Georgian Mathematical Journal Volume 9 (2002), Number 4, 683–690

ROOTS OF UNITY AS A LIE ALGEBRA

A. DAVYDOV AND R. STREET

Dedicated to Professor Hvedri Inassaridze

Abstract. This note gives a categorical development arising from a theorem of A. A. Klyachko relating the Lie operad to roots of unity. We examine the “substitude” structure on the groupoid C whose homsets are the cyclic groups. The roots of unity representations of the cyclic groups form a Lie algebra for a certain oplax monoidal structure on the category of linear representations of C.

2000 Mathematics Subject Classification: 18D10, 17B01, 18D50. Key words and phrases: Lie algebra, operad, substitude, species, cyclic group.

1. Introduction

Given a functor J : 𝒞 → 𝒜 into a monoidal category 𝒜, there is a substitude structure [5] obtained on 𝒞 by restriction along J. If J is fully faithful, the substitude 𝒞 is a multicategory in the sense of [9]. While a substitude structure is fairly weak, we can actually define Lie algebras in any additive symmetric substitude. The particular functor J : C → P we wish to consider is the union of the inclusions of the cyclic groups in the symmetric groups; this J is faithful and bijective on objects, but not full.

Now P is the free symmetric strict monoidal category on a single generating object; the tensor product is denoted by +. The category 𝒱 ^{P} of linear representations of the symmetric groups is the category of tensorial species in the sense of [7]. Linear symmetric operads are monoids in 𝒱 ^{P} equipped with the substitution tensor product. The Lie operad lie is a Lie algebra in 𝒱 ^{P} equipped with the convolution tensor product coming from P with +. We are interested in a relaxedly associative tensor operation on the category 𝒱 ^{C} of linear representations of the cyclic groups, which is derived by convolution and restriction from + on P. The object ω in 𝒱 ^{C} made up of all the roots of unity representations of the cyclic groups turns out to be a Lie algebra for this lax tensor structure. By a theorem of Klyachko [8], the representation of the symmetric groups induced along J by ω is lie. While the monoidal structure on 𝒱 ^{P} is traditional, the object lie is rather complicated. In contrast, we have only a lax monoidal structure on 𝒱 ^{C}, yet the object ω is quite easily understood.

2. The Symmetric and Cyclic Groupoids

Write P for the symmetric groupoid; it is the category whose objects are natural numbers and whose morphisms are permutations; so the homset P(m, n) is

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empty for m ≠ n and the endomorphism monoid P(n, n) is the permutation (or symmetric) group P_{n} on the set {1, 2, …, n}. Write C for the cyclic groupoid; its objects are natural numbers and the endomorphism monoid C(n, n) is the cyclic group C_{n} of order n. Up to isomorphism there is a unique faithful functor J : C → P which is the identity on objects; to be explicit we choose the J that takes a distinguished generator of C_{n} to the permutation i ↦ i + 1 (mod n).

3. Substitudes

Following [5] we use the term “substitude” for a slight weakening of the notion of multicategory. More precisely, a substitude is a category 𝒜 together with:

• for each integer n ≥ 0, a functor

P_{n} : 𝒜^{op} × ⋯ × 𝒜^{op} × 𝒜 ⟶ Set

whose value at (A_{1}, …, A_{n}, A) is denoted by P_{n}(A_{1}, …, A_{n}; A);

• for each partition^{1} ξ : m_{1} + ⋯ + m_{n} = m, a natural family of functions

μ_{ξ} : P_{n}(A_{•}; A) × P_{m_{1}}(A_{1•}; A_{1}) × ⋯ × P_{m_{n}}(A_{n•}; A_{n}) ⟶ P_{ξ}(A_{••}; A)

called substitution, where we use the shorthand

P_{k}(X_{•}; X) = P_{k}(X_{1}, …, X_{k}; X) and

P_{ξ}(X_{••}; X) = P_{m}(X_{11}, …, X_{1m_{1}}, …, X_{n1}, …, X_{nm_{n}}; X); and

• a natural family of functions

η : 𝒜 (A, B) ⟶ P_{1}(A; B)

called the unit; subject to three conditions that can be found in [5].

