Niemeier lattices and homological algebra
Leo Jiang Supervised by Zsuzsanna Dancso University of Sydney
Vacation Research Scholarships are funded jointly by the Department of Education and Training and the Australian Mathematical Sciences Institute. 1 Introduction to lattices and lattice gluing
A (Euclidean) lattice is a finitely generated free abelian group L with a positive definite sym- metric bilinear pairing L × L → R. It is integral if the pairing is integer-valued. In this report, unless otherwise specified, all lattices will be integral. Integral lattices have applications in areas of mathematics such as group theory, number theory, Lie theory, and coding theory.
Let v1, . . . , vn be a basis for L as a free abelian group. The pairing of L is determined by its values on the basis elements. This information is contained in the Gram matrix GL, which
L has entries Gij = hvi, vji. For every lattice we define its determinant to be the determinant of its Gram matrix. A lattice L is unimodular if det L = 1. The norm of a vector x ∈ L is hx, xi. An integral lattice is even if the norm of each vector is even, and odd otherwise. Geometrically, every lattice can be considered (by extension of scalars) as being embedded in the R-vector space L ⊗ R. The pairing then extends to an inner product on L ⊗ R. Hence by choosing an orthonormal basis for L ⊗ R we can identify it with Rn, where n is the rank of L. The lattice L is then a discrete subgroup of Rn with the pairing inherited from the Euclidean inner product. We will often assume such an embedding when considering examples.
With this interpretation, we can write a basis v1, . . . , vn for L in terms of an orthonormal basis of the underlying Euclidean space. The generator matrix M is then the matrix with rows given by the basis vectors, and we have GL = MM T .
Example 1. The cubic lattice of rank n is Zn. A basis is given by the standard basis, so the Gram matrix is the identity matrix.
Example 2. The hexagonal lattice in Figure 1 has generator and Gram matrices √ 2 0 T 2 −1 M = √ ,G = MM = . − √1 √3 −1 2 2 2
1.1 Lattice gluing
One of the basic problems in studying algebraic structures is the construction and classification of examples.
1 Figure 1: Examples of lattices: Z2 and the hexagonal lattice.
A typical construction is the (orthogonal) direct sum: for lattices, the direct sum L1⊕...⊕Lk L of lattices L1,...,Lk is the direct sum of the corresponding groups, with Gram matrix G =
GL1 ⊕ ... ⊕ GLk . This gives lattices of higher rank; conversely, decomposing lattices as direct sums of indecomposable sublattices (potentially) gives lower rank lattices.
Lattices can also have sublattices of the same rank: for example, consider 2Z as a sublattice of Z. It is then natural to ask about the converse construction: given a lattice L, how can we construct integral lattices M of the same rank containing L as a sublattice? To answer this question, we consider the dual lattice L∗ of L. This is the set of vectors in the ambient vector space L ⊗ R which pair integrally with all the vectors in L, that is,
∗ L = {v ∈ L ⊗ R | hv, `i ∈ Z for all ` ∈ L}.
Dual lattices are in general not integral, since vectors in the dual are not required to pair integrally with each other. However, the integrality condition implies that any M must be contained in L∗. Explicitly, M must be generated by L together with certain vectors from L∗ which have integral inner products with each other and themselves. In fact, it suffices to consider particular representatives for cosets in the dual quotient L∗/L, usually the vectors of minimal norm. Combining this idea with direct sums gives a construction called lattice gluing; given lattices
L1,...,Lk we take the direct sum L1 ⊕...⊕Lk and adjoin vectors from the dual lattice. In this ∗ case, the adjoined vectors are called glue vectors, and they belong to (L1 ⊕ ... ⊕ Lk) /(L1 ⊕
... ⊕ Lk). The advantage of using direct sums is given by the following simple lemma:
2 √ 5 3
√ √ 5 2 4 3
√ 4 2 √ 3 3
√ 3 2 √ 2 3 √ 2 2
√ √ 3 2
√ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ −5 2 −4 2 −3 2 −2 2 − 2 0 2 2 2 3 2 4 2 5 2 − 3 0 3 2 3 3 3 4 3 5 3 √ − 2 √ − 3
√ −2 2
√ −3 2 √ √ √ √ Figure 2: Gluing 2Z ⊕ 2Z and 3Z ⊕ 3Z. √ −4 2 ∗ ∗ ∗ Lemma 1. (L1 ⊕ ... ⊕ Lk) /(L1 ⊕ ... ⊕ Lk) = L1/L1 ⊕ ... ⊕ Lk/Lk. √ −5 2 ∗ ∗ ∗ Proof. Firstly, we note that (L1 ⊕ ... ⊕ Lk) = L1 ⊕ ... ⊕ Lk. The lemma follows from applying ∗ ∗ ∗ ∗ the first isomorphism theorem to the projection of L1 ⊕ ... ⊕ Lk onto L1/L1 ⊕ ... ⊕ Lk/Lk.
