n-dimensional Lie algebras over F. We say
that g1 degenerates to g2 (respectively, g1 properly degenerates to g2) if
there exist structure vectors λ1 of g1 and λ2 of g2, relative to some bases
of g1 and g2, such that λ2 ∈ O(λ1) (respectively, λ2 ∈ O(λ1) \O(λ1)).
It is immediate from Remark 4(ii) that if λ ∈ O(µ) and ν ∈ O(λ),
then ν ∈ O(µ), (λ,µ,ν ∈ Ln(F)). In other words, the transitivity
property holds in the case of degenerations.
Finally for this subsection we remark that there are no proper dege-
nerations over finite fields as finite subsets of Fm are closed in the Zariski
topology.
3. Lie algebra orbit closures via polynomial equations. We
continue with our assumption that F is an arbitrary infinite field.
Below, h3 will denote the Heisenberg (Lie) algebra, g2 will denote the
2-dimensional non-Abelian Lie algebra and ak, for k ≥ 1, the Abelian
Lie algebra of dimension k.
We will make use of the action of G = GL(n,F) on Ln(F) described
in Remark 3 in order to perform some explicit computations concerning
the orbits (and their closure in the Zariski topology) of certain given
Lie algebra structure vectors corresponding to h3 and g2 ⊕ a1 respec-
tively. This will allow us to give descriptions of the orbit closures of
these structure vectors as algebraic sets (via polynomial equations) and,
in addition, obtain descriptions of the particular orbits we investigate
here as intersections of a Zariski-closed with a Zariski-open set.
We will also show how these explicit descriptions of the orbits enable
us to provide an alternative way of obtaining all degenerations of h3 and
g2 ⊕ a1 over F.
3.1. The Heisenberg algebra. We consider the Heisenberg alge-
bra h3. This (3-dimensional) algebra has an F-basis ê = (e1, e2, e3)
relative to which the only non-zero products (commutation relations) are
[e2, e3] = e1 = −[e3, e2]. The structure vector of h3 relative to ê is η =
(ηijk) ∈ F27 where η231 and η321 (with η231 = 1, η321 = −1) are the only
nonzero components of η. First we determine O(η) as a subset of F27.
For this, let g = (gij) ∈ GL(3,F) and suppose that Mij (i, j = 1, 2, 3) is
the determinant of the matrix obtained from g by deleting its i-th row
and j-th column. Assume further that g is the transition matrix from
Describing certain Lie algebra orbits via polynomial equations 91
basis (e1, e2, e3) to the basis (e′1, e
′
2, e
′
3) of h3. (So (e′1, e
′
2, e
′
3) is the basis
of h3 given by e′j =
∑3
i=1 gijei for 1 ≤ j ≤ 3.) An easy computation then
shows that, relative to this new basis, the multiplication in h3 is given by
[e′1, e
′
2] = (det g)−1M13(M11e
′
1 −M12e
′
2 +M13e
′
3),
[e′1, e
′
3] = (det g)−1M12(M11e
′
1 −M12e
′
2 +M13e
′
3),
[e′2, e
′
3] = (det g)−1M11(M11e
′
1 −M12e
′
2 +M13e
′
3).
It follows that there exist α, β, γ, δ ∈ F such that
[e′1, e
′
2] = γδ(αe′1 − βe′2 + γe′3),
[e′1, e
′
3] = βδ(αe′1 − βe′2 + γe′3),
[e′2, e
′
3] = αδ(αe′1 − βe′2 + γe′3).
The above relations motivate the following definition. For α, β, γ, δ ∈ F,
let η′(α, β, γ, δ) ∈ F27 be defined by η′(α, β, γ, δ) = (0, 0, 0, αγδ,−βγδ,
γ2δ, αβδ,−β2δ, βγδ,−αγδ, βγδ,−γ2δ, 0, 0, 0, α2δ,−αβδ, αγδ,−αβδ, β2δ,
−βγδ,−α2δ, αβδ,−αγδ, 0, 0, 0).
