ON LIE ALGEBRAS IN THE CATEGORY OF YETTER-DRINFELD MODULES

BODO PAREIGIS

Abstract. The category of Yetter-Drinfeld modules YD_{K}^{K} over a Hopf algebra K (with bijektive antipode over a field k) is a braided monoidal category. If H is a Hopf algebra in this category then the primitive elements of H do not form an ordinary Lie algebra anymore. We introduce the notion of a (generalized) Lie algebra in YD_{K}^{K} such that the set of primitive elements P (H) is a Lie algebra in this sense. Also the Yetter-Drinfeld module of derivations of an algebra A in YD_{K}^{K} is a Lie algebra. Furthermore for each Lie algebra in YD_{K}^{K} there is a universal enveloping algebra which turns out to be a Hopf algebra in YD_{K}^{K}.

Keywords: braided category, Yetter-Drinfeld module, Lie algebra, universal enveloping algebra.

1991 Mathematics Subject Classification 16W30, 17B70, 16W55, 16S30, 16S40

1. Introduction

The concept of Hopf algebras in braided categories has turned out to be very important in the context of understanding the structure of quantum groups and noncommutative noncocommutative Hopf algebras. In particular the work of Radford [7], Majid [3], Lusztig [2], and Sommerhäuser [8] show the importance of the decomposition of quantum groups into a product of ordinary Hopf algebras and of Hopf algebras in braided categories.

Since by the work of Yetter [9] Hopf algebras in braided categories that are defined on an underlying (finite-dimensional) vector space can be considered as Hopf algebras in some category of Yetter-Drinfeld modules, we will restrict our attention to Hopf algebras H in a category of Yetter-Drinfeld modules YD_{K}^{K} over a Hopf algebra K with bijective antipode.

There are two structurally interesting and important concepts that survive in this generalized situation, the concept of group-like elements (Δ(g) = g⊗g, ε(g) = 1) and the concept of primitive elements (Δ(x) = x ⊗ 1 + 1 ⊗ x, ε(x) = 0).

Date: March 11, 1996.

1991 Mathematics Subject Classification. Primary 16W30, 17B70, 16W55, 16S30, 16S40.

For ordinary Hopf algebras H the set of primitive elements P (H) of H forms a Lie algebra. This result (in a somewhat generalized form) still holds for Hopf algebras in a symmetric monoidal category. This is, however, not true for braided monoidal categories.

There have been various attempts to generalize the notion of Lie algebras to braided monoidal categories. The main obstruction for such a generalization is the assumption that the category is only braided and not symmetric. One of the most important examples of such braided categories is given by the category of Yetter-Drinfeld modules YD_{K}^{K} over a Hopf algebra K with bijective antipode which is always properly braided (except for K = k, the base field) [6].

We introduce a concept of Lie algebras in YD_{K}^{K} that generalizes the concepts of ordinary Lie algebras, Lie super algebras, Lie color algebras, and (G, χ)-Lie algebras as given in [5].

The Lie algebras defined on Yetter-Drinfeld modules have partially defined n-ary bracket operations for every n ∈ ℕ and every primitive n-th root of unity. They satisfy generalizations of the (anti-)symmetry and Jacabi identities.

Our main aim is to show that these Lie algebras have universal enveloping algebras which turn out to be Hopf algebras in YD_{K}^{K}. Conversely the set of primitive elements of a Hopf algebra in YD_{K}^{K} is such a generalized Lie algebra. We also give an example that generalizes the concept of orthogonal or symplectic Lie algebras.

2. Braid Symmetrization

We begin with two simple module theoretic observations. The following is well known: if A, B are algebras and M is an A-B-bimodule, then Hom_{A}(.P, .M) is a right B-module for every A-module P. We need a comodule analogue of this.

Let A be an algebra, C be a coalgebra, and _{A}M^{C} be an A-C-dimodule, i.e. a left A-module and a right C-comodule such that δ(am) = (a ⊗ 1)δ(m). Proposition 2.1. Let P be a finitely generated left A-module. Then Hom_{A}(.P, .M) is a right C-comodule with the canonical comodule structure such that

(Hom_{A}(P, M) ⟶^{δ} Hom_{A}(P, M)⊗C ⟶ Hom_{A}(P, M⊗C)) = Hom_{A}(P, δ).

Proof. Let p_{1}, …, p_{n} be a generating set of P and let f ∈ Hom_{A}(.P, .M). Let m_{i} := f(p_{i}). Then by the structure theorem on comodules the m_{i} are contained in a finite dimensional subcomodule M_{0} ⊆ M which is even a comodule over a finite dimensional subcoalgebra C_{0} ⊆ C, i.e. the diagram

[figure: M_{0}; M_{0} ⊗ C_{0}; δ; M; M ⊗ C; δ]

commutes. Furthermore M_{1} := AM_{0} is a C_{0}-comodule contained in M, since M is a dimodule, and f : P ⟶ M obviously factors through M_{1}. Since M and M_{1} are dimodules the diagram

[figure: Hom_{A}(.P, .M_{1}); Hom_{A}(.P, .M_{1} ⊗ C_{0}); δ_{∗}; Hom_{A}(.P, .M); Hom_{A}(.P, .M ⊗ C); δ_{∗}; Hom_{A}(.P, .M_{1}) ⊗ C_{0}; ≅; Hom_{A}(.P, .M) ⊗ C]

commutes, so each f has a uniquely defined image δ_{∗}(f) ∈ Hom_{A}(.P, .M)⊗ C. Now it is easy to check that this map induces a comodule structure on Hom_{A}(P, M). □

The second observation is the following. We consider k-algebras A and B. Let α : B ⟶ A be an algebra homomorphism. α induces an underlying functor V_{α} : A-Mod ⟶ B-Mod with right adjoint Hom_{B}(A, -) : B-Mod ⟶ A-Mod. If α : B ⟶ A is surjective then Hom_{B}(A, M) ⟶ Hom_{B}(B, M) ≅ M is injective, so that we can identify Hom_{B}(A, M) = {m ∈ M|Ker(α)m = 0}.

Let B_{n} be the Artin braid group with generators τ_{i}, i = 1, …, n − 1 and relations

\begin{aligned}τ_{i}τ_{j} & = τ_{j}τ_{i} & if |i − j| ≥ 2; \\ τ_{i}τ_{i+1}τ_{i} & = τ_{i+1}τ_{i}τ_{i+1}. \end{aligned}  (1)

Let ζ ∈ k be invertible. Then kB_{n} ∋ τ_{i} ↦ ζτ_{i} ∈ kB_{n} (for the generators τ_{i} of B_{n}) is an algebra automorphism denoted again by ζ : kB_{n} ⟶ kB_{n}. This holds true since the relations for B_{n} are homogeneous.

(Observe that this construction can be performed for every group algebra if the group is given by generators and homogeneous relations. The given construction of an automorphism for every ζ ∈ U(k) defines a group homomorphism U(k) ⟶ Aut(kB_{n}) ⟶ Aut(kB_{n}-Mod).)