For constructing lattices of large rank, the adjoined glue vectors can then be more easily
∗ found in terms of the smaller component lattices. In this context the dual quotients Li /Li are called glue groups, and their elements are also called glue vectors. √ √ √ √1 Example 3. Consider lattices of the form nZ ⊕ nZ (Figure 2). The dual of nZ is n Z. √ √ Hence any glue vectors must be of the form (a/ n, b/ n) with a, b ∈ Z. For integrality we √ √ √ √ require (a2 + b2)/n ∈ Z. So 2Z ⊕ 2Z can be glued with (1/ 2, 1/ 2), while no nontrivial √ √ glue vector exists for 3Z ⊕ 3Z. This shows that not all lattices can be glued.
1.2 Root lattices
Root lattices are important examples of integral lattices arising from Lie theory. Informa- tion about a root lattice is encoded in a graph called a Dynkin diagram. Each vertex of the Dynkin diagram corresponds to a basis vector of the root lattice, and the adjacency of vertices determines the angle between them.
3 An Dn
E6 E7 E8
Figure 3: Dynkin diagrams for the ADE root lattices.
We are interested in the lattices with simply-laced diagrams (Figure 3), where each basis vector has the same norm. In this case each basis vector has norm 2, nonadjacent vertices are orthogonal, and adjacent vertices correspond to an angle of 2π/3 between the basis vectors. Indecomposable root lattices therefore have connected Dynkin diagrams. These root lattices are denoted by An, Dn, E6, E7, and E8, and their properties are listed below (following [1]).
1.2.1 An
The rank n lattice An is given by
n+1 An = (x0, . . . , xn) ∈ Z | x0 + ... + xn = 0 .
A generator matrix is −1 1 0 0 ... 0 0 0 −1 1 0 ... 0 0 0 0 −1 1 ... 0 0 ...... ...... 0 0 0 0 ... −1 1
∗ and det An = n + 1. The glue group An/An is the cyclic group of order n + 1, with elements [0], [1],..., [n] given by ! i j −j i [i] = , . n + 1 n + 1 Here j = n + 1 − i.
4 1.2.2 Dn
For n ≥ 4, the lattice Dn is defined as
n Dn = {(x1, . . . , xn) ∈ Z | x1 + ... + xn ∈ 2Z} .
