Automorphism groups of nilpotent Lie algebras associatedto certain graphs

Debraj Chakrabarti, Meera Mainkar, and Savannah Swiatlowski

Department of Mathematics, Central Michigan University, Mount Pleasant, Michigan, USA

ABSTRACTWe consider a family of 2-step nilpotent Lie algebras associated to uniformcomplete graphs on odd number of vertices. We prove that the symmetrygroup of such a graph is the holomorph of the additive cyclic group Zn:Moreover, we prove that the (Lie) automorphism group of the correspond-ing nilpotent Lie algebra contains the dihedral group of order 2n asa subgroup.

ARTICLE HISTORYReceived 2 February 2019Revised 14 June 2019Communicated by MatejBresar

KEYWORDSAutomorphism group;edge-colored graphs;nilpotent Lie algebras

2010 MATHEMATICSSUBJECTCLASSIFICATION17B30; 05C15; 05C25

1. Introduction

Many classes of 2-step nilpotent Lie algebras associated with various types of graphs have beenstudied recently from different points of view, see, e.g., [1, 2, 6–8, 10–15]. A 2-step nilpotent Liealgebra is a Lie algebra where each 3-fold Lie bracket ½X; ½Y;Z�� of elements X;Y;Z of the Liealgebra is 0. The 3-dimensional Heisenberg Lie algebra is well-studied example of a 2-step nilpo-tent Lie algebra. In [1], the authors studied the automorphism group of a 2-step nilpotent Liealgebra associated with a simple graph and then classified the graphs which correspond to the 2-step nilpotent Anosov Lie algebras. These Lie algebras give rise to interesting hyperbolic dynamicson nilmanifolds. For some related constructions, see [9].

In this paper, we consider a similar problem for an interesting class of edge-colored directedsimple graphs Hn where n is an odd integer. We begin by considering the underlying undirectededge-colored graph Gn of Hn: The example G5 occurred in the recent paper [13, Example 5.7], inconnection with uniform Lie algebras. The graphs Gn are remarkable for having a large amountof symmetry which can be used for constructing other objects associated with it with nontrivialsymmetry, e.g., Einstein solvmanifolds [5], infranilmanifolds, etc. Interestingly, we found that, Gn

arises naturally when we consider the cyclic group Zn of n elements as a space on which Zn actsby translations, i.e., as a torsor without a distinguished identity element. In Section 3, we will givethe algebraic definition of Gn but now we introduce it geometrically. For every odd integer n weconstruct the edge-colored simple graph Gn by beginning with the complete graph on n verticesv1; :::; vn and thinking of these vertices as the vertices of a regular n-gon in the plane. We colorthe edges with n colors c1; :::cn in such a way that for every vertex vk the n�1

2 edges which are

CONTACT Meera Mainkar maink1m@cmich.edu Department of Mathematics, Central Michigan University, Pearce Hall,Mount Pleasant, 48859 MI.Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/lagb.� 2019 Taylor & Francis Group, LLC

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https://doi.org/10.1080/00927872.2019.1640239

perpendicular to the axis of symmetry of the n-gon passing through vk are colored with the samecolor ck: Below we illustrate this for n ¼ 5:

In our first result, we compute explicitly the group of symmetries CPAðGnÞ of Gn preservingthe coloring structure which we call the color permuting automorphisms, see Definition 2.1.

Theorem 1.1. CPAðGnÞ ffi Zn 3 AutðZnÞ:Here AutðZnÞ is the group of group-automorphisms of the cyclic group Zn: The semidirect

product Zn 3 AutðZnÞ is known as the holomorph of Zn:This result is interesting because the graph Gn is constructed out of the cyclic group Zn and

therefore the result expresses an aspect of the combinatorics of this familiar object.As already mentioned, the real motivation for considering these graphs comes from the theory