4. Restriction

As a particular case of Proposition 4.1 of [4], given any functor J : 𝒞 → 𝒜, we can restrict the substitude structure on 𝒜 to 𝒞 by defining

P_{n}(C_{1}, …, C_{n}; C) = P_{n}(JC_{1}, …, JC_{n}; JC),

by defining the substitution operation to be that of 𝒜 restricted to objects which are values of J, and by defining the unit to be the composite

𝒞 (C, D) ⟶^{J} 𝒜 (JC, JD) ⟶^{η} P_{1}(JC, JD).

In particular, addition of natural numbers gives a monoidal structure on P which restricts along J to yield a substitude structure on C.

1The summands in our partitions are allowed to be zero and in non-monotone order.

5. Lax and Oplax Monoidal Categories

Here we use the term linear to mean enrichment (in the sense of [6]) over the base monoidal category 𝒱 of vector spaces over an algebraically closed field k of characteristic zero. Recall from [4] that a lax monoidal structure on a linear category 𝒳 consists of linear functors

⊗_{n} : 𝒳 ⊗ ⋯ ⊗ 𝒳 ⟶ 𝒳

(called n-fold tensor product) together with natural substitutions

μ_{ξ} : ⊗_{n}(⊗_{m_{1}}(X_{11}, …, X_{1m_{1}}), …, ⊗_{m_{n}}(X_{n1}, …, X_{nm_{n}})) ⟶ ⊗_{m}(X_{11}, …, X_{nm_{n}})

and unit η : X → ⊗_{1}(X), satisfying three axioms. An oplax monoidal structure on the linear category 𝒳 is a lax monoidal structure on 𝒳 ^{op}; again we have linear functors

⊗_{n} : 𝒳 ⊗ ⋯ ⊗ 𝒳 ⟶ 𝒳

however, we have cosubstitutions

δ_{ξ} : ⊗_{m}(X_{11}, …, X_{nm_{n}}) ⟶ ⊗_{n}(⊗_{m_{1}}(X_{11}, …, X_{1m_{1}}), …, ⊗_{m_{n}}(X_{n1}, …, X_{nm_{n}}))

and counit ε : ⊗_{1}(X) → X.

6. Standard Convolution

For any small substitude 𝒜 and any cocomplete lax monoidal linear category 𝒳 , the linear category of functors from 𝒜 to 𝒳 , with natural transformations as morphisms, supports the standard convolution lax monoidal structure (see [5]) in which the n-fold tensor product is defined by

⊗_{n}(F_{1}, …, F_{n})(A) = ∫ ^{A_{1},...,A_{n}} P_{n}(A_{1}, …, A_{n}; A) × ⊗_{n}(F_{1}A_{1}, …, F_{n}A_{n})

where S × V denotes the coproduct of S copies of the object V and the integral notation denotes the “coend” (see [10]).

7. Symmetry

A symmetry on a substitude 𝒜 is a natural family of isomorphisms

γ_{σ} : P_{n}(A_{1}, …, A_{n}; A) ⟶ P_{n}(A_{σ(1)}, …, A_{σ(n)}; A)

for each permutation σ satisfying certain axioms documented in [5]. For example, the symmetry on the monoidal category P induces a symmetry on the substitude C. A symmetry on a lax or an oplax monoidal linear category 𝒳 is a natural family of isomorphisms

c_{σ} : ⊗_{n}(X_{1}, …, X_{n}) ⟶ ⊗_{n}(X_{σ(1)}, …, X_{σ(n)})

for each permutation σ satisfying certain axioms documented in [5]. If 𝒜 and 𝒳 are symmetric, there is an induced symmetry on the standard convolution lax linear monoidal category 𝒳 ^{𝒜} .