Now consider the canonical quotient homomorphism B_{n} ⟶ S_{n} from the braid group onto the symmetric group. It induces a surjective homomorphism γ : kB_{n} ⟶ kS_{n} with kernel

Ker(γ) = ⟨ψ(τ_{i}^{2} − 1)φ|φ, ψ ∈ B_{n}, i = 1, …, n − 1⟩.

The composition α : kB_{n} ⟶^{ζ} kB_{n} ⟶^{γ} kS_{n} defines a functor kB_{n}-Mod ⟶ kS_{n}-Mod by

\begin{aligned}M(ζ) : & = Hom_{kB_{n}}(_{ζ}kS_{n}, M) \\ & = \{m ∈ M|φ^{−1}τ_{i}^{2}φ(m) = ζ^{2}m ∀φ ∈ B_{n}, i = 1, …, n − 1\}. \\ & = \{m ∈ M|τ_{i}^{2}φ(m) = ζ^{2}φ(m) ∀φ ∈ B_{n}, i = 1, …, n − 1\}. \end{aligned} (2)

This holds since the map γ : kB_{n} ⟶ kS_{n} has as kernel the twosided ideal generated as a k-subspace by {ψ(τ_{i}^{2} − 1)φ |ψ, φ ∈ B_{n}, i = 1, …, n − 1}. So f ∈ Hom_{kB_{n}}(_{ζ}kS_{n}, M) with f(1) = m ∈ M, iff ζ^{−1}(ψ(τ_{i}^{2} − 1)φ)m = 0 for all ψ, φ, i, iff ζ^{−1}(τ_{i}^{2} − 1)φm = 0 for all φ, i, iff τ_{i}^{2}φm = ζ^{2}φm for all φ, i, iff φ^{−1}τ_{i}^{2}φ(m) = ζ^{2}m for all φ, i.

If the action of B_{n} on M is given by an action of S_{n} and the canonical epimorphism B_{n} ⟶ S_{n}, then the construction of the M(ζ) becomes trivial, since M(ζ) = {m ∈ M|τ_{i}^{2}φ(m) = ζ^{2}φ(m) = φ(m)} = 0 if ζ^{2} ≠ 1 and M(−1) = M(1) = M. Observe that the module M(ζ) depends only on ζ^{2}, but that the action of kS_{n} on M(ζ) depends on ζ.

M(1) gives a solution of the following universal problem. Proposition 2.2. For every kB_{n}-module M the subspace

M(1) := {m ∈ M|φ^{−1}τ_{i}^{2}φ(m) = m ∀φ ∈ B_{n}, i = 1, …, n − 1}

is a kS_{n}-module and the inclusion M(1) ⟶ M is a kB_{n}-module homomorphism, such that for every kS_{n}-module T and every kB_{n}-module homomorphism f : T ⟶ M there is a unique kS_{n}-module homomorphism g : T ⟶ M(1) such that the diagram

[figure: M(1); M; ι; T; g; f]

commutes. Definition 2.3. We call the functor -(ζ) : kB_{n}-Mod ⟶ kS_{n}-Mod the ζ-symmetrization of kB_{n}-modules.

The definition gives

M(ζ) = {m ∈ M|φ^{−1}τ_{i}^{2}φ(m) = ζ^{2}m ∀φ ∈ B_{n}, i = 1, …, n − 1}.

The action of S_{n} on M(ζ) is given by

σ_{i}(m) = ζ^{−1}τ_{i}(m),  (3)

where σ_{i} resp. τ_{i} are the canonical generators of S_{n} resp. B_{n}. Thus M(ζ) is also a kB_{n}-submodule of M. Since the functor M ↦ M(ζ) is a rightadjoint functor, it preserves limits. Like for eigenspaces we have that the sum of the subspaces M(ζ) for all ζ with different ζ^{2} is a direct sum. On M(ζ) we have two distinct kS_{n}-structures σ_{i}(m) = −ζ^{−1}τ_{i}(m) and σ_{i} = ζ^{−1}τ_{i}(m), since ζ and −ζ define the same subspace M(ζ) = M(−ζ) ⊆ M.

The ζ-symmetrization M(ζ) of M can also be calculated by Lemma 2.4.

M(ζ) = {m ∈ M|τ_{i}^{−1}τ_{i+1}^{−1} … τ_{j−1}^{−1}τ_{j}^{2}τ_{j−1} … τ_{i+1}τ_{i}(m) = ζ^{2}m ∀1 ≤ i ≤ j ≤ n−1}

which reduces the number of conditions to be imposed on the m ∈ M in order to be in M(ζ).

Proof. Given in Appendix. □

One of the interesting kB_{n}-structures, for which we will apply the previous construction, occurs on n-fold tensor products M^{n} := M ⊗ …⊗M of an object M in a braided monoidal category of vector spaces.

Let K be a Hopf algebra. Let M be an K-module such that M is a kB_{n}-K-bimodule. The functoriality of our construction then makes M(ζ) again an K-module and in fact a kS_{n}-K-bimodule.

Let M be an K-comodule such that M is a kB_{n}-K-dimodule. Then by Proposition 2.1 M(ζ) is an K-comodule and in fact a kS_{n}-K-dimodule.

Let K be a Hopf algebra with bijective antipode. Let YD_{K}^{K} denote the category of Yetter-Drinfeld modules over K, i.e. of right K-modules and right K-comodules M such that ∑(x · h)_{0} ⊗ (x · h)_{1} = ∑(x_{0} · h_{2}) ⊗ S(h_{1})x_{1}h_{3} for all x ∈ M. The usual tensor product makes YD_{K}^{K} a monoidal category. YD_{K}^{K} has a braiding given by τ_{X,Y} : X ⊗ Y ⟶ Y ⊗ X, τ(x ⊗ y) = ∑ y_{0} ⊗ xy_{1}. We assume that the reader is familiar with the properties of the B_{n}-action that is induced by the braiding τ on n-fold tensor products ([4] 10.6). Theorem 2.5. Let K be a Hopf algebra with bijective antipode. Then for each ζ ∈ k^{∗} and each n ≥ 2 the construction given above defines a (non-additive) functor

YD_{K}^{K} ∋ M ↦ (M ⊗ … ⊗ M)(ζ) ∈ YD_{K}^{K}.

Proof. If M ∈ YD_{K}^{K} then the n-fold tensor product M ⊗ … ⊗ M is a Yetter-Drinfeld module on which B_{n} and thus kB_{n} acts in such a way, that M is a (kB_{n}, K)-bimodule and a (kB_{n}, K)-dimodule. The ζsymmetrization functor -(ζ) preserves the module and comodule structures hence the Yetter-Drinfeld structure.