of 2-step nilpotent Lie algebras. This idea goes back to [1] for simple graphs and has beenextended by many authors [1, 2, 6–8, 10–15]. In [13, 15], with each directed edge-colored graphG; a 2-step nilpotent Lie algebra N G was associated and its properties were studied. This con-struction is recalled in Section 4.1. These Lie algebras can be thought as a quotient of the 2-stepnilpotent Lie algebras associated with simple graphs as in [1]. To obtain a Lie algebra from anedge-colored graph, we will further need that the edges are directed. We assign a certain naturalorientation of the edges to Gn to obtain the directed edge-colored simple graphs Hn in Section4.7. We are interested in understanding the group AutðN HnÞ of Lie algebra automorphisms of thecorresponding Lie algebra N Hn : In the situation considered in [1], each graph automorphism givesrise to a Lie algebra automorphism of the corresponding 2-step nilpotent Lie algebra. However, if weallow the repetition of the edge-colors, then only certain type of graph automorphisms or symme-tries can be extended to the automorphisms of the associated Lie algebra. We call those automor-phisms as graph Lie automorphisms and we denote the group of all such automorphisms of a graphG by GLAðGÞ:We compute explicitly the group GLAðGnÞ and prove the following theorem.

Theorem 1.2. GLAðHnÞ ffi Dn, dihedral group of order 2n. Consequently AutðN HnÞ contains a sub-group isomorphic to the dihedral group of order 2n:

If Hn is thought as above to be a regular n-gon in the plane along with all the diagonals whichare colored and directed in a certain way, then GLAðHnÞ can be thought of the Euclidean groupof symmetries of this polygon, which is well-known to be the dihedral group of order 2n:

The automorphism group of a nilpotent Lie algebra plays an important role in studying certainEinstein homogeneous spaces and infranilmanifolds (see, e.g., [3, 4, 10]). It would be very inter-esting to find a complete description of the automorphism group of N Hn:

2 D. CHAKRABARTI ET AL.

2. Edge-colored graphs and their automorphisms

In this section we recall some definitions (see [13] for example). Let ðS;EÞ denote a finite simplegraph where S is the set of vertices and E is the set of edges. We denote an edge by a 2-setfa; bg: Let C denote a finite set of colors. An edge-coloring is a surjective function c : E ! C: Wecall a graph G ¼ ðS; E; c : E ! CÞ an edge-colored graph.

Recall that a bijection r : S ! S is a graph automorphism of ðS;EÞ if the following holds: Forall a; b 2 S; frðaÞ; rðbÞg 2 E if and only if fa; bg 2 E: In this case, we extend r on the set E bydefining rðfa; bgÞ ¼ frðaÞ; rðbÞg:Definition 2.1. Let G ¼ ðS; E; c : E ! CÞ be an edge-colored graph. A graph automorphism v ofðS;EÞ is called a color permuting automorphism of G if there exists a permutation / of the set ofcolors C such that / � c ¼ c � v on E: The set of all color permuting automorphisms form agroup which we denote by CPAðGÞ:Example 2.2. Let C4 denote a cycle graph on 4 vertices where the vertex set S ¼ fa; b; c; dg andE ¼ ffa; bg; fb; cg; fc; dg; fa; dgg: Let C ¼ f1; 2g: We define the edge-coloring c : E ! C by

c a; bf gð Þ ¼ c b; cf gð Þ ¼ 1

c c; df gð Þ ¼ c d; af gð Þ ¼ 2:

Let v be the permutation of S given by v ¼ ða cÞðb dÞ; which is a graph automorphism ofðS;EÞ: Then the permutation of colors / ¼ ð1 2Þ satisfies / � c ¼ c � v on E and hence v is acolor permuting automorphism of C4 with the above coloring c:

Note that s ¼ ða b c dÞ is not a color permuting automorphism of C4 because

c s að Þ; s bð Þ� �� � ¼ 1 6¼ c s bð Þ; s cð Þ� �� �

:

It can be seen that CPAðC4Þ ¼ fid; ða cÞ; ðb dÞ; ða cÞðb dÞg: w

A uniform graph is a special type of an edge-colored graph.

Definition 2.3. We say that an edge-colored graph ðS;E; c : E ! CÞ is a uniform graph if it satis-fies the following properties.

(1) No two edges incident on the same vertex have the same color, i.e., cðfa; bgÞ 6¼ cðfa; cgÞif b 6¼ c:

(2) Each color occurs the same number of times, i.e., jc�1ðfcigj ¼ jc�1ðfcjgj for all ci; cj 2 C:

Example 2.4. Consider the same uncolored graph ðS; EÞ as in Example 2.2 and the same set ofcolors C ¼ f1; 2g: We define a new edge-coloring c : E ! C by

c a; bf gð Þ ¼ c c; df gð Þ ¼ 1

c b; cf gð Þ ¼ c d; af gð Þ ¼ 2:

We can see that G ¼ ðS; E; c : E ! CÞ is a uniform graph and CPAðGÞ ffi D8; the dihedral groupwith 8 elements.