A generator matrix is
−1 −1 0 ... 0 0 1 −1 0 ... 0 0 0 1 −1 ... 0 0 ...... ...... 0 0 0 ... 1 −1
∗ and det Dn = 4. The glue group Dn/Dn is C2 × C2 if n is even and C4 if n is odd. In both cases, the glue vectors are ! 1n 1n−1 1 [0] = (0n), [1] = , [2] = (0n−1, 1), [3] = , − 2 2 2
1.2.3 E6
The E6 lattice is defined as ( ) 1 8 E = (x , . . . , x ) ∈ | x + x = x + ... + x = 0 6 1 8 2Z 1 8 2 7
A generator matrix is 0 −1 1 0 0 0 0 0 0 0 −1 1 0 0 0 0 0 0 0 −1 1 0 0 0 0 0 0 0 −1 1 0 0 0 0 0 0 0 −1 1 0 1 1 1 1 1 1 1 1 2 2 2 2 − 2 − 2 − 2 − 2
∗ and det E6 = 3. The glue group E6 /E6 must therefore be C3 with glue vectors ! 22 14 [0] = (08), [1] = 0, − , , 0 , [2] = −[1] 3 3
5 1.2.4 E7
The E7 lattice is defined as
( 8 8 ) 1 X E = (x , . . . , x ) ∈ | x = 0 7 1 8 2Z i i=1 A generator matrix is −1 1 0 0 0 0 0 0 0 −1 1 0 0 0 0 0 0 0 −1 1 0 0 0 0 0 0 0 −1 1 0 0 0 0 0 0 0 −1 1 0 0 0 0 0 0 0 −1 1 0 1 1 1 1 1 1 1 1 2 2 2 2 − 2 − 2 − 2 − 2
∗ and det E6 = 2. The glue group E7 /E7 is C2 with nonzero glue vector ! 16 32 [1] = , − . 4 4
1.2.5 E8
The E8 lattice is defined as
( 8 8 ) 1 X E = (x , . . . x ) ∈ | x ∈ 2 8 1 8 2Z i Z i=1 A generator matrix is 2 0 0 0 0 0 0 0 −1 1 0 0 0 0 0 0 0 −1 1 0 0 0 0 0 0 0 −1 1 0 0 0 0 0 0 0 −1 1 0 0 0 0 0 0 0 −1 1 0 0 0 0 0 0 0 −1 1 0 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 and det E8 = 1. Hence E8 is unimodular and the glue group is trivial.
6 1.3 Niemeier lattices
An important application of lattice gluing is the construction of even unimodular lattices, which only exist in dimensions divisible by 8. They are completely classified in ranks 8, 16, and 24.
In rank 8 the unique even unimodular lattice is the root lattice E8, and the two examples of + rank 16 are E8 ⊕E8 and D16. The rank 24 even unimodular lattices were classified by Niemeier:
Theorem 1 (Niemeier). A rank 24 even unimodular lattice L is uniquely determined by its root system {v ∈ L | hv, vi = 2}. Specifically, there are 24 such lattices, corresponding to the root systems
∅,
24A1, 12A2, 8A3, 6A4, 4A6, 3A8, 2A12,A24,
6D4, 4D6, 3D8, 2D12,D24, 4E6, 3E8,
4A5 + D4, 2A7 + 2D5, 2A9 + D6,A15 + D9,
E8 + D16, 2E7 + D10,E7 + A17,E6 + D7 + A11.
Remark 1. The empty root system corresponds to the well-known Leech lattice. This lattice is famous for its connections to various interesting parts of mathematics, such as the Monster group and monstrous moonshine.
The remaining 23 Niemeier lattices can be constructed by adding glue vectors to the root lat- tice generated by the corresponding root system. We give some simple examples: the complete list of glue vectors for each lattice is given in [1].
Example 4. Since E8 is unimodular and has trivial glue group, the direct sum E8 ⊕ E8 ⊕ E8 is a Niemeier lattice.
Example 5. The Niemeier lattices corresponding to the root systems A24 and D24 are formed by adjoining the glue vectors [5] = ((1/5)20(−4/5)5) and [1] = ((1/2)24) respectively.
Example 6. To construct a Niemeier lattice from A15 + D9, we adjoin to A15 ⊕ D9 the glue vector ! 2 14 −142 19 [21] = . 16 16 2
7 2 Categorification of lattices
Categorification refers to a process which replaces classical notions with categorical analogues. The idea is to use powerful tools available on the categorical level to obtain new results, which can then be translated to the original setting using an inverse decategorification process. Decategorification ‘forgets’ information about a category by identifying isomorphic objects. For example, isomorphisms in FinSet, the category of finite sets, are bijections. Hence decat- egorifying FinSet gives the natural numbers N by sending each finite set to its cardinality. When the category has extra structure, this can be preserved under the decategorification. An important example of this is given by the (split) Grothendieck group.
Definition 1. The split Grothendieck group of an additive category C is the abelian group generated by isomorphism classes of objects of C modulo relations [A ⊕ B] = [A] + [B] for all objects A and B in C.