8. Induced Representations

Let H be a subgroup of the group G. Then the inclusion i : H → G can be regarded as a functor between one-object categories. Right Kan extension along i defines a linear functor

Ind_{H}^{G} : 𝒱 ^{H} ⟶ 𝒱 ^{G}

between the categories of linear representations; it is right adjoint to restriction along i which takes the representation V of G to the representation V_{H} of H with the same vector space and restricted action. For any S we write k[S] for the vector space with basis S; if S is a G-set then k[S] is a linear representation of G. For any representation M of H, we have the induced representation of G given by

Ind_{H}^{G}(M) = 𝒱 ^{H}(k[G]_{H}, M).

The functor Ind_{H}^{G} : 𝒱 ^{H} ⟶ 𝒱 ^{G} is faithful: to see this we need to observe that the counit Ind_{H}^{G}(M)_{H} = 𝒱 ^{H}(k[G]_{H}, M)_{H} → 𝒱 ^{H}(k[H], M)_{H} ≅ M is an epimorphism, but this follows from the fact that the inclusion k[H] → k[G]_{H} is a split monomorphism in 𝒱 ^{H}.

9. Duality for Induced Representations

In Part 8, if G is finite then Ind_{H}^{G} is also left Kan extension Lan_{i} along i : H → G. We have

Ind_{H}^{G}(M) = 𝒱 ^{H}(k[G]_{H}, M) ≅ k[G]^{∗} ⊗_{k[H]} M
≅ k[G^{op}] ⊗_{k[H]} M ≅ k[G] ⊗_{k[H]} M
= Lan_{i}(M),

since k[G]_{H} has a dual in 𝒱 ^{H} and G^{op} ≅ G as groups.

10. Kan Extensions along J : C → P

The functor 𝒱 ^{J} : 𝒱 ^{P} → 𝒱 ^{C} defined by restriction along J has a left adjoint Lan_{J} and a right adjoint Ran_{J}. There is a natural bijection between natural transformations Lan_{J}(M) → F and natural transformations M → F J. Also there is a natural bijection between natural transformations F → Ran_{J}(M) and natural transformations F J → M. The formulas for left and right Kan extension are respectively:

Lan_{J}(M)(p) = ∫ ^{r} P (J(r), p) × M(r) ≅ k[P_{p}]_{C_{p}} ⊗_{k[C_{p}]} M(p)  and
Ran_{J}(M)(p) ≅ ∫_{r} M(r)^{P (p,J(r))}  ≅ Hom_{C_{p}}(k[P_{p}]_{C_{p}}, M(p)),

where V^{S} denotes the vector space of functions from the set S to the vector space V. It follows for this J, using Parts 8 and 9, that Lan_{J} and Ran_{J} are isomorphic faithful functors.

11. Left Kan Extension along a Product

Suppose we have functors F : 𝒜 → 𝒱 , F′ : 𝒜′ → 𝒱 , J : 𝒜 → ℬ, and J′ : 𝒜′ → ℬ′. We write F ⊗ F′ : 𝒜 ⊗ 𝒜′ → 𝒱 for the functor defined by

(F ⊗ F′)(A, A′) = F A ⊗ F′A′.

An easy calculation shows that the left Kan extension of the functor F ⊗ F′ along J × J′ : 𝒜 × 𝒜′ → ℬ × ℬ′ is given by

Lan_{J×J′}(F ⊗ F′) ≅ Lan_{J}(F) ⊗ Lan_{J′}(F′).

12. Restriction Followed by Convolution

Given any functor J : 𝒞 → 𝒜 into a monoidal category 𝒜, as a particular case of Part 4 we obtain a restriction substitude structure on 𝒞 defined by

P_{n}(C_{1}, …, C_{n}; C) = 𝒜 (JC_{1} ⊗ ⋯ ⊗ JC_{n}, JC).