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The edge-colored graph in Example 2.2 is not uniform.

3. A uniform graph associated to Zn

In this section, we associate an edge-colored graph to a cyclic group of odd order and computeits symmetries. We note however, that this construction and the computation are very generaland can be done for any abelian group of odd order.

Throughout we assume that n is an odd integer. We let Zn denote the cyclic group of order nwritten additively and denote the elements of Zn as f0; :::; n� 1g: Let Gn be an edge-coloredgraph where the underlying uncolored graph is a complete graph with vertex set Zn and wherethe set of colors C is also Zn: We let the color of an edge fi; jg be ðiþ jÞ 2 Zn where of coursethe addition þ means addition modulo n in Zn: More precisely, the edge-coloring in Gn is givenby c : E ! C is defined by cðfi; jgÞ ¼ iþ j for all i; j 2 Zn:

Proposition 3.1. The edge-colored graph Gn is a uniform graph.

Proof. Let i; j; k 2 Zn be distinct. Then if cðfi; jgÞ ¼ cðfi; kgÞ; then iþ j ¼ iþ k: Hence j ¼ k:This shows that no two edges incident on the same vertex have the same color.

Consider the group homomorphism f : Zn ! Zn defined by f ðiÞ ¼ iþ i ¼ 2i: Since n is odd, fis injective. For, if i 2 Zn with 2i ¼ 0; then the order of i is either 1 or 2 and divides the oddnumber n: Hence i ¼ 0 and f is a group isomorphism.

For m 2 Zn; we denote the set of all edges with color m by Am: Equivalently,

Am ¼ i; jf g : i; j 2 Zn;m ¼ iþ j; i 6¼ j� �

:

We will prove that jAmj ¼ n�12 : Since f is a bijection, there is a unique l 2 Zn such that 2l ¼

m: Hence Am ¼ ffi;m� ig : i 2 Zn; i 6¼ lg: Note that for each i 2 Zn; we have fi;m� ig ¼fm� i; ig: Therefore, jAmj ¼ n�1

2 : In other words, the number of edges with color m is constantfor all m: This proves that the edge-colored graph Gn is uniform. w

3.2. Color permuting automorphism group of Gn. In this section, we study the structure of thecolor permuting automorphism group CPAðGnÞ of the edge-colored graph Gn and proveTheorem 1.1.

Definition 3.3. We call a bijection s : Zn ! Zn special if for all a; b; c; d 2 Zn with a 6¼ b; c 6¼ d;and aþ b ¼ cþ d; we have sðaÞ þ sðbÞ ¼ sðcÞ þ sðdÞ:

We first observe the following.

Proposition 3.4. The following statements are equivalent for a bijection s : Zn ! Zn:

(1) s is special.(2) For all a; b; c; d 2 Zn with c 6¼ d, and aþ b ¼ cþ d, we have sðaÞ þ sðbÞ ¼ sðcÞ þ sðdÞ:(3) For all a; b; c; d 2 Zn with a� c ¼ d � b, we have sðaÞ � sðcÞ ¼ sðdÞ � sðbÞ:(4) s 2 CPAðGnÞ:

4 D. CHAKRABARTI ET AL.

Proof. Assume (1). Let a; c; d 2 Zn and assume that aþ a ¼ cþ d: Let l ¼ sðcÞ þ sðdÞ Wewill prove that sðaÞ þ sðaÞ ¼ l: Let B ¼ Zn n fag: If x 2 B; then x 6¼ 2a� x as n is odd, andxþ ð2a� xÞ ¼ cþ d: By our assumption (1), we have sðxÞ þ sð2a� xÞ ¼ sðcÞ þ sðdÞ ¼ l: Fromthis, we can conclude that sðBÞ ¼ fl� sð2a� xÞ : x 6¼ ag ¼ Zn n fl � sðaÞg: Since s is a bijection,this implies that sðaÞ ¼ l � sðaÞ and hence sðaÞ þ sðaÞ ¼ l which proves (2).