We aim to show that the subset V of F27 defined by
V =
{
η′(α, β, γ, δ) ∈ F27 : α, β, γ, δ ∈ F
}
is in fact the (disjoint) union of O(η) and O(0) (recall that 0, the zero
vector of F27, is the unique structure vector corresponding to the 3-
dimensional Abelian Lie algebra). It is clear from the above discussion
that O(η) ⊆ V , hence it suffices to show that any nonzero vector v ∈ V
belongs to O(η). For this, it will be convenient to consider the decom-
position V = V1 ∪ V2 ∪ V3 where the subsets V1, V2, V3 of V are defined
as follows: First, for µ, ν, λ, σ, τ, κ ∈ F, define the elements η1(µ, ν, λ),
η2(τ, σ) and η3(κ) of F27 by
η1(µ, ν, λ) = (0, 0, 0, νλ,−µνλ, ν2λ, µλ,−µ2λ, µνλ,−νλ, µνλ,
− ν2λ, 0, 0, 0, λ,−µλ, νλ,−µλ, µ2λ,−µνλ,−λ, µλ,
− νλ, 0, 0, 0),
η2(τ, σ) = (0, 0, 0, 0, στ,−στ2, 0, σ,−στ, 0,−στ, στ2, 0, 0, 0, 0,
0, 0, 0,−σ, στ, 0, 0, 0, 0, 0, 0),
η3(κ) = (0, 0, 0, 0, 0, κ, 0, 0, 0, 0, 0,−κ, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0).
92 N.M. Ivanova, C.A. Pallikaros
We then let V1 = {η1(µ, ν, λ) : µ, ν, λ ∈ F}, V2 = {η2(τ, σ) : τ, σ ∈ F}
and V3 = {η3(κ) : κ ∈ F}.
In order to establish that V is indeed the union of the three sets
above, it suffices to verify that V1 = {η′(α, β, γ, δ) ∈ V : α 6= 0}, V2 =
{η′(α, β, γ, δ) ∈ V : α = 0 and β 6= 0} and V3 = {η′(α, β, γ, δ) ∈ V : α =
0 and β = 0}. That the above equalities of sets in fact hold is immedia-
te from the relations η′(1, µ, ν, λ) = η1(µ, ν, λ), η1(βα−1, γα−1, δα2) =
η′(α, β, γ, δ) (for α 6= 0), η′(0, 1, τ,−σ) = η2(τ, σ), η2(γβ−1,−δβ2) =
η′(0, β, γ, δ) (for β 6= 0) and η′(0, 0, 1, κ) = η3(κ), η3(δγ2)=η′(0, 0, γ, δ).
Since V = V1 ∪ V2 ∪ V3, we can see that any nonzero element of V
has one of the following forms: η1(µ, ν, λ) (with λ 6= 0), η2(τ, σ) (with
σ 6= 0) or η3(κ) (with κ 6= 0).
Moreover, for λ 6= 0 we have ηg1(µ, ν, λ) = η1(µ, ν, λ), for σ 6= 0 we
have ηg2(τ, σ) = η2(τ, σ) and finally for κ 6= 0 we have ηg3(κ) = η3(κ)
where, for λ 6= 0, σ 6= 0, κ 6= 0 respectively, the matrices
g1(µ, ν, λ) =
λ−1 0 0
µ 1 0
−ν 0 1
, g2(τ, σ) =
0 σ−1 0
1 0 0
0 τ 1
,
g3(κ) =
0 0 κ−1
1 0 0
0 1 0
all belong to G = GL(3,F). This establishes that V \ {0} = O(η).
Our next aim is to show that V is an irreducible algebraic set. For
this, let S = S1 ∪ S2 ∪ S3 where S1, S2, S3 are the following subsets of
F[Xijk : 1 ≤ i, j, k ≤ 3]:
S1 = {Xiik : 1 ≤ i, k ≤ 3}, S2 = {Xijk +Xjik : 1 ≤ i, j, k ≤ 3},
S3 = {X121 −X233, X131 +X232, X122 +X133, X
2
122 +X123X132,
X2
121−X123X231, X
2
131+X132X231, X121X131 +X122X231}.