For our purposes, categorification of an algebraic structure refers to the construction of a category C such that its Grothendieck group is the original structure. Such a category is not unique, but a judicious choice may provide more information about the original object. If C is a category of finitely generated projective modules over a sufficiently nice algebra, its Grothendieck group is free and finitely generated. Often it can be given a natural lattice structure with the Hom pairing
h[A], [B]i := dim hom (A, B).
This suggests that results in lattice theory might be obtained by applying homological methods to the categories of modules. A long term aim is therefore to categorify useful notions of lattice theory. In particular, we are interested in lifting the lattice gluing construction (and specific examples such as the Niemeier lattices) to the categorical level. It is natural to consider categorifications of the ADE root lattices, since they are most usefully glued. Such a categorification was given by Huerfano and Khovanov [2]. Given a simply laced Dynkin diagram Γ, they construct an associated zigzag algebra A(Γ). Then A(Γ)–pmod, the category of finite dimensional graded projective left modules over A(Γ), categorifies the corresponding root lattice. The following sections provide the details of this categorification.
8 2.1 Zigzag algebras
Let Γ be a connected graph without loops or multiple edges. The double quiver of Γ, denoted by DΓ, is the quiver with the same vertices as Γ and directed edges a → b and b → a for each pair of vertices a, b connected by an edge in Γ.
The path algebra P (DΓ) of DΓ is the C–algebra with underlying C–vector space basis consisting of oriented paths in DΓ. The product xy of paths x and y is given by concatenation of paths from left to right if the source of y is the target of x, and is zero otherwise. Since Γ does not have multiple edges, a path is determined by the vertices it travels through. Hence we denote the path passing through vertices a1, . . . , an (in that order) by (a1| ... |an). If Γ has more than two vertices, its zigzag algebra A(Γ) is P (DΓ) modulo the following relations:
• (i|j|k) = 0 if i 6= k;
• (i|j|i) = (i|k|i) for all j, k connected to i in Γ.
If Γ has one or two vertices, it must be A1 or A2 respectively. We define A(A1) to be the 2 algebra generated by 1 and X, with X = 0. The algebra A(A2) is defined as P (DΓ) modulo the two-sided ideal generated by paths of length greater than 2. A path algebra, and hence the corresponding zigzag algebra, is naturally graded by path length; paths of length k are of degree k. For the case Γ = A1, we set X to be of degree 2, so the following proposition holds for all Γ:
Proposition 1. The homogeneous components of A(Γ) of degrees 0, 1, and 2 have dimensions |V (Γ)|, 2|E(Γ)|, and |V (Γ)| respectively, where V (Γ) and E(Γ) are the sets of vertices and edges of Γ. All other components are zero.
Proof. For Γ = A1, this immediately follows from the defined grading. So assume that Γ has more than one vertex. Since there are no relations on paths of length 0 or 1, the dimensions of the homogeneous components A(Γ)0 and A(Γ)1 are equal to the number of vertices and edges of DΓ respectively. The double quiver DΓ has the same vertices as Γ, and two edges for each edge in Γ (one for each orientation); hence dim A(Γ)0 = |V (Γ)| and dim A(Γ)1 = 2|E(Γ)|.
9 By the relations, the only nonzero paths of length 2 are loops of the form (i|j|i), and all loops based at the same vertex are equal. Since Γ is connected, there exists a loop for each vertex. Hence dim A(Γ)2 = |V (Γ)|. Finally, we show that all homogeneous components of degree greater than two are zero. For
Γ = A2, this is true by definition. Now consider Γ with more than two vertices. There are two cases for a path p of length 3. If p passes through at least 3 distinct vertices, then it is zero by (i|j|k) = 0. If p only passes through 2 vertices, then it must be of the form (i|j|i|j). At least one of i or j, say i, is connected to another vertex k. Since (i|j|i) = (i|k|i), we have (i|j|i|j) = (i|k|i|j). But (k|i|j) = 0, so (i|j|i|j) = 0. Hence all paths of length three are zero. Since all paths of length greater than three contain subpaths of length three, all paths of length greater than three are also zero.