Assume (2). Then the statement (3) is clear for elements a; b; c; d 2 Zn with c 6¼ d: The casea 6¼ b follows similarly. If a ¼ b and c ¼ d and a� c ¼ d � b; then a ¼ b ¼ c ¼ d as n is odd.Hence sðaÞ � sðcÞ ¼ sðcÞ � sðaÞ ¼ 0: This proves (3).

It is clear that (3) ) (1).Assume (1). We define / : Zn ! Zn as /ðmÞ ¼ rðiÞ þ rðjÞ; where m ¼ iþ j and i 6¼ j: We

note that / is well-defined function because s is special. Let l 2 Zn: We write l ¼ aþ b wherea 6¼ b: Then /ðs�1ðaÞ þ s�1ðbÞÞ ¼ aþ b ¼ l: Hence / is surjective and hence it is a bijection.Also the color of an edge fsðiÞ; sðjÞg is the same as /ðiþ jÞ; i.e., /(color of the edge fi; jgÞ: Thisproves that s 2 CPAðGnÞ: Hence (1) ) (4).

Suppose now s 2 CPAðGnÞ: Then there exists a permutation / of Zn such that the color ofthe edge fsðaÞ; sðbÞg is the same as /(color of the edge fa; bg). Equivalently, sðaÞ þ sðbÞ ¼/ðaþ bÞ: In particular, if aþ b ¼ cþ d; then sðaÞ þ sðbÞ ¼ sðcÞ þ sðdÞ: Hence s is special.Hence (4) ) (1). w

Next, we define a notion of an affine bijection on an abelian group.

Definition 3.5. Let ðA;þÞ denote an abelian group. A bijection f : A ! A is called affine if thereexists a group automorphism cf of A such that

f aþ xð Þ ¼ cf að Þ þ f xð Þfor all a; x 2 A: We denote the set of all affine bijections on A by AffðAÞ:

It is not difficult to check that AffðAÞ is a group under composition.

Proposition 3.6. Let f : Zn ! Zn be a bijection. Then the following statements are equivalent.

(1) f 2 CPAðGnÞ:(2) f is special.(3) f 2 AffðZnÞ:

Proof. (1) () (2) by Proposition 3.4.Assume that f is special. We define cf : Zn ! Zn by cf ðaÞ ¼ f ðaÞ � f ð0Þ for all a 2 Zn: First,

we prove that cf is a group homomorphism. Let i; j 2 Zn: As f is special, by Proposition 3.4, forall i; j 2 Zn; we have f ðiþ jÞ � f ðjÞ ¼ f ðiÞ � f ð0Þ:

Hence for all i; j 2 Zn;

cf iþ jð Þ ¼ f iþ jð Þ � f 0ð Þ¼ f iþ jð Þ � f jð Þ þ f jð Þ � f 0ð Þ¼ f ið Þ � f 0ð Þ þ f jð Þ � f 0ð Þ¼ cf ið Þ þ cf jð Þ:

Suppose that cf ðiÞ ¼ 0: This implies that f ðiÞ � f ð0Þ ¼ 0 ) f ðiÞ ¼ f ð0Þ: As f is one-to-one,i ¼ 0: Hence kercf ¼ f0g: This proves that cf is a bijection and hence a group automorphismof Zn:

Also for a; x 2 Zn; we have f ðaþ xÞ � f ðxÞ ¼ f ðaÞ � f ð0Þ by Proposition 3.4. Hencef ðaþ xÞ � f ðxÞ ¼ cf ðaÞ and f ðaþ xÞ ¼ cf ðaÞ þ f ðxÞ: This proves that f 2 AffðZnÞ and (2) ) (3).

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We will prove that (3) ) (2). For, we assume that f 2 AffðZnÞ and let cf 2 AutðZnÞ such thatf ðaþ xÞ ¼ cf ðaÞ þ f ðxÞ for all a; x 2 Zn: Let i; j; k; l 2 Zn with i� j ¼ k� l: We will prove thatf ðiÞ � f ðjÞ ¼ f ðkÞ � f ðlÞ:

f ið Þ � f jð Þ ¼ f i� jð Þ þ jð Þ � f jð Þ¼ cf i� jð Þ¼ cf k� lð Þ¼ f k� lð Þ þ lð Þ � f lð Þ¼ f kð Þ � f lð Þ:

By Proposition 3.4, f is special. w

The following fact is well-known but we give a proof for completeness.