Observe that S ⊆ I(V ). We claim that V = V(S). It is clear that
V ⊆ V(S). To establish the reverse inclusion V(S) ⊆ V , let γ =
(γijk) ∈ F27 be a common zero of the elements of S. Since γ is a com-
mon zero of the elements of S1 ∪ S2, we see that the shape of γ is
determined once we determine the shape of the auxiliary vector γ̂ =
(γ121, γ122, γ123, γ131, γ132, γ133, γ231, γ232, γ233) ∈ F9. Invoking now the
fact that γ is a common zero of the polynomials of degree 1 in S3 we
Describing certain Lie algebra orbits via polynomial equations 93
see that in fact γ̂ has shape (γ121, γ122, γ123, γ131, γ132,−γ122, γ231,−γ131,
γ121). We will consider the cases γ231 6= 0 and γ231 = 0 separately. If
λ = γ231 6= 0 we can set µ = γ131λ
−1 and ν = γ121λ
−1 from which we can
deduce that γ123 = ν2λ (since γ2
121 − γ123γ231 = 0), γ132 = −µ2λ (since
γ2
131 + γ132γ231 = 0) and γ122 = −µνλ (since γ121γ131 + γ122γ231 = 0).
Hence γ ∈ V1 whenever γ231 6= 0.
For the case γ231 = 0, by similar argument, one can show that if
γ132 6= 0, then γ ∈ V2 and if γ132 = 0 then γ ∈ V3. We conclude that
V = V(S) and hence V is an algebraic set.
Next, we consider the map Φ: F4 → F27: (α, β, γ, δ) 7→ η′(α, β, γ, δ).
Clearly Φ is a regular map having V as its image. Thus, Φ(F4) = V =
V . Invoking Remarks 1 and 2, we see that V is irreducible. It follows
that O(η) is not closed in the Zariski topology. (Note that if O(η)
were Zariski-closed this would imply that V = O(η) ∪ {0} is reducible,
being the union of two nonempty closed sets.) Hence, O(η) is properly
contained in O(η). Also O(η) ⊆ V since O(η) ⊆ V and V is an algebraic
set. We conclude that O(η) = V = O(η) ∪ {0}. In other words, over
an arbitrary infinite field, the only proper degeneration of h3 is to the
Abelian Lie algebra a3. We remark here that this is a well-known fact
and has been proved using different methods over various fields, see for
example [1, 3, 8, 12]. In the discussion above we presented an alternative
way of obtaining it, based on the definition of an irreducible algebraic
set.
3.2. The algebra g2 ⊕ a1. In this subsection we perform a similar
investigation for the algebra g = g2 ⊕ a1. Note that this algebra has an
F-basis b̂ = (b1, b2, b3) relative to which the only non-zero commutation
relations are given by [b1, b2] = b1 = −[b2, b1]. Let ρ = (ρijk) ∈ F27 be
the structure vector of g relative to the basis b̂. Suppose now that g ∈
G = GL(3,F) is the transition matrix from b̂ to the basis b̂′ = (b′1, b
′
2, b
′
3)
of g. It is then easy to show that
[b′1, b
′
2] = (det g)−1M33(M11b
′
1 −M12b
′
2 +M13b
′
3),
[b′1, b
′
3] = (det g)−1M32(M11b
′
1 −M12b
′
2 +M13b
′
3),
[b′2, b
′
3] = (det g)−1M31(M11b
′
1 −M12b
′
2 +M13b
′
3),
where, as before, Mij denotes the determinant of the matrix obtained
from g by deleting its i-th row and j-th column (in particular, theMij are
elements of our field F). It follows that there exist χ1, ψ1, ω1, χ2, ψ2, ω2, δ
94 N.M. Ivanova, C.A. Pallikaros
∈ F such that
[b′1, b
′
2] = δχ2(χ1b
′
1 − ψ1b
′
2 + ω1b
′
3),
[b′1, b
′
3] = δψ2(χ1b
′
1 − ψ1b
′
2 + ω1b
′
3),
[b′2, b
′
3] = δω2(χ1b
′
1 − ψ1b
′
2 + ω1b
′
3).
This prompts us to define ρ′(χ1, ψ1, ω1, χ2, ψ2, ω2, δ) ∈ F27 by
ρ′(χ1, ψ1, ω1, χ2, ψ2, ω2, δ) = (0, 0, 0, χ1χ2δ,−ψ1χ2δ, ω1χ2δ,
χ1ψ2δ,−ψ1ψ2δ, ω1ψ2δ,−χ1χ2δ, ψ1χ2δ,−ω1χ2δ, 0, 0, 0,
χ1ω2δ,−ψ1ω2δ, ω1ω2δ,−χ1ψ2δ, ψ1ψ2δ,−ω1ψ2δ,−χ1ω2δ, ψ1ω2δ,
−ω1ω2δ, 0, 0, 0),
and the subset U of F27 by U = {ρ′(χ1, ψ1, ω1, χ2, ψ2, ω2, δ) : χ1, ψ1, ω1,
χ2, ψ2, ω2, δ ∈ F}.