The functor is not additive since the “diagonal” functor M ↦ M ⊗M is not additive. □

We abbreviate M^{n}(ζ) := M^{⊗n}(ζ) = M ⊗ … ⊗ M(ζ). Then M^{n}(ζ) is a submodule of M^{n} in the category of Yetter-Drinfeld modules and the elements in M^{n}(ζ) are of the form z = ∑_{k} x_{k,1} ⊗ … ⊗ x_{k,n}. We often suppress the summation index and summation sign and simply write z = z_{1} ⊗ … ⊗ z_{n} ∈ M^{n}(ζ) although M^{n}(ζ) does not decompose into a tensor product.

3. Symmetric Multiplication and Jacobi Identities

For the rest of the paper let C be the category YD_{K}^{K} of Yetter-Drinfeld modules over a Hopf algebra K with bijective antipode over a field k. We study objects P ∈ C together with (partially defined) operations in C

[., .] : P ⊗ … ⊗ P (ζ) = P^{n}(ζ) ⟶ P

for all n ∈ ℕ and all primitive n-th roots of unity ζ.

Occasionally we write [., .]_{n} for such an operation [., .]. By composing such operations certain additional operations may be constructed as follows. Proposition 3.1. Let ζ be a primitive n-th roots of unity. Then the operations

[., [., .]_{n}]_{2} : P^{n+1}(ζ) ∋ x_{1} ⊗ … ⊗ x_{n+1} ↦ [x_{1}, [x_{2}, …, x_{n+1}]] ∈ P

and

[[., .]_{n}, .]_{2} : P^{n+1}(ζ) ∋ x_{1} ⊗ … ⊗ x_{n+1} ↦ [[x_{1}, …, x_{n}], x_{n+1}] ∈ P

are well defined.

Proof. Given in Appendix. □

We will have to consider objects

P^{n+1}(−1, ζ) := P ⊗P^{n}(ζ)∩{z ∈ P^{n+1}|∀φ ∈ S_{n} : (1⊗φ)^{−1}τ_{1}^{2}(1⊗φ)(z) = z}.

Since this is a kernel (limit) construction in C, P^{n+1}(−1, ζ) is again an object in C. Proposition 3.2. Let ζ be a primitive n-th roots of unity. Then the operations

[., [., .]_{n}]_{2} : P^{n+1}(−1, ζ) ∋ x ⊗ y_{1} ⊗ … ⊗ y_{n} ↦ [x, [y_{1}, …, y_{n}]] ∈ P

and

[., [., .]_{2}, .]_{n}τ_{i−1} … τ_{1} : P^{n+1}(−1, ζ) ∋ x⊗y_{1}⊗…⊗y_{n} ↦ [y_{1}, …, [x, y_{i}], …, y_{n}] ∈ P

are well defined.

Proof. Given in Appendix. □

We introduce special bracket multiplications which then lead to the definition of a Lie algebra on a Yetter-Drinfeld module. Definition 3.3. Let A be an algebra in C = YD_{K}^{K} and let ∇^{n} : A ⊗ … ⊗ A ⟶ A denote the n-fold multiplication. We define a bracket or symmetric multiplication

[., .] : A^{n}(ζ) ⟶ A  by  [z] := ∑_{σ∈S_{n}} ∇^{n}σ(z)

where the action of S_{n} on A^{n}(ζ) is given as in (3).

We will only use those bracket operations which are defined with ζ a primitive n-th root of unity (for all n ∈ ℕ and all ζ).

We consider these bracket operations as a generalization of the Liebracket [-, -] : L×L ⟶ L or [–] : L⊗L ⟶ L. Observe that our bracket operation is only partially defined and should not be considered as a multilinear operation, since A^{n}(ζ) ⊆ A^{n} is just a submodule in C and does not necessarily decompose into an n-fold tensor product. The elements in A^{n}(ζ) are, however, of the form z = ∑_{k} x_{k,1} ⊗ … ⊗ x_{k,n}.

If we suppress the summation index and the summation sign then we may write the bracket operation on z = x_{1} ⊗ … ⊗ x_{n} also as [z] = [x_{1}, …, x_{n}]. If we define

σ(z) =: x_{σ^{−1}(1)} ⊗ … ⊗ x_{σ^{−1}(n)}  (4)

then we get

[x_{1}, …, x_{n}] = ∑_{σ∈S_{n}} x_{σ(1)} · . . . · x_{σ(n)}.

Observe that the components x_{1}, …, x_{n} in this expression are interchanged according to the action of the braid group resp. the symmetric group on A^{n}(ζ), so x_{σ(1)} ⊗ … ⊗ x_{σ(n)} is only a symbolic expression.

The bracket operation obviously satisfies the “anti”-symmetry identity

[σ(z)] = [z]  ∀σ ∈ S_{n}.  (5)

We apply Proposition 3.1 to an algebra A in C with the operations given in Definition 3.3 and get Theorem 3.4. (1. Jacobi identity) For all n ∈ ℕ, for all primitive n-th roots of unity ζ, and for all z ∈ A^{n+1}(ζ) we have

∑_{i=1}^{n+1}[x_{i}, [x_{1}, …, x̂_{i}, …, x_{n+1}]] = ∑_{i=1}^{n+1}[., [., .]_{n}]_{2}(1 … i)(z) = 0, (6)

where we use the notation (4).

Proof. We define (S_{n+1})_{(i)} := {σ ∈ S_{n+1}|σ(i) = 1}. Then S_{n+1} = ⋃_{i}(S_{n+1})_{(i)}. For σ ∈ (S_{n+1})_{(i)} let ρ̄:= σ(i … 1). Since ρ̄(1) = 1 there is a unique ρ ∈ S_{n} with ρ̄ = 1 ⊗ ρ and σ = (1 ⊗ ρ)(1 … i). So we obtain a bijection

S_{n} ∋ ρ ↦ (1 ⊗ ρ)(1 … i) ∈ (S_{n+1})_{(i)}.

Analogously we define (S_{n+1})^{(i)} := {σ ∈ S_{n+1}|σ(i) = n + 1} and get a bijection

S_{n} ∋ ρ ↦ (ρ ⊗ 1)(n + 1 … i) ∈ (S_{n+1})^{(i)}.