Proposition 3.7. AffðZnÞ ffi Zn 3 AutðZnÞ:

Proof. We define / : AffðZnÞ ! AutðZnÞ by /ðf Þ ¼ cf where cf 2 AutðZnÞ such that f ðaþ xÞ ¼cf ðaÞ þ f ðxÞ for all a; x 2 Zn: We note that if f 2 AffðZnÞ; then for all x 2 Zn; cf ðxÞ ¼f ðxÞ � f ð0Þ: We first prove that / is a group homomorphism. Let f ; g 2 AffðZnÞ: We need toprove cf �g ¼ cf � cg : Let a 2 Zn: Then

cf cg að Þ� � ¼ cf g að Þ � g 0ð Þ� �¼ cf g að Þð Þ � cf g 0ð Þ� �

as cf 2 Aut Znð Þ¼ f g að Þð Þ � f 0ð Þ � f g 0ð Þ� �� f 0ð Þ� �¼ f � g að Þ � f � g 0ð Þ¼ cf �g að Þ:

This shows that /ðf � gÞ ¼ /ðf Þ � /ðgÞ and hence / is a group homomorphism.It is clear that ker/ ¼ ff 2 AffðZnÞ : f ðxÞ ¼ xþ f ð0Þ for all x 2 Zng: Equivalently, ker / ¼

fTa : a 2 Zng where Ta : Zn ! Zn is a translation by a given by TaðxÞ ¼ xþ a: For, if f 2ker /; then f ¼ Tf ð0Þ: Also given a 2 Zn; we have Ta 2 AffðZnÞ as Taðxþ yÞ ¼ xþ yþ a ¼xþ TaðyÞ and hence Ta 2 ker /: We note that ker / ffi Zn:

If c 2 AutðZnÞ; then cðaþ xÞ ¼ cðaÞ þ cðxÞ for all a; x 2 Zn ) c 2 AffðZnÞ and /ðcÞ ¼ c:Hence / is surjective and / � i ¼ id where i : AutðZnÞ ! AffðZnÞ is the inclusion and id :AutðZnÞ ! AutðZnÞ is the identity map. This means that the following exact sequencesplits.

0 ! Zn ! Aff Znð Þ !/ Aut Znð Þ ! 1

This proves that AffðZnÞ ffi Zn 3 AutðZnÞ: w

Proof of Theorem 1.1. By Proposition 3.6, CPAðGnÞ ¼ AffðZnÞ and by Proposition 3.7, AffðZnÞ isisomorphic to Zn 3 AutðZnÞ: w

Remark 1. We note that CPAðGnÞ acts transitively on the set of vertices Zn as all rotations arethe color permuting automorphisms of Gn: The stabilizer of 0 under this action is precisely theautomorphisms group of Zn; AutðZnÞ: For if f 2 CPAðGnÞ and f ð0Þ ¼ 0; then there exists cf 2AutðZnÞ such that f ðaþ xÞ ¼ cf ðaÞ þ f ðxÞ for all a; x 2 Zn: Then for a 2 Zn; we have f ðaÞ ¼f ðaþ 0Þ ¼ cf ðaÞ þ f ð0Þ ¼ cf ðaÞ: This shows that f ¼ cf 2 AutðZnÞ:

6 D. CHAKRABARTI ET AL.

4. 2-step nilpotent Lie algebras

4.1. Associating a Lie algebra with a graph. In this section, we recall the construction of a 2-step nilpotent Lie algebra associated with an edge-colored directed graph (see [15] and also [13]).Consider an edge-colored directed simple graph H ¼ ðS; E; c : E ! CÞ where S is the set of verti-ces, E is the set of directed edges, and c is a surjective edge-coloring function from the set of(directed) edges to the set of colors C: We will denote a directed edge from a to b by an orderedpair ða; bÞ: By abuse of notation, we will denote the color of the directed edge ða; bÞ 2 E; by sim-ply cða; bÞ rather than the more accurate cðða; bÞÞ:

We associate with H a 2-step nilpotent Lie algebra N H over R in the following way. Theunderlying vector space of N H is V�W; where V is the R-vector space consisting of formalR-linear combinations of elements of S (so that S is a basis of V), and W is the R-vector spaceconsisting of formal R-linear combinations of elements of C: The Lie bracket structure on N H isgiven by the following

(1) If ða; bÞ 2 E and cða; bÞ ¼ Z; then ½a; b� ¼ �½b; a� ¼ Z:(2) If ða; bÞ 62 E; then ½a; b� ¼ ½b; a� ¼ 0:(3) ½Y;Z� ¼ 0 for all Y 2 N H and Z 2 W:

We say that N H is the 2-step nilpotent Lie algebra associated with the graph H: Note that thederived Lie algebra ½N H;N H� is the span of C and the dimension of N H is jSj þ jCj:

The above construction is a generalization of the construction of 2-step nilpotent Lie algebrasassociated with simple graphs as in [1, 11] where the edge-coloring c is a bijection.

Example 4.2. Consider the following directed edge-colored graph H ¼ ðS;E; c : E ! CÞ; whereS ¼ fa; b; c; dg; E ¼ fða; bÞ; ðb; cÞ; ðc; dÞ; ðd; aÞg; C ¼ fZ1;Z2g and edge-coloring c : E ! C isgiven by

c a; bð Þ ¼ c c; dð Þ ¼ Z1;

c b; cð Þ ¼ c a; dð Þ ¼ Z2:

Then the associated 2-step nilpotent Lie algebra N H is of dimension 6. The only non-zero Liebrackets among the basis vectors of N H are given by

a; b½ � ¼ c; d½ � ¼ Z1 ¼ � b; a½ � ¼ � d; c½ �;b; c½ � ¼ a; d½ � ¼ Z2 ¼ � c; b½ � ¼ � d; a½ �:

4.3. Automorphism group. Recall that a linear isomorphism s : N ! N is called a Lie automor-phism of the Lie algebra N if for all X;Y 2 N ;

s X;Y½ � ¼ s Xð Þ; s Yð Þ� �:

The group of all automorphisms of the Lie algebra N is called the automorphism group of N andis denoted by AutðN Þ:

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Given an edge-colored directed graph H ¼ ðS; E; c : E ! CÞ; we will characterize those graphautomorphisms of the simple graph ðS;EÞ which can be extended to Lie automorphisms of theassociated 2-step nilpotent Lie algebra N H :

Let H ¼ ðS;E; c : E ! CÞ be an edge-colored simple directed graph. Let Eu be the collection ofassociated undirected edges;

Eu ¼ a; bf g : a; bð Þ 2 E� �

:

We color the undirected edges by the same colors, i.e., we use the coloring cu : Eu ! C given by

cu a; bf gð Þ ¼ c a; bð Þ if a; bð Þ 2 E:

The edge-colored undirected simple graph Hu ¼ ðS; Eu; cu : Eu ! CÞ will called the underlyingundirected graph of H:

Let E� denote the set fðb; aÞ : ða; bÞ 2 Eg and C� denote the set f�Z 2 W : Z 2 Cg: Weextend the edge-coloring function c on E [ E� as follows: If cða; bÞ ¼ Z; then we define cðb; aÞby �Z:

Definition 4.4. Let H ¼ ðS;E; c : E ! CÞ be an edge-colored simple directed graph. We say thata color permuting automorphism v of the underlying undirected graph Hu ¼ ðS;Eu; cu : Eu ! CÞis a graph Lie automorphism of H if it induces a permutation on C [ C�; i.e., if there exists apermutation / of C [ C� such that cðvðaÞ; vðbÞÞ ¼ /ðcða; bÞÞ for all ða; bÞ 2 E [ E�: We denotethe group of all such automorphisms by GLAðHÞ:

Example 4.5. Consider the following directed edge-colored graph H and the associated 2-step nil-potent Lie algebra N H as in Example 4.2.