It is then clear that O(ρ) ⊆ U . We want to show that U is an
algebraic set containing V = O(η) ∪ {0} (we keep the notation for V ,
η, η′ and also for S, S1, S2, S3 introduced in the previous subsection).
The inclusion V ⊆ U is immediate from the fact that η′(α, β, γ, δ) =
ρ′(α, β, γ, γ, β, α, δ).
Next, we define the subset T of I(U) by T = S1 ∪ S2 ∪ T3 where
T3 = {X121X132 −X122X131, X121X232 −X122X231,
X131X232 −X132X231, X121X133 −X123X131,
X121X233 −X123X231, X232X123 −X122X233,
X122X133 −X123X132, X132X233 −X133X232,
X233X131 −X133X231}
(recall the definition of S1 and S2 in Section 3.1).
Now let S′ = T ∪{X121−X233, X131 +X232, X122 +X133} ⊆ T ∪S3.
It is easy to check that V ⊆ V(S′). We also have V(S′) = V(T ∪ S3).
To see this last equality of sets, note first that V(T ∪ S3) ⊆ V(S′) since
S′ ⊆ T ∪ S3. On the other hand, any ν ∈ V(S′) is a common zero of
every polynomial in T ∪ S3. Hence, we also have V(S′) ⊆ V(T ∪ S3).
Since V ⊆ V(S′), we get V ⊆ V(T∪S3). But T∪S3 ⊇ S, so V(T∪S3) ⊆
V(S) = V . We conclude that V (= V(S)) = V(T ∪ S3) = V(S′).
We aim to show that U = V(T ). This would imply that U is an
algebraic set (and also provide an alternative way of seeing that V ⊆ U
in view of the observation above).
Describing certain Lie algebra orbits via polynomial equations 95
Clearly, U ⊆ V(T ). In order to establish the reverse inclusion, it
will be convenient to decompose U as a union of three subsets which
contain among them all elements of V(T ). With α, β, γ, µ, ν, φ, ρ, σ, τ ,
ζ, θ, ξ, κ ∈ F, define the elements ρ1(α, β, γ, µ, ν, φ), ρ2(σ, τ, ρ, ζ) and
ρ3(θ, ξ, κ) ∈ F27 by
ρ1(α, β, γ, µ, ν, φ) = (0, 0, 0, µα,−µβ, µγ, να,−νβ, νγ,−µα, µβ,
− µγ, 0, 0, 0, φα,−φβ, φγ,−να, νβ,−νγ,
− φα, φβ,−φγ, 0, 0, 0),
ρ2(σ, τ, ρ, ζ) = (0, 0, 0, 0, σ,−σζ, 0, τ,−τζ, 0,−σ, σζ, 0, 0, 0, 0, ρ,
− ρζ, 0,−τ, τζ, 0,−ρ, ρζ, 0, 0, 0),
ρ3(θ, ξ, κ) = (0, 0, 0, 0, 0, θ, 0, 0, ξ, 0, 0,−θ, 0, 0, 0, 0, 0, κ, 0, 0,−ξ,
0, 0,−κ, 0, 0, 0).
Also define the subsets U1, U2 and U3 of F27 by U1 = {ρ1(α, β, γ, µ, ν, φ) :
α, β, γ, µ, ν, φ ∈ F and α 6= 0}, U2 = {ρ2(σ, τ, ρ, ζ) : σ, τ, ρ, ζ ∈ F} and
U3 = {ρ3(θ, ξ, κ) : θ, ξ, κ ∈ F}.
It is then immediate from the relations ρ1(α, β, γ, µ, ν, φ) = ρ′(χ1 =
α, ψ1 = β, ω1 = γ, χ2 = µ, ψ2 = ν, ω2 = φ, δ = 1), ρ2(σ, τ, ρ, ζ) =
ρ′(χ1 = 0, ψ1 = −1, ω1 = −ζ, χ2 = σ, ψ2 = τ, ω2 = ρ, δ = 1) and
ρ3(θ, ξ, κ) = ρ′(χ1 = 0, ψ1 = 0, ω1 = 1, χ2 = θ, ψ2 = ξ, ω2 = κ, δ = 1)
that Ui ⊆ U for i = 1, 2, 3.