2.2 Representation theory of zigzag algebras
In this section we show how A(Γ)–pmod categorifies the root lattice with Dynkin diagram Γ. We fix some notation for the elements of zigzag algebras. Given a labelling of the vertices of Γ, we denote by ei the degree 0 element of A(Γ) corresponding to the path of length 0 at the vertex i. The degree 1 element corresponding to the length 1 path from i to j is denoted by dij, and yi denotes the degree 2 element corresponding to the loop based at i.
From the construction of the path algebra, it follows that the ei are orthogonal idempotents: they satisfy eiei = ei and eiej = 0 for i 6= j. Furthermore, their sum is the identity for A(Γ).
2.2.1 Simple A(Γ)–modules
The simple modules over A(Γ) are classified as follows:
Theorem 2. The simple A(Γ)–modules are in bijection with vertices of Γ, and they are all one dimensional as C-vector spaces. Furthermore, if Sj is the simple module corresponding to the vertex j, then ej acts as the identity and all other generators of A(Γ) act as zero.
To prove this theorem we consider the action of A(Γ)+, the positively graded subalgebra of A(Γ), separately from that of the degree 0 orthogonal idempotents. Let S be a simple A(Γ)–module.
10 Lemma 2. A(Γ)+ · S := spanZ{as | a ∈ A(Γ)+, s ∈ S} = 0.
Proof. The nonnegative grading of A(Γ) implies A(Γ) · A(Γ)+ = A(Γ)+, so A(Γ)+ · S is closed under the action of A(Γ) and therefore a submodule. Since S is simple, any submodule must
3 be 0 or S. But the homogeneous components of degree greater than 2 are zero, so (A(Γ)+) =
A(Γ)>2 = 0. It follows that A(Γ)+ · S = 0.
P Since 1 = i∈V (Γ) ei must act nontrivially on S, at least one orthogonal idempotent, say ej, must act nontrivially.
Lemma 3. The action of ei on S is the identity for i = j and zero otherwise.
Proof. The previous proposition shows that any positively graded component acts as zero on ejS ⊆ S. Since the ei are orthogonal idempotents, we have ei(ejS) = (eiej)S = 0 if i 6= j and ej(ejS) = (ejej)S = ejS. So ejS is closed under the A(Γ) action and is therefore a submodule of S.
Since ej was assumed to act nontrivially, we must have ejS = S. It follows that eiS = P ei(ejS) = (eiej)S = 0 for i 6= j. As 1 = i∈V (Γ) ei, the ej action on S must be the identity.
Proof of Theorem 2. The above lemmas show that S must be one dimensional as a C-vector space: if not, any strict subspace would be closed under the action of A(Γ) and hence be a nonzero strict submodule.
It follows that we can associate to every vertex j of Γ a simple module Sj with nontrivial ej action, and that this simple module is unique up to isomorphism.
2.2.2 Indecomposable projective A(Γ)–modules
Definition 2. A projective A(Γ)–module is a direct summand of a free A(Γ)–module.
Every projective module can be decomposed as a direct sum of indecomposable projective modules. To find these, we consider A(Γ) itself as a free A(Γ)–module.
∼ Proposition 2. A(Γ) = ⊕i∈V (Γ)A(Γ)ei
Proof. The sets A(Γ)ei can be interpreted as the submodules consisting of the paths ending at the vertex i. The isomorphism is then clear.
11 Let Pi := A(Γ)ei be the projective modules obtained in the previous proposition.
Proposition 3. The Pi are indecomposable projective modules.
Proof. Assume Pi can be written as Q1 ⊕ Q2. Then ei = x + (ei − x), where x ∈ Q1 and ei − x ∈ Q2. We can write these in terms of the C-vector space basis for Pi as
X aei ei + ayi yi + adji dji. j−i
At least one of x and ei − x, say x, has nonzero coefficient aei . Consider z ∈ A(Γ) defined by x = aei ei + z. Then z · z = 0 and z · x = aei z ∈ Q1. Hence x − z = aei ei ∈ Q1, and
A(Γ)ei = Pi ⊂ Q1. It follows that Q1 = Pi and Q2 = 0, so there is no nontrivial direct sum decomposition of Pi.