The color permuting automorphism group of the underlying undirected graph Hu; CPAðHuÞ ffiD8 (see Example 2.4). If v ¼ ða bÞðc dÞ; then we can see that v 2 CPAðHuÞ: Note that C [ C� ¼fZ1;Z2;�Z1;�Z2g: We define a permutation / of C [ C� as follows:

/ Z1ð Þ ¼ �Z1; / Z2ð Þ ¼ Z2; / �Z1ð Þ ¼ Z1; / �Z2ð Þ ¼ �Z2:

Then cðvðaÞ; vðbÞÞ ¼ cðb; aÞ ¼ �Z1 ¼ /ðZ1Þ ¼ /ðcða; bÞÞ: Similarly, one can check thatcðvðxÞ; vðyÞÞ ¼ /ðcðx; yÞÞ for all ðx; yÞ 2 E [ E�: Hence v 2 GLAðHÞ:

Now if r ¼ ða b c dÞ; then r 2 CPAðHuÞ: Note that r 62 GLAðHÞ: This is becausecðrðaÞ; rðbÞÞ ¼ cðb; cÞ ¼ Z2 and cðrðcÞ; rðdÞÞ ¼ cðd; aÞ ¼ �Z2: However, cða; bÞ ¼ cðc; dÞ ¼ Z1

and hence there is no permutation w of C [ C� such that cðrðaÞ; rðbÞÞ ¼ wðcða; bÞÞ:We now show that the elements of GLAðHÞ give rise to automorphisms of the associated Lie

algebra N H:

Lemma 4.6. Let H ¼ ðS;E; c : E ! CÞ be an edge-colored simple directed graph. If v 2 GLAðHÞ,then v can be uniquely extended to a Lie automorphism of N H. Therefore, the group GLAðHÞ canbe realized as a subgroup of AutðN HÞ:

8 D. CHAKRABARTI ET AL.

Proof. Note that N H ¼ V�W where V is the R-vector space with S as a basis and W is theR-vector space with C as a basis. To extend v to a Lie automorphism of N H; we first extend vlinearly on V and then linearly on W by defining vðZÞ ¼ cðvðaÞ; vðbÞÞ if Z ¼ cða; bÞ: We willdenote the extended linear map from N H to N H by v as well. Now v is well defined on N H

because v 2 GLAðHÞ (see Definition 4.4). It can be seen that v is onto and hence it is a linearisomorphism.

If ða; bÞ 2 E and cða; bÞ ¼ Z; then vð½a; b�Þ ¼ vðZÞ ¼ cðvðaÞ; vðbÞÞ ¼ ½vðaÞ; vðbÞ� by definitionof the Lie bracket on N H: As v is linear, we have vð½X1;X2�Þ ¼ ½vðX1Þ; vðX2Þ� for all X1;X2 2 V:Recall ½Y;U� ¼ 0 for all Y 2 N H and U 2 W: Using the linearity of v again, we have

v X;Y½ �ð Þ ¼ v Xð Þ; v Yð Þ� �for all X;Y 2 N H and v 2 AutðN HÞ: The uniqueness of the extension is clear from the definitionof N H: w

Remark 2. We note here that the following converse of Lemma 4.6 holds: If v 2 CPAðHuÞ can beextended to a Lie automorphism of N H; then v 2 GLAðHÞ:

4.7. The directed graph Hn. Throughout we assume that n is an odd integer. We define thedirected edge-colored graph Hn whose underlying undirected graph is Gn as introduced inSection 3. We define the vertex set of Hn to be Zn ¼ f0; 1; :::; n� 1g and the directed edge set Eas follows:

E ¼ mþ i; m� ið Þ : 0 � m � n� 1; 1 � i � n� 12

� :

The set of colors C is denoted by fZi : i 2 Zng and the edge-coloring c : E ! C is defined bycði; jÞ ¼ Ziþj for all ði; jÞ 2 E:

Geometrically the orientation of edges of Hn can be visualized as follows. Note that the under-lying undirected graph ðHnÞu is nothing but the graph Gn; which can be pictured as a regularn-gon in the plane along with all possible diagonals. To obtain Hn; we orient the n edges of theregular polygon clockwise and then orient the diagonals in such a way that the edges and diago-nals which are parallel receive the same orientation.

For example, the directed edge-colored graph G5 is as below.