We now show that V(T ) ⊆ U1 ∪ U2 ∪ U3. Let γ = (γijk) ∈ F27
be a common zero of all polynomials in T . As T ⊇ S1 ∪ S2, similarly
to the Heisenberg algebra case, we will work with the auxiliary vector
γ̂ = (γ121, γ122, γ123, γ131, γ132, γ133, γ231, γ232, γ233) ∈ F9. Again, we will
need to consider different subcases. We begin by considering the case
γ121 6= 0. Since γ ∈ V(T ), we get γ̂ = (γ121, γ122, γ123, γ131, γ122γ131γ
−1
121,
γ123γ131γ
−1
121, γ231, γ122γ231γ
−1
121, γ123γ231γ
−1
121). For example, to see that
γ132 = γ122γ131γ
−1
121, note that γ is a zero of the polynomial X121X132 −
X122X131 which belongs to T . On setting µ = 1, ν = γ131γ
−1
121, φ =
γ231γ
−1
121, α = γ121 (6= 0), β = −γ122, γ = γ123, we see that γ =
ρ1(α, β, γ, µ, ν, φ) where α 6= 0, so γ ∈ U1. Next we consider the case
γ121 = 0. We split this case into the subcases γ122 6= 0 (where, by similar
argument as above, we can show that γ ∈ U2) and γ122 = 0. It remains to
consider the case when both γ121 and γ122 are equal to zero and the next
step is to split this case into subcases according to whether γ123 6= 0 (we
can show then that γ ∈ U3) or γ123 = 0. Continuing in a similar fashion,
96 N.M. Ivanova, C.A. Pallikaros
we finally deduce that V(T ) is indeed a subset of U1∪U2∪U3. Summing
up the above discussion, we see that U ⊆ V(T ) ⊆ U1∪U2∪U3 ⊆ U . This
forces U = U1 ∪U2 ∪U3 = V(T ). Recall now that V = O(η)∪{0} ⊆ U .
In order to show that U = O(ρ)∪O(η)∪{0}, we find, for each δ ∈ U \V ,
an invertible matrix g(δ) ∈ G such that δ = ρg(δ).
In the table below we summarize the results of this computation,
listing also the corresponding matrices g = g(δ). We first split into
subcases according to whether δ ∈ U \V is of the form ρ1 (with α 6= 0),
ρ2 or ρ3 and as it turns out, depending on the values of the elements of
F involved, we need to split into further subcases.
It is now useful to recall that V = V(S′) where S′ = T ∪ {X121 −
X233, X131 + X232, X122 + X133} ⊆ T ∪ S3. Let ρ′ = (ρ′ijk) ∈ U . It
follows that ρ′ ∈ V if, and only if all three conditions ρ′121 − ρ′233 = 0,
ρ′131 + ρ′232 = 0 and ρ′122 + ρ′133 = 0 are satisfied. In particular, in the
case ρ′ = ρ1(α, β, γ, µ, ν, φ), we have ρ′ ∈ V if, and only if, all of the
conditions µα − φγ = 0, να − φβ = 0 and −µβ + νγ = 0 are satisfied.
For ρ1 to be an element of U1 we have the restriction α 6= 0, so in
this case, the third of the last three conditions follows from the other
two (this is because the conditions µα − φγ = 0 and να − φβ = 0 are
equivalent to the conditions µ = φγα−1 and ν = φβα−1 if α 6= 0). For
simplicity, in the table below we will write A1 = µα−φγ, A2 = να−φβ.
Similar observations can be made in the cases ρ′ has form ρ2 or ρ3 (as
it can also be seen from the table). Moreover, in the table below, vector
ρ1 = ρ1(α, β, γ, µ, ν, φ) will always be considered under the restriction
α 6= 0, compare with the definition of set U1.