Now observe that τ_{n} … τ_{1}(z) = ζ^{n}σ_{n} … σ_{1}(z) = (n + 1 … 1)(z) (by ζ^{n} = 1) for z ∈ P^{n+1}(ζ) to get

∑_{i=1}^{n} [., [., .]](1 … i)(z) =

= ∑_{i=1}^{n} ∇(1 ⊗ [., .])(1 … i)(z) − ∇([., .] ⊗ 1)τ_{n} … τ_{1}(1 … i)(z)

= ∑_{i=1}^{n} ∇(1 ⊗ [., .])(1 … i)(z) − ∇([., .] ⊗ 1)(n + 1 … i)(z)

= ∑_{i=1}^{n} ∑_{ρ∈S_{n}} ∇^{n+1}(1 ⊗ ρ)(1 … i)(z) − ∇^{n+1}(ρ ⊗ 1)(n + 1 … i)(z)

= ∑_{σ∈S_{n+1}} ∇^{n+1}σ(z) − ∇^{n+1}σ(z) = 0.
□

Theorem 3.5. (2. Jacobi identity) For all n ∈ ℕ, for all primitive n-th roots of unity ζ, and for all z = x ⊗ y_{1} ⊗ … ⊗ y_{n} ∈ A^{n+1}(−1, ζ) we have

[x, [y_{1}, …, y_{n}]] = ∑_{i=1}^{n}[y_{1}, …, [x, y_{i}], …, y_{n}]  (7)

where y_{1} ⊗ … ⊗ y_{i−1} ⊗ x ⊗ y_{i} ⊗ … ⊗ y_{n} := τ_{i−1} … τ_{1}(z) and

[y_{1}, …, y_{i−1}, [x, y_{i}], …, y_{n}] = [., [., .]_{2}, .]_{n}τ_{i−1} … τ_{1}(z).  (8)

Proof. The equation in the Theorem can also be written as

[., [., .]_{n}]_{2}(z) = ∑_{i=1}^{n}[., [., .]_{2}, .]_{n}τ_{i−1} … τ_{1}(z).

Lemma 8.1 together with φ̃(i) = j shows

∇^{n}φ(1 ⊗ … ⊗ ∇ ⊗ … ⊗ 1)τ_{i−1} … τ_{1}(z)
= ∇^{n}(1 ⊗ … ⊗ ∇ ⊗ … ⊗ 1)φ_{(i)}τ_{i−1} … τ_{1}(z)
= ∇^{n+1}τ_{j−1} … τ_{1}(1 ⊗ φ)(z);
∇^{n}φ(1 ⊗ … ⊗ ∇ ⊗ … ⊗ 1)τ_{i} … τ_{1}(z)
= ∇^{n}(1 ⊗ … ⊗ ∇ ⊗ … ⊗ 1)φ_{(i)}τ_{i} … τ_{1}(z)
= ∇^{n+1}τ_{j} … τ_{1}(1 ⊗ φ)(z);

hence

\begin{aligned}∇^{n}φ(1 ⊗ … ⊗ ∇ & ⊗ … ⊗ 1)τ_{i−1} … τ_{1}(z) \\ & = ∇^{n+1}τ_{k−1} … τ_{1}(1 ⊗ φ)(z) \\ & = ∇^{n+1}τ_{l} … τ_{1}(1 ⊗ φ)(z) \\ & = ∇^{n}φ(1 ⊗ … ⊗ ∇ ⊗ … ⊗ 1)τ_{j} … τ_{1}(z). \end{aligned} (9)

for all i, j = 1, …, n with φ̃(i) = φ̃(j) + 1, i.e. for all i except φ̃^{−1}(1) and all j except φ̃^{−1}(n). The other i’s and j’s used in (9) are in bijective correspondence.

To prove the equation of the theorem we write each σ ∈ S_{n} as ζ^{r}φ with a representative φ ∈ B_{n} and a suitable power ζ^{r} according to (3) and use (9). Then we get

∑_{i=1}^{n}[., [., .]_{2}, .]_{n}τ_{i−1} … τ_{1}(z) =
= ∑_{i=1}^{n}[., .]_{n}(1 ⊗ … ⊗ (∇ − ∇τ) ⊗ … ⊗ 1)τ_{i−1} … τ_{1}(z) =
= ∑_{i=1}^{n} ∑_{σ∈S_{n}} ∇^{n}ζ^{r}φ(1 ⊗ … ⊗ ∇ ⊗ … ⊗ 1)τ_{i−1} … τ_{1}(z)
− ∑_{j=1}^{n} ∑_{σ∈S_{n}} ∇^{n}ζ^{r}φ(1 ⊗ … ⊗ ∇ ⊗ … ⊗ 1)τ_{j}τ_{j−1} … τ_{1}(z)

= ∑_{σ∈S_{n}} ∇^{n+1}(1 ⊗ ζ^{r}φ)(z) − ∑_{σ∈S_{n}} ∇^{n+1}τ_{n} … τ_{1}(1 ⊗ ζ^{r}φ)(z) = ∑_{σ∈S_{n}} ∇^{n+1}(1 ⊗ ζ^{r}φ)(z) − ∑_{σ∈S_{n}} ∇^{n+1}(ζ^{r}φ ⊗ 1)τ_{n} … τ_{1}(z)

= ∇(1 ⊗ ∑_{σ∈S_{n}} ∇^{n}σ)(z) − ∇(∑_{σ∈S_{n}} ∇^{n}σ ⊗ 1)τ_{P,P^{n}}(z)
= (∇ − ∇τ)(1 ⊗ [., .]_{n})(z)
= [., [., .]_{n}]_{2}(z).
□

Clearly there are symmetric right sided identities.

4. Lie Algebras on Yetter-Drinfeld Modules

Now we can define the notion of a Lie algebra in the category of Yetter-Drinfeld modules. Definition 4.1. A Yetter-Drinfeld module P together with operations in YD_{K}^{K}

[., .] : P ⊗ … ⊗ P (ζ) = P^{n}(ζ) ⟶ P

for all n ∈ ℕ and all primitive n-th roots of unity ζ is called a Lie algebra if the following identities hold:

(1) for all n ∈ ℕ, for all primitive n-th roots of unity ζ, for all σ ∈ S_{n}, and for all z ∈ P^{n}(ζ)

[z] = [σ(z)],

(2) for all n ∈ ℕ, for all primitive n-th roots of unity ζ, and for all z ∈ P^{n+1}(ζ)

∑_{i=1}^{n+1}[x_{i}, [x_{1}, …, x̂_{i}, …, x_{n+1}]] = ∑_{i=1}^{n+1}[., [., .]](1 … i)(z) = 0,

where we use the notation (4),

(3) for all n ∈ ℕ, for all primitive n-th roots of unity ζ, and for all z = x ⊗ y_{1} ⊗ … ⊗ y_{n} ∈ P^{n+1}(−1, ζ) we have

[x, [y_{1}, …, y_{n}]] = ∑_{i=1}^{n}[y_{1}, …, [x, y_{i}], …, y_{n}]

where we use the notation (8). Corollary 4.2. Let A be an algebra in YD_{K}^{K}. Then A carries the structure of a Lie algebra A^{L} with the symmetric multiplications

[–] : A^{n}(ζ) ⟶ A  by  [z] := ∑_{σ∈S_{n}} ∇^{n}σ(z).

for all n ∈ ℕ and all roots of unity ζ ∈ k^{∗}.