Now we can apply standard convolution as in Part 6 to obtain a \underline{lax monoidal} \underline{structure} on the linear category 𝒱 ^{𝒞} ; explicitly we have

⊗_{n}(M_{1}, …, M_{n})(C) = ∫ ^{C_{1},...,C_{n}} 𝒜 (JC_{1}, …, JC_{n}; JC) × M_{1}C_{1} ⊗ ⋯ ⊗ M_{n}C_{n}.

13. Convolution Followed by Restriction

Given our same functor J : 𝒞 → 𝒜 into the monoidal category 𝒜 as in Part 12, we can restrict the convolution monoidal structure on 𝒱 ^{𝒜} to 𝒱 ^{𝒞} by means of the left Kan extension functor K = Lan_{J} : 𝒱 ^{𝒞} → 𝒱 ^{𝒜} . This leads to a substitude structure on 𝒱 ^{𝒞} defined by

P_{n}(M_{1}, …, M_{n}; M) = 𝒱 ^{𝒜} (⊗_{n}(KM_{1}, …, KM_{n}), KM)

≅ ∫_{A} 𝒱 ( ∫ ^{A_{1},...,A_{n}}𝒜 (A_{1} ⊗ ⋯ ⊗ A_{n}, A)×(KM_{1})A_{1} ⊗ ⋯ ⊗ (KM_{n})A_{n}, (KM)A)

≅ ∫_{A} 𝒱 ( ∫ ^{A_{1},...,A_{n}} 𝒜 (A_{1} ⊗ ⋯ ⊗ A_{n}, A)×

× ∫ ^{C_{1},...,C_{n}} 𝒜 (JC_{1}, A_{1}) × ⋯ × 𝒜 (JC_{n}, A_{n}) ⊗ M_{1}C_{1} ⊗ ⋯ ⊗ M_{n}C_{n}, (KM)A)

≅ ∫_{A;C_{1},...,C_{n}} 𝒱 (𝒜 (JC_{1} ⊗ ⋯ ⊗ JC_{n}, A) ⊗ M_{1}C_{1} ⊗ ⋯ ⊗ M_{n}C_{n}, (KM)A)

≅ 𝒱 ^{𝒜} ( ∫ ^{C_{1},...,C_{n}} 𝒜 (JC_{1} ⊗ ⋯ ⊗ JC_{n}, −) ⊗ M_{1}C_{1} ⊗ ⋯ ⊗ M_{n}C_{n}, KM).

In the case such as in Part 10 where KM is also a right Kan extension of M along J, this last vector space is isomorphic (naturally in all variables) to

𝒱 ^{𝒞} (⊗_{n}(M_{1}, …, M_{n}), M),

where ⊗_{n}(M_{1}, …, M_{n}) is defined in Part 12. In other words, the substitude structure on 𝒱 ^{𝒞} is representable and so defines an \underline{oplax monoidal structure} on 𝒱 ^{𝒞} . So we have both an oplax and a lax monoidal structure on 𝒱 ^{𝒞} in which the multiple tensor product functors agree. In the situation of Part 10, in fact, the cosubstitution operation for the oplax structure is left inverse to the substitution operation for the lax structure and the counit is left inverse to the unit. To reiterate, the cosubstitution and counit of the oplax monoidal structure on 𝒱 ^{𝒞} are not invertible (as would occur if we had a true monoidal structure) but they are split epimorphisms for which the splittings are natural and satisfy the axioms for a lax monoidal structure. We are particularly interested in this symmetric oplax monoidal structure on 𝒱 ^{C} coming from J : C → P.

14. Lie Algebras

In the paper [5], Lie algebras were defined in any braided additive substitude. We now make this definition explicit in the case of a symmetric oplax monoidal linear category 𝒳 . A Lie algebra in 𝒳 is an object L together with a morphism

β : ⊗_{2}(L, L) ⟶ L,

called the bracket, satisfying the two conditions

β(1 + c_{τ_{2}}) = 0 and λ(1 + c_{τ_{3}} + c_{τ_{3}}^{2}) = 0,

where τ_{n} is the permutation i ↦ i + 1( mod n) and λ is the composite

\xymatrix{⊗_{3}(L, L; L) \ar[r]^{δ_{3=2+1}} & ⊗_{2}(⊗_{2}(L, L), ⊗_{1}(L)) \ar[r]^{⊗_{2}(β,ε)} & ⊗_{2}(L, L) \ar[r]^{β} & L.}

In particular, a Lie algebra in 𝒱 , with the usual tensor product of vector spaces, is a Lie algebra over k in the usual sense. The purpose of [1] was to study Lie algebras in a non-standard symmetric linear substitude of representations of a Hopf algebra.