ρi conditions transition matrix g det g
ρ1 A1 6= 0, A2 6= 0
ν φ 0
µβ − νγ A1 A2
−γ 0 α
A1A2
ρ1
A1 = 0, A2 6= 0,
φγ 6= 0
β α αφ−1γ−1A2
−γα−1A2 0 A2
φβα−1 φ 0
−A2
2
ρ1
A1 = 0, A2 6= 0,
φ = 0
(⇒ µ = 0, ν 6= 0)
1 0 0
−γν 0 αν
−γ + β α α
−α2ν
ρ1
A1 = 0, A2 6= 0,
φ 6= 0, γ = 0
(⇒ µ = 0)
ν φ 0
0 0 A2
β α 0
−A2
2
Describing certain Lie algebra orbits via polynomial equations 97
ρ1
A1 6= 0, A2 = 0,
β 6= 0, γ 6= 0
0 α2µγ αγφβ
α−1βA1 A1 0
−α−1A1 0 γ−1A1
−βA3
1
ρ1
A1 6= 0, A2 = 0,
γ = 0 (⇒ µ 6= 0)
−µ 0 φ
βµ µα 0
β α 1
−µ2α
ρ1
A1 6= 0, A2 = 0,
β = 0 (⇒ ν = 0)
µ 0 −φ
0 A1 0
−γ 0 α
A2
1
ρ1 A1 = 0, A2 = 0 ρ1 ∈ O(η) —
ρ2 ρ 6= 0, τζ − σ 6= 0
0 −σ −τ
τζ − σ ρζ ρ
1 0 0
ρ(τζ − σ)
ρ2 ρ 6= 0, τζ − σ = 0
τ ρ 0
0 ρζ ρ
1 0 0
ρ2
ρ2 ρ = 0, τζ − σ 6= 0
0 σ τ
τζ − σ 0 0
0 ζ 1
(τζ − σ)2
ρ2 ρ = 0, τζ − σ = 0 ρ2 ∈ O(η) —
ρ3 κ 6= 0, ξ 6= 0
1 ξ−1(κ+ θ) 1
−ξ −κ 0
1 0 0
κ
ρ3 κ 6= 0, ξ = 0
−θκ−1 0 1
0 −κ 0
1 0 0
κ
ρ3 κ = 0, ξ 6= 0
0 θξ−1 1
−ξ 0 0
0 1 0
−ξ
ρ3 κ = 0, ξ = 0 ρ3 ∈ O(η) —
The computation above establishes that U = O(ρ)∪O(η)∪{0}. Now
recall that U (= V(T )) is Zariski-closed. In fact, by similar argument as
in the case of the set V , we can show that U is irreducible, considering
now the regular map Φ: F7 → U = Ū ⊆ F 27: (χ1, ψ1, ω1, χ2, ψ2, ω2, δ) 7→
(0, 0, 0, χ1χ2δ, −ψ1χ2δ, ω1χ2δ, χ1ψ2δ, −ψ1ψ2δ, ω1ψ2δ, −χ1χ2δ, ψ1χ2δ,
−ω1χ2δ, 0, 0, 0, χ1ω2δ,−ψ1ω2δ, ω1ω2δ,−χ1ψ2δ, ψ1ψ2δ,−ω1ψ2δ,−χ1ω2δ,
ψ1ω2δ, −ω1ω2δ, 0, 0, 0) = ρ′(χ1, ψ1, ω1, χ2, ψ2, ω2, δ).
Since U is Zariski-closed and O(ρ) ⊆ U , we get O(ρ) ⊆ U . Invoking
the fact that U = O(ρ) ∪ O(η) ∪ {0} we can deduce that h3 and a3
98 N.M. Ivanova, C.A. Pallikaros
are the only possible Lie algebras which g2⊕ a1 can properly degenerate
to. In order to establish that g2 ⊕ a1 in fact degenerates to both h3
and a3 it suffices to show that O(ρ) = U . Since U is irreducible and
O(η) = O(η)∪{0} we get that O(ρ) is not Zariski-closed. It follows that
O(ρ) is properly contained in O(ρ). If η 6∈ O(ρ), then O(η)∩O(ρ) = ∅
since O(ρ) is a union of orbits (see Remark 4(ii)). It would then follow
that O(ρ) = O(ρ) ∪ {0}, contradicting the fact that U is irreducible.
We conclude that η ∈ O(ρ). It follows that O(η) ⊆ O(ρ) and hence
O(η) ⊆ O(ρ). Since 0 ∈ O(η), we get that 0 ∈ O(ρ). Summing up, we
have shown O(ρ) ⊆ U = O(ρ) ∪ O(η) ∪ {0} ⊆ O(ρ). Hence, U = O(ρ)
as required.
We remark here that it is well-known that, over an infinite field, any
Lie algebra degenerates to the abelian Lie algebra of the same dimension.