Proof. This is a rephrasing of the “anti”-symmetry identity (5) and the Jacobi identities (6) and (7) in Theorems 3.4 and 3.5. □

5. The Lie Algebra of Primitive Elements

Let A be an algebra in C = YD_{K}^{K}. Then A⊗A is an algebra with the

multiplication A⊗A⊗A⊗A ⟶^{1⊗τ⊗1} A⊗A⊗A⊗A ⟶^{∇⊗∇} A⊗A. Let p : A

⟶ A ⊗ A be the map p(x) := x ⊗ 1 + 1 ⊗ x. Then p(= 1 ⊗ η + η ⊗ 1) is in C but p is not an algebra morphism. Let p^{n} : A^{n} ⟶ (A ⊗ A)^{n} be the n-fold tensor product of p with itself. Lemma 5.1. Let H be a Hopf algebra in C. Then P (H) := {x ∈ H|Δ(x) = x ⊗ 1 + 1 ⊗ x} is a Yetter-Drinfeld submodule of H in C.

Proof. P (H) = Ker(Δ − p). □

In particular we have δ(x) ∈ P (H) ⊗ K and xλ ∈ P (H) for all x ∈ P (H) and all λ ∈ K. Lemma 5.2. p^{n}(A^{n}(ζ)) ⊆ (A ⊗ A)^{n}(ζ).

Proof. By Theorem 2.5 p : A ⟶ A ⊗ A induces p^{n} : A^{n}(ζ) ⟶ (A ⊗ A)^{n}(ζ). □

Theorem 5.3. Let ζ be a primitive n-th root of unity and let z ∈ A^{n}(ζ). Then

[p^{n}(z)] = p([z]).

Proof. If z = ∑_{k} x_{k,1} ⊗ … ⊗ x_{k,n} ∈ A^{n}(ζ) then the equation of the theorem reads as

\begin{matrix}[∑_{k}(x_{k,1} ⊗ 1 + 1 ⊗ x_{k,1}) ⊗ … ⊗ (x_{k,n} ⊗ 1 + 1 ⊗ x_{k,n})] = \\ [∑_{k} x_{k,1} ⊗ … ⊗ x_{k,n}] ⊗ 1 + 1 ⊗ [∑_{k} x_{k,1} ⊗ … ⊗ x_{k,n}] . \end{matrix} (10)

We want to evaluate

\begin{matrix}[∑_{k}(x_{k,1} ⊗ 1 + 1 ⊗ x_{k,1}) ⊗ … ⊗ (x_{k,n} ⊗ 1 + 1 ⊗ x_{k,n})] \\ = ∑_{σ∈S_{n}} ∇^{n}σ(∑_{i}(x_{k,1} ⊗ 1 + 1 ⊗ x_{k,1}) ⊗ … ⊗ (x_{k,n} ⊗ 1 + 1 ⊗ x_{k,n})) \end{matrix}

where σ ∈ S_{n} operates on p^{n}(z) ∈ (A ⊗ A)^{n}(ζ) as described in section 2.

Let x, y ∈ A and τ(x ⊗ y) = ∑_{i} u_{i} ⊗ v_{i}. Then (1 ⊗ x) · (y ⊗ 1) = (∇⊗∇)(∑_{i} 1⊗u_{i} ⊗v_{i} ⊗1) = ∑_{i} u_{i} ⊗v_{i} = τ(x⊗y) = ∑_{i}(u_{i} ⊗1)·(1⊗v_{i}). So we have

\begin{matrix}(x ⊗ 1)(y ⊗ 1) = (xy ⊗ 1), \\ (x ⊗ 1)(1 ⊗ y) = (x ⊗ y), \\ (1 ⊗ x)(1 ⊗ y) = (1 ⊗ xy), \\ (1 ⊗ x)(y ⊗ 1) = τ(x ⊗ y). \end{matrix}  (11)

We expand a product (x_{1} ⊗ 1 + 1 ⊗ x_{1}) · . . . · (x_{n} ⊗ 1 + 1 ⊗ x_{n}). It produces after multiplication 2^{n} summands, each a product of n terms. A typical product is (x_{1} ⊗ 1)(1 ⊗ x_{2})(x_{3} ⊗ 1) …, some of the factors being of the form x_{j} ⊗ 1, the others of the form 1 ⊗ x_{j}. To evaluate such a product we use the rule of multiplication in A ⊗ A given by ∇_{A⊗A} = (∇ ⊗ ∇)(1 ⊗ τ ⊗ 1).

To explain the following calculation we consider as an example the product (x_{1} ⊗ 1)(1 ⊗ x_{2})(x_{3} ⊗ 1)(1 ⊗ x_{4})(x_{5} ⊗ 1). It is calculated with the following braid diagram

[figure]

The second and fourth factors are pulled over to the right and then all factors are multiplied according to (11). Thus we have (x_{1} ⊗ 1)(1 ⊗ x_{2})(x_{3} ⊗1)(1⊗x_{4})(x_{5} ⊗1) = (∇^{3} ⊗∇^{2})φ(x_{1} ⊗x_{2} ⊗x_{3} ⊗x_{4} ⊗x_{5}), where φ = τ_{3}τ_{4}τ_{2} as defined by the given braid diagram.

We prove now by induction on n that for every product (x_{1} ⊗ 1)(1 ⊗ x_{2})(x_{3} ⊗ 1) … with i factors of the form x_{j} ⊗ 1 and n − i factors of the form 1 ⊗ x_{j} there is an element φ ∈ B_{n} such that

(x_{1} ⊗ 1)(1 ⊗ x_{2})(x_{3} ⊗ 1) … = (∇^{i} ⊗ ∇^{n−i})φ(x_{1} ⊗ … ⊗ x_{n}).

Furthermore if t denotes the number of pairs of factors f_{1}, f_{2} in the product (x_{1} ⊗ 1)(1 ⊗ x_{2})(x_{3} ⊗ 1) … where f_{1} is to the left of f_{2}, f_{1} is of the form (1 ⊗ x_{j}) and f_{2} is of the form (x_{j} ⊗ 1), or briefly the number of factors in reverse position, then φ is composed of t generators τ_{j} of B_{n}. Observe that φ and the number t are uniquely determined by the properties of the multiplication of A⊗A and the braid group B_{n}, which has homogeneous relations.

For n = 1 we have the trivial cases x ⊗ 1 = (∇^{1} ⊗ ∇^{0})(x) and 1 ⊗ x = (∇^{0} ⊗ ∇^{1})(x), where ∇^{1} = id and ∇^{0} = 1. For the induction nothing is to be proved if i = n or i = 0. In these cases we have t = 0.