15. The Lie Operad

Let E : P → 𝒱 denote the functor which takes each object n of P to the vector space k^{n} and takes each permutation to the linear function represented by the corresponding permutation matrix. This E is a symmetric strong monoidal (symmetry and tensor-preserving) functor for the + structure on P and the direct sum structure on 𝒱 . It follows (see [3]) that left Kan extension

T = Lan_{E} : 𝒱 ^{P} → 𝒱 ^{𝒱}

is symmetric strong monoidal with respect to the convolution structures. The convolution structure on 𝒱 ^{𝒱} is merely pointwise tensor product. The functor T takes each tensorial species F to its “Taylor series” T F defined by

(T F)V = ⊕_{n=0}^{∞} F_{n} ⊗ V^{⊗n}/P_{n}.

It is also true that T is strong monoidal with respect to the substitution structure on 𝒱 ^{P} and the composition structure on 𝒱 ^{𝒱} . Each species F ∈ 𝒱 ^{P} gives a representation F_{n} = F n of the symmetric group P_{n} since F is defined as a functor on permutations of every n. Let LV denote the free Lie algebra on the vector space V. This gives an object L of 𝒱 ^{𝒱} which is a monoid for the composition structure (that is, L is a monad on the category 𝒱 ) and a Lie algebra for the pointwise tensor product. By a very general argument, it is shown in [7] that there is a Lie algebra lie in the convolution symmetric monoidal linear category (called “une algèbre de Lie tordue”) providing the Taylor coefficients for L; that is, T lie ≅ L as Lie algebras. Less important for our purpose here is that lie is also a monoid for the substitution structure on 𝒱 ^{P} and so is a symmetric linear operad [11]. The representation lie_{n} of P_{n} has underlying vector space spanned by those elements of the free Lie algebra Lk^{n} in which each of the canonical basis vectors e_{1}, …, e_{n} of k^{n} occurs precisely once; the permutations act by applying them to these basis vectors; so lie is a subobject of LE. The bracket and substitution operations on lie are easily guessed.

16. The Lie Algebra ω

Let ω_{n} denote the linear representation of C_{n} whose supporting vector space is k and whose action by the generator of C_{n} is multiplication by a primitive n-th root of unity; any choice of generator and primitive root gives an isomorphic representation. A theorem of Klyachko [8] (also see [2]) is that the representation induced on P_{n} by ω_{n} is lie_{n}; that is,

Ind_{C_{n}}^{P_{n}}(ω_{n}) ≅ lie_{n}.

In other words, the roots of unity representations make up an object ω ∈ 𝒱 ^{C} satisfying

Kω ≅ lie

where K = Lan_{J} ≅ Ran_{J}. From Part 13 we have the isomorphisms

𝒱 ^{C}(⊗_{n}(ω, …, ω), ω) ≅ 𝒱 ^{P}(⊗_{n}(lie, …, lie), lie)

which are compatible with the substitution and unit operations. So the bracket on lie corresponds to a bracket on ω and we have our result.

Theorem. The roots of unity representation ω is a Lie algebra in the symmetric oplax monoidal linear category 𝒱 ^{C} obtained by restriction along K of the convolution structure on 𝒱 ^{P}. There is an isomorphism of Lie algebras Kω ≅ lie.

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(Received 17.04.2001; revised 05.07.2002)

Authors’ address: Macquarie University New South Wales 2109 Australia E-mail: street@ics.mq.edu.au