Figure 1. H5:

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4.8. Graph Lie automorphism group of Hn First, note that GLAðHnÞ acts transitively on the setof vertices Zn: This is because rotations (or translations) are the graph Lie automorphism. To seethis, suppose that v : Zn ! Zn is given by vðiÞ ¼ iþ k for all i 2 Zn: We define a permutation /on C by /ðZlÞ ¼ Zlþ2k for all l 2 Zn: Then if 0 � m � n� 1 and 1 � i � n�1

2 ; we have

c � vððmþ i; m� iÞ ¼ cðmþ iþ 2k; m� iþ 2kÞ¼ Z2mþ4k

¼ /ðZ2mþ2kÞ¼ / � cððmþ i; m� iÞ:

We can extend / on C [ C� by defining /ð�ZlÞ ¼ �Zlþ2k: Hence v 2 GLAðHnÞ:Let Stabð0Þ denote the stabilizer of 0 under the action of GLAðHnÞ on Zn; i.e., let Stabð0Þ ¼

fr 2 GLAðHnÞ : rð0Þ ¼ 0g: Then by Orbit-stabilizer theorem, we have

jGLA Hnð Þj ¼ njStab 0ð Þj: (1)

We will prove that Stabð0Þ ¼ f6idg: In other words, we will prove that if id 6¼ v 2 GLAðHnÞand vð0Þ ¼ 0; then v is a reflection, i.e., vðiÞ ¼ n� i for all i 2 Zn:

Let R ¼ f1; :::; n�12 g � Zn and L ¼ fnþ1

2 ; :::; n� 1g � Zn: Note that in Figure 1 for H5; R (resp.L) consists of the vertices to the right (resp. left) of the vertical line through 0:

Lemma 4.9. If s 2 AutðZnÞ and s 6¼ id, then sðRÞ 6¼ R:

Proof. Let sð1Þ ¼ k: If k 2 L; we are done as 1 2 R: Now we assume that k 2 R; i.e., we assumethat 2 � k � n�1

2 : We claim that there exists q with 1 � q � n�12 such that nþ1

2 � qk � n� 1: InZ; we divide n� 1 by k: Let q 2 N and r with 0 � r � k� 1 such that n� 1 ¼ qkþ r: Then q ¼n�1k � r

k � n�1k � n�1

2 as k � 2: Hence q � n�12 : Also n� 1 � n� 1� r > n�1

2 as r < n�12 : Hence

n� 1 � qk > n�12 : This proves our claim.

As s 2 AutðZnÞ and sð1Þ ¼ k; we have sðqÞ ¼ qk: Hence sðRÞ 6¼ R as q 2 R and qk 2 L: w

Lemma 4.10. Suppose that v 2 GLAðHnÞ with vð0Þ ¼ 0. If vðkÞ 2 R for some k 2 R, then vðRÞ ¼ R:

Proof. Assume that vðkÞ ¼ i 2 R where k 2 R: We note that v is a color permutingautomorphism of the undirected edge-colored complete graph. By Remark 1, v 2 AutðZnÞ:Hence vðn� kÞ ¼ n� i 2 L: Assume that vðjÞ 2 L for some j 2 R: Then vðn� jÞ 2 R: Hencec � vðk; n� kÞ ¼ cði; n� iÞ ¼ Z0 and c � vðj; n� jÞ ¼ cðvðjÞ; vðn� jÞÞ ¼ �Z0: We note thatcðk; n� kÞ ¼ cðj; n� jÞ as both k and j are in R: This is a contradiction to our assumption thatv 2 GLAðHnÞ: w

Proposition 4.11. If v 2 GLAðHnÞ and vð0Þ ¼ 0, then v2 ¼ id:

Proof. If v 2 GLAðHnÞ and vð0Þ ¼ 0; then v 2 AutðZnÞ by Remark 1. By Lemmas 4.9 and 4.10,we have vðLÞ ¼ R and vðRÞ ¼ L: We note that v2 2 GLAðHnÞ and hence v2ðLÞ ¼ vðRÞ ¼ L;v2ðRÞ ¼ vðLÞ ¼ R: By Lemma 4.9, v2 ¼ id: w

As noted before, GLAðHnÞ contains all n rotations. Also Proposition 4.11 implies that the onlynonidentity group automorphism which is a graph Lie automorphism must be the reflectionabout 0: This proves Theorem 1.2.

Acknowledgments

The authors gratefully acknowledge the helpful comments of the referee and of Dave Morris.

10 D. CHAKRABARTI ET AL.

Funding

Debraj Chakrabarti and Savannah Swiatlowski were partially supported by NSF grant DMS-1600371.

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