We assume now that the claim is true for n. The induction step for i ≠ 0, n + 1 is given by

(x_{1} ⊗ 1) ·(1 ⊗ x_{2}) · (x_{3} ⊗ 1) · . . . · (1 ⊗ x_{n+1}) =
= {(∇^{i} ⊗ ∇^{n−i})φ(x_{1} ⊗ … ⊗ x_{n})} · (1 ⊗ x_{n+1})
= {(∇^{i} ⊗ ∇^{n−i}) ∑_{k}(u_{k,1} ⊗ … ⊗ u_{k,n})} · (1 ⊗ x_{n+1})
= (∑_{k} u_{k,1} · . . . · u_{k,i} ⊗ u_{k,i+1} · . . . · u_{k,n}) · (1 ⊗ x_{n+1})

= ∑_{k}(u_{k,1} ⊗ 1) · . . . · (u_{k,i} ⊗ 1) · (1 ⊗ u_{k,i+1}) · . . . · (1 ⊗ u_{k,n}) · (1 ⊗ x_{n+1})

= (∇^{i} ⊗ ∇^{n−i+1}) ∑_{k}(u_{k,1} ⊗ … ⊗ u_{k,n} ⊗ x_{n+1})
= (∇^{i} ⊗ ∇^{n−i+1})(φ ⊗ 1)(x_{1} ⊗ … ⊗ x_{n+1})

where t, the number of factors in reverse position, does not change, neither does the number of generators τ_{i} used in the representation of φ ⊗ 1. The second possibility is

(x_{1} ⊗ 1) ·(1 ⊗ x_{2}) · (x_{3} ⊗ 1) · . . . · (x_{n+1} ⊗ 1)
= {(∇^{i} ⊗ ∇^{n−i})φ(x_{1} ⊗ … ⊗ x_{n})} · (x_{n+1} ⊗ 1)
= (∑_{k} u_{k,1} · . . . · u_{k,i} ⊗ u_{k,i+1} · . . . · u_{k,n}) · (x_{n+1} ⊗ 1)

= (∑_{k} u_{k,1} · . . . · u_{k,i} ⊗ u_{k,i+1} · . . . · u_{k,n−1}) · (1 ⊗ u_{k,n}) · (x_{n+1} ⊗ 1)

= (∑_{k} u_{k,1} · . . . · u_{k,i} ⊗ u_{k,i+1} · . . . · u_{k,n−1}) · (v_{k,n} ⊗ 1) · (1 ⊗ v_{k,n+1}) = (∇^{i+1} ⊗ ∇^{n−i−1})ρ(∑_{k} u_{k,1} ⊗ … ⊗ u_{k,n−1} ⊗ v_{k,n}) · (1 ⊗ v_{k,n+1}) = (∇^{i+1} ⊗ ∇^{n−i})(ρ ⊗ 1)(∑_{k} u_{k,1} ⊗ … ⊗ u_{k,n−1} ⊗ v_{k,n} ⊗ v_{k,n+1}) = (∇^{i+1} ⊗ ∇^{n−i})(ρ ⊗ 1)(1^{n−1} ⊗ τ)(φ ⊗ 1)(x_{1} ⊗ … ⊗ x_{n+1}).

where φ(x_{1} ⊗ … ⊗ x_{n}) = ∑_{k} u_{k,1} ⊗ … ⊗ u_{k,n}, τ(u_{k,n} ⊗ x_{n+1}) = ∑ v_{k,n} ⊗ v_{k,n+1}, and (1^{n−1} ⊗ τ)(φ ⊗ 1)(x_{1} ⊗ … ⊗ x_{n} ⊗ x_{n+1}) = ∑_{k} u_{k,1} ⊗ … ⊗ u_{k,n−1} ⊗ v_{k,n} ⊗ v_{k,n+1}. We determine the number t(ψ) of generators τ_{i} occurring in ψ = (ρ ⊗ 1)(1^{n−1} ⊗ τ)(φ ⊗ 1). We have by induction t(φ) = t_{n} the number of factors in (x_{1} ⊗ 1) · (1 ⊗ x_{2}) · (x_{3} ⊗ 1) · . . . in reverse position. Also we have t_{n+1} = t_{n} +(n−i) the number of factors in (x_{1} ⊗ 1) · (1 ⊗ x_{2}) · (x_{3} ⊗ 1) · . . . · (x_{n+1} ⊗ 1) in reverse position. Then t(ψ) = t((ρ ⊗ 1)(1^{n−1} ⊗ τ)(φ ⊗ 1)) = t(ρ ⊗ 1) + t(1^{n−1} ⊗ τ) + t(φ ⊗ 1) = (n − i − 1) + 1 + t_{n} = t_{n+1}.

If we sum up we obtain

(x_{1}⊗1+1⊗x_{1})·. . .·(x_{n}⊗1+1⊗x_{n}) = ∑_{i} ∑_{φ_{i}}(∇^{i}⊗∇^{n−i})φ_{i}(x_{1}⊗…⊗x_{n}),

for certain φ_{i} ∈ B_{n} which arise in the evaluation given above.

Now let z ∈ A^{n}(ζ). We expand the products in ∇^{n}p^{n}z = ∑_{k}(x_{k,1} ⊗ 1 + 1 ⊗ x_{k,1}) · . . . · (x_{k,n} ⊗ 1 + 1 ⊗ x_{k,n}). Each of these products in the sum is treated in the same way as described above. Using (3) we get

∇^{n}p^{n}(z) = ∑_{k}(x_{k,1} ⊗ 1 + 1 ⊗ x_{k,1}) · . . . · (x_{k,n} ⊗ 1 + 1 ⊗ x_{k,n})
= ∑_{k} ∑_{i} ∑_{φ_{i}}(∇^{i} ⊗ ∇^{n−i})φ_{i}(x_{k,1} ⊗ … ⊗ x_{k,n})
= ∑_{i} ∑_{φ_{i}}(∇^{i} ⊗ ∇^{n−i})φ_{i}(z)
= ∑_{i} ∑_{φ_{i}}(∇^{i} ⊗ ∇^{n−i})ζ^{−t(φ_{i})}ρ_{i}(z)

where ρ_{i} ∈ S_{n} are the canonical images of the φ_{i} ∈ B_{n} and t(φ_{i}) is the number of factors τ_{j} in the representation of φ_{i}.

This gives us

[p^{n}(z)] = ∑_{σ∈S_{n}} ∇^{n}p^{n}σ(z)
= ∑_{σ} ∑_{i} ∑_{φ_{i}} ζ^{−t(φ_{i})}(∇^{i} ⊗ ∇^{n−i})ρ_{i}σ(z)
= ∑_{σ} ∑_{i} ( ∑_{φ_{i}} ζ^{−t(φ_{i})})(∇^{i} ⊗ ∇^{n−i})σ(z)
= ∑_{i} c_{i}(∇^{i} ⊗ ∇^{n−i}) ∑_{σ} σ(z)

where the factors c_{i} = ∑_{φ_{i}} ζ^{−t(φ_{i})} ∈ k. We want to show that the c_{i} are zero for all 0 < i < n.