∼ Proposition 4. Pi 6= Pj if i 6= j.
Proof. For example, consider the action of yi on Pi and Pj. We have yiPi = span{yi} and yiPj = 0. Hence Pi and Pj cannot be isomorphic.
The fact that the Pi are all of the indecomposable projective modules follows from a classical theorem on Artinian algebras.
Definition 3. An algebra A is Artinian if it satisfies the descending chain condition: every descending chain of ideals eventually terminates.
Proposition 5. A(Γ) is Artinian.
Proof. A(Γ) is finite dimensional as a vector space over C. Since ideals are subspaces, an infinite descending chain of ideals would have strictly decreasing dimensions, which is impossible.
Theorem 3. If A is an Artinian algebra, then there is a bijection between the simple and indecomposable A–modules.
Proof. See for example [3, Proposition 1.9.3].
Since the number of simple A(Γ)–modules is the number of vertices of Γ, the theorem implies that the Pi are all of the indecomposable projective modules.
12 2.3 Decategorification
The Grothendieck group of A(Γ)–pmod is the free abelian group with a basis given by the isomorphism classes of the Pi. For the pairing, we consider the homomorphisms between indecomposable projective modules.
Proposition 6. 0 i 6= j, i not adjacent to j q dim hom(Pi,Pj) = q i 6= j, i adjacent to j 2 1 + q i = j
Proof. Since Pi is generated by ei, any homomorphism f : Pi → Pj is determined by f(ei). But f(ei) = f(eiei) = eif(ei) ∈ eiA(Γ)ej, and all elements of eiA(Γ)ej determine homomorphisms of the same degree by concatenation. If i = j, this submodule is generated by ei and yi, which are of degree 0 and 2 respectively. If i is adjacent to j, then it is generated by the degree 1 element dij. Otherwise, eiA(Γ)ej is 0.
Defining h[Pi], [Pj]i = q dim hom(Pi,Pj)|q=−1 recovers the pairing of the root lattice L cor- responding to Γ. The isomorphism between the Grothendieck group and L sends [Pi] to the basis vector of L associated with the vertex i.
2.4 Quantised lattices
As an intermediate step towards a categorified lattice gluing, we consider the structures obtained by taking the Grothendieck group of A(Γ)–pmod while retaining the grading information [4].
−1 These quantised lattices are free Z[q, q ]-modules generated by the [Pi]. L To see this, recall the degree shift operation ·{k}, which takes a graded module M = i Mi to the graded module with homogeneous components M{k}i := Ai−k. Under the ‘graded’ Grothendieck group, these shifts correspond to multiplication by q:[M{k}] = qk[M].
Quantised lattices have a Z[q, q−1]-valued pairing, defined on the basis elements as
h[Pi], [Pj]iq = q dim hom(Pi,Pj).
13 This pairing is q-semilinear: since degree n homomorphisms P → Q correspond to degree n − 1 maps P {1} → Q and degree n + 1 maps P → Q{1}, we have
−1 −1 hq[P ], [Q]iq = q dim hom(P {1},Q) = q q dim hom(P,Q) = q h[P ], [Q]iq
h[P ], q[Q]iq = q dim hom(P,Q{1}) = qq dim hom(P,Q) = qh[P ], [Q]iq.
We are currently investigating the properties of the quantised pairing, with a view to defining the appropriate analogues of classical lattice notions for the gluing construction. The necessary concepts include quantised versions of the direct sum and dual lattice, with corresponding glue groups and glue vectors.
References
[1] Conway, JH & Sloane, NJA 1999, Sphere packings, lattices and groups, 3rd edn., Springer- Verlag, New York.
[2] Huerfano, RS & Khovanov, M 2001, ‘A category for the adjoint representation’, Journal of Algebra, vol. 246, no. 2, pp. 514-542.
[3] Zimmermann, A 2014, Representation Theory, Springer, Cham.
[4] Dancso, Z & Licata, A, ‘A categorification of the cut and flow lattices of graphs’, draft.
14