Also note that already in [1] it is shown that g degenerates to h3 in the
case the ground field is R. In view of [8, Lemma 3.9] the technique
used in [1] can be extended to obtain a degeneration from g to h3 now
over an arbitrary infinite field. In the discussion above we provided an
alternative way of obtaining this particular degeneration using the notion
of an irreducible algebraic set.
We close this subsection with some general comments regarding our
sets above. First, we can observe that O(ρ) = U \ O(η) = O(ρ) \ O(η)
so O(ρ) is open in its closure (compare [2, Proposition 2.5.2] for the
case of an algebraically closed field). Now let W be the union of the
three principal open sets {α ∈ Fn3
: fi(α) 6= 0} for i = 1, 2, 3 where
f1 = X121 − X233, f2 = X131 + X232 and f3 = X122 + X133. Since
O(ρ) = V(T ) and O(η) = V(S′) where S′ = T ∪{f1, f2, f3}, we see that
O(ρ) = V(T ) ∩W . This in fact verifies that O(ρ) consists of precisely
those points in U (= O(ρ)) which do not correspond to unimodular Lie
algebras (compare, for example, with [8, Remark 4.12]).
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|
| id | oai:trim.imath.kiev.ua:article-370 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-08-04T01:06:15Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/a8/4255e033f851f6f7a3dee8e477fc44a8.pdf |
| spelling | oai:trim.imath.kiev.ua:article-3702020-08-13T08:52:53Z Describing certain Lie algebra orbits via polynomial equations Опис орбіт певних алгебр Лі за допомогою поліноміальних рівнянь Ivanova, N. Pallikaros, C. Іванова, Н. Паллікарос, К. Let $\mathfrak{h}_3$ be the Heisenberg algebra and let $\mathfrak g$ be the 3-dimensional Lie algebra having $[e_1,e_2]=e_1$ $(=-[e_2,e_1])$ as its only non-zero commutation relations. We describe the closure of the orbit of a vector of structure constants corresponding to $\mathfrak{h}_3$ and $\mathfrak g$ respectively as an algebraic set giving in each case a set of polynomials for which the orbit closure is the set of common zeros.Working over an arbitrary infinite field, this description enables us to give an alternative way, using the definition of an irreducible algebraic set, of obtaining all degenerations of $\mathfrak{h}_3$ and $\mathfrak g$ (the degeneration from $\mathfrak g$ to $\mathfrak{h}_3$ being one of them. Розглядаються алгебра Гейзенберга $\mathfrak{h}_3$ та тривимірна алгебра Лі $\mathfrak g$ з ненульовими комутаційними співвідношеннями $[e_1,e_2]=e_1$ $(=-[e_2,e_1])$. Описано алгебраїчні множини, що є замиканням орбіт векторів структурних сталих, що відповідають $\mathfrak{h}_3$ і $\mathfrak g$, а саме: у кожному з випадків побудовано набір поліномів, таких, що множина їх спільних нулів є замиканням орбіти вектора структурних сталих. Такий опис дозволяє надати альтернативний підхід до знаходження всіх можливих вироджень $\mathfrak{h}_3$ та $\mathfrak g$ у довільному нескінченному полі за допомогою означення незвідної алгебраїчної множини. Інститут математики НАН України 2019-09-03 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/370 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 16 No. 1 (2019): Symmetry and Integrability of Equations of Mathematical Physics; 84-99 Сборник Трудов Института математики НАН Украины; Том 16 № 1 (2019): Симетрія та інтегровність рівнянь математичної фізики; 84-99 Збірник Праць Інституту математики НАН України; Том 16 № 1 (2019): Симетрія та інтегровність рівнянь математичної фізики; 84-99 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/370/365 Авторське право (c) 2019 Н. Іванова, К. Паллікарос |
| spellingShingle | Ivanova, N. Pallikaros, C. Іванова, Н. Паллікарос, К. Describing certain Lie algebra orbits via polynomial equations |
| title | Describing certain Lie algebra orbits via polynomial equations |
| title_alt | Опис орбіт певних алгебр Лі за допомогою поліноміальних рівнянь |
| title_full | Describing certain Lie algebra orbits via polynomial equations |
| title_fullStr | Describing certain Lie algebra orbits via polynomial equations |
| title_full_unstemmed | Describing certain Lie algebra orbits via polynomial equations |
| title_short | Describing certain Lie algebra orbits via polynomial equations |
| title_sort | describing certain lie algebra orbits via polynomial equations |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/370 |
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