So fix n and i. Consider one product (x_{1} ⊗1)·(1⊗x_{2})·(x_{3} ⊗1)·. . . in the development of (x_{1}⊗1+1⊗x_{1})·. . .·(x_{n}⊗1+1⊗x_{n}) = ∑_{i} ∑_{φ_{i}}(∇^{i}⊗ ∇^{n−i})φ_{i}(x_{1}⊗…⊗x_{n}) and its corresponding φ_{i}. The chosen summand is completely determined by giving the positions in {1, …, n} of the n−i factors of the form (1 ⊗ x_{j}). The first of these factors has λ_{1} factors of the form (x_{j} ⊗ 1) to its right with 0 ≤ λ_{1} ≤ i. So it contributes λ_{1} pairs of factors in reverse position. The second factor of the form (1⊗x_{j}) contributes λ_{2} (with 0 ≤ λ_{2} ≤ λ_{1} ≤ i) pairs of factors in reverse position, and so on. We obtain t = λ_{1} + λ_{2} + … + λ_{n−i} pairs in reverse position. If we know the λ_{i} with 0 ≤ λ_{n−i} ≤ … ≤ λ_{2} ≤ λ_{1} ≤ i then they also determine uniquely the position of the factors of the form (1 ⊗ x_{j}). Each partition of t = λ_{1} + λ_{2} + … + λ_{n−i} into (at most) n − i parts each ≤ i gives one term ζ^{−t} in c_{i} = ∑_{φ_{i}} ζ^{−t(φ_{i})} and we find p(i, n − i, t) partitions of t into at most n − i parts each ≤ i. So we get

c_{i} = ∑_{t≥0} p(i, n − i, t)ζ^{−t}.

By a theorem of Sylvester ([1] Theorem 3.1) we have

∑_{t≥0} p(i, n − i, t)q^{t} = \frac{(1 − q^{n})(1 − q^{n−1}) … (1 − q^{n−i+1})}{(1 − q^{i})(1 − q^{i−1}) … (1 − q)}

hence c_{i} = 0 for 0 < i < n since ζ and also ζ^{−1} are primitive n-th roots of unity.

So we have shown

[p^{n}(z)] = ∑_{σ∈S_{n}} ∇^{n}p^{n}σ(z)
= ∑_{i} c_{i}(∇^{i} ⊗ ∇^{n−i}) ∑_{σ} σ(z)
= ∑_{σ} ∇^{n}σ(z) ⊗ 1 + 1 ⊗ ∑_{σ} ∇^{n}σ(z)
= p[z].
□

Corollary 5.4. Let H be a Hopf algebra in C. Then the set of primitive elements P (H) forms a Lie algebra in C.

Proof. By Lemma 5.1 P (H) is a Yetter-Drinfeld submodule of H. Let z ∈ P (H)^{n}(ζ). Then p([z]) = [p^{n}(z)] = [Δ^{n}(z)] = Δ([z]) since Δ is an algebra homomorphism. Hence [z] ∈ P (H). So P (H) is a Lie subalgebra of H^{L}. □

Definition 5.5. Let A be an algebra in C and let end(A) be the inner endomorphism object of A in C, i.e. the Yetter- Drinfeld module end(A) satisfying C(X ⊗ A, A) ≅ C(X, end(A)) for all X ∈ C. It can be shown that

end(A) := {f ∈ Hom(A, A)|∃ ∑ f_{(0)} ⊗ f_{(1)} ∈ Hom(A, A) ⊗ K∀a ∈ A :
∑ f_{(0)}(a) ⊗ f_{(1)} = ∑ f(a_{(0)})_{(0)} ⊗ f(a_{(0)})_{(1)}S(a_{(1)})}

is the Yetter-Drinfeld module with the required universal property. end(A) operates on A by a canonical map ev : end(A) ⊗ A ⟶ A with ev(f ⊗ a) = f(a).

A derivation from A to A is a linear map (d : A ⟶ A) ∈ end(A) such that

d(ab) = d(a)b + (1 ⊗ d)(τ ⊗ 1)(d ⊗ a ⊗ b)

for all a, b ∈ A. Observe that in the symmetric situation this means d(ab) = d(a)b + ad(b).

It is clear that all derivations from A to A form an object Der(A) in C and that there is an operation Der(A) ⊗ A ⟶ A. Corollary 5.6. Der(A) is a Lie algebra.

Proof. Let m denote the multiplication of A. An endomorphism x : A ⟶ A in end(A) is a derivation iff m(x ⊗ 1 + 1 ⊗ x) = xm where (x ⊗ y)(a ⊗ b) = (ev ⊗ ev)(1 ⊗ τ ⊗ 1)(x ⊗ y ⊗ a ⊗ b) for elements a and b in A and elements x and y in end(A). So x ∈ end(A) is a derivation iff mp(x) = xm.

To show that Der(A) is a Lie algebra it suffices to show that it is closed under Lie multiplication since it is a subobject of end(A), which is an algebra in the category C. Let ζ be a primitive n-th root of unity. Let ∇ : end(A) ⊗ end(A) ⟶ end(A) be the multiplication of end(A).

If x_{1}, x_{2} ∈ Der(A) then mp(x_{1})p(x_{2}) = x_{1}mp(x_{2}) = x_{1}x_{2}m or more generally m(∇^{n}p^{n})(x_{1} ⊗ … ⊗ x_{n}) = ∇^{n}(x_{1} ⊗ … ⊗ x_{n})m for all x_{1} ⊗ … ⊗ x_{n} ∈ Der(A)^{n}. Thus we get for z ∈ Der(A)^{n}(ζ)

\begin{matrix}mp([z]) & = m[p^{n}(z)] = ∑ m(∇^{n}σ(p^{n}(z))) \\ = ∑ m(∇^{n}p^{n}σ(z)) = ∑ ∇^{n}σ(z)m = [z]m \end{matrix}

hence [z] ∈ Der(A). □

6. The Universal Enveloping Algebra of a Lie Algebra

As in [5] we can now construct the universal enveloping algebra of a Lie algebra P in C as U(P) := T (P)/I where T (P) is the tensor algebra over P, which lives again in C, and where I is the ideal generated by the relations [z]−∑ ∇^{n}σ(z) for all z ∈ P^{n}(ζ), for all n and for all primitive n-th roots of unity ζ. Then U(P) clearly is a universal solution for the following universal problem

[figure: P; U(P); ι; f; A; g]

where for each morphism of Lie-algebras f there is a unique morphism of algebras g such that the diagram commutes. Theorem 6.1. Let P be a Lie algebra in C. Then the universal enveloping algebra U(P) is a Hopf algebra in C.

Proof. It is easily seen that δ : P ⟶ (U(P) ⊗ U(P))^{L} in M^{kG} given by δ(x) := \overline{x} ⊗ 1 + 1 ⊗ \overline{x} where \overline{x} is the canonical image of x ∈ P in U(P) and the counit ε : U(P) ⟶ k given by the zero morphism 0 : P ⟶ k define the structure of a bialgebra on U(P) in C.

Now we want to define S : U(P) ⟶ U(P)^{op+} by the Lie homomorphism S : P ⟶ U(P)^{op+}, S(x) = −x̄. Here A^{op+} is the algebra obtained from the algebra A by the multiplication A⊗A ⟶^{τ} A⊗A ⟶^{∇} A. Then for z ∈ P^{n}(ζ) we have

S([z]) = −\overline{[z]} = − ∑_{σ} ∇^{n}σ(\overline{z}) = − ∑_{σ} ∇^{n}ππ^{−1}σ(\overline{z})

= − ∑_{σ}(∇^{op})^{n}π^{−1}σ(\overline{z}) (by (3)) = − ∑_{σ}(∇^{op})^{n}ζ^{\frac{n(n−1)}{2}} ρ^{−1}σ(\overline{z})

= −ζ^{\frac{n(n−1)}{2}} [\overline{z}] = (−1)^{n}[\overline{z}] = [\overline{S^{n}(z)}]

where π ∈ B_{n} is the braid map given by the twist of all n strands with source {1, …, n} and domain {n, …, 1}, π = (τ_{1}) … (τ_{n−2} … τ_{2}τ_{1})(τ_{n−1} … τ_{2}τ_{1}) and ζ^{− \frac{n(n−1)}{2}} π = ρ ∈ S_{n}.

Hence S is a Lie homomorphism and factorizes through U(P). Since U(P) is generated as an algebra by P we prove that S is the antipode by complete induction:

∇(1 ⊗ S)Δ(1) = 1S(1) = 1 = ε(1),
∇(1 ⊗ S)Δ(\overline{x}) = \overline{x} + S(\overline{x}) = 0 = ε(\overline{x}).

Before we prove the general induction step we observe that Δ : U(P) ⟶ U(P) ⊗ U(P) is a morphism in C = YD_{K}^{K} so that we have in particular

∑(a^{0})_{1} ⊗ (a^{0})_{2} ⊗ a^{1} = ∑(a_{1})^{0} ⊗ (a_{2})^{0} ⊗ (a_{1})^{1}(a_{2})^{1} ∈ U(P) ⊗ U(P) ⊗ K

for a ∈ U(P). (Here we use δ(a) = ∑ a^{0} ⊗ a^{1} to denote the comodule structure in YD_{K}^{K}.) Assume now that a is writte as a product of n ≥ 1 elements in P and that ∑ a_{1}S(a_{2}) = 0. Then for all x ∈ P we have ∑(a_{1})^{0}S((a_{2})^{0})S(x(a_{1})^{1}(a_{2})^{1}) = ∑(a^{0})_{1}S((a^{0})_{2})S(xa^{1}) = 0 since δ(a) = ∑ a^{0} ⊗ a^{1} ∈ \overline{P ⊗ … ⊗ P} ⊗ K ⊆ U(P) ⊗ K. So we have

∇(1 ⊗ S)Δ(xa) = ∇(1 ⊗ S) ∑(xa_{1} ⊗ a_{2} + (a_{1})^{0} ⊗ (x(a_{1})^{1})a_{2})
= ∑ xa_{1}S(a_{2}) + ∑(a_{1})^{0}S((x(a_{1})^{1})a_{2})

= ∑(a_{1})^{0}S((a_{2})^{0})S(x(a_{1})^{1}(a_{2})^{1}) = 0 = ηε(xa).

The second condition ∇(S ⊗ 1)Δ = ηε is proved in a similar way (by using elements of the form ax and the equation ∑ S((aκ)_{1})(aκ)_{2} = 0 for a written as a product of n elements in P and κ ∈ K). So S is an antipode and U(P) is a Hopf algebra in C. □

7. (G, χ) Lie algebras

In [5] we introduced and studied the concept of G-graded Lie algebras or (G, χ)-Lie algebras for an abelian group G with a bicharacter χ generalizing the concepts of Lie algebras, Lie super algebras, and Lie color algebras. The reader may find examples of such (G, χ)-Lie algebras in [5]. A generalization of this concept of Lie algebras to the group graded case for a noncommutative group requires the use of Yetter-Drinfeld modules over kG. We show that (G, χ)-Lie algebras are Lie algebras on Yetter-Drinfeld modules in the sense of this paper. We use the notation of [5].

Let G be an abelian group with a bicharacter χ : G ⊗_{ℤ} G ⟶ k^{∗}. Let P be a kG-comodule. Then P is a Yetter-Drinfeld module over kG [4] with the module structure x · g = χ(h, g)x for homogeneous elements x = x_{h} ∈ M with δ(x) = x ⊗ h. The braid map is τ(x_{h} ⊗ y_{g}) = y_{g}⊗x_{h}·g = χ(h, g)y_{g}⊗x_{h}, hence the braiding given in [5] after Example 2.3.

Let ζ ∈ k^{∗} be given. Let (g_{1}, …, g_{n}) be a ζ-family, i.e. χ(g_{i}, g_{j})χ(g_{j}, g_{i}) = ζ^{2}. Let Q := ∑_{σ∈S_{n}} P_{g_{σ(1)}} ⊗ … ⊗ P_{g_{σ(n)}}. Lemma 7.1. Q is a right S_{n}-module by

(x_{1} ⊗ … ⊗ x_{n})σ = ρ(σ, (g_{1}, …, g_{n}))x_{σ(1)} ⊗ … ⊗ x_{σ(n)}

for σ ∈ S_{n} and x_{1} ⊗ … ⊗ x_{n} ∈ P_{g_{1}} ⊗ … ⊗ P_{g_{n}}.

Proof. We have to show the compatibility of this operation with the composition of permutations. Let σ, τ ∈ S_{n}. We use Lemma 2.2 of [5]. Then

(x_{1}⊗… ⊗ x_{n})(στ) =
= ρ(στ, (g_{1}, …, g_{n}))x_{στ(1)} ⊗ … ⊗ x_{στ(n)}
= ρ(σ, (g_{1}, …, g_{n}))ρ(τ, (g_{σ(1)}, …, g_{σ(n)}))x_{στ(1)} ⊗ … ⊗ x_{στ(n)}
= (ρ(σ, (g_{1}, …, g_{n}))x_{σ(1)} ⊗ … ⊗ x_{σ(n)})τ
= ((x_{1} ⊗ … ⊗ x_{n})σ)τ.
□

Q becomes a left S_{n}-module by σ(x_{1}⊗…⊗x_{n}) = ρ(σ^{−1}, (g_{1}, …, g_{n}))x_{σ^{−1}(1)}⊗ … ⊗ x_{σ^{−1}(n)}. Thus ⨁_{{(g_{1},... ,g_{n}) ζ-family}} P_{g_{1}} ⊗ … ⊗ P_{g_{n}} is also a left S_{n}- module.

This action is connected with the action of B_{n} on ⨁_{{(g_{1},... ,g_{n}) ζ- family}} P_{g_{1}}⊗ … ⊗ P_{g_{n}} by

ζ^{−1}τ_{i}(x_{1} ⊗ … ⊗ x_{n}) = σ_{i}(x_{1} ⊗ … ⊗ x_{n})  (12)

for the canonical generators τ_{i} of B_{n} resp. σ_{i} of S_{n}, since

ζ^{−1}τ_{i}(x_{1} ⊗ … ⊗ x_{n}) =
= ζ^{−1}χ(g_{i}, g_{i+1})x_{1} ⊗ … ⊗ x_{i+1} ⊗ x_{i} ⊗ … ⊗ x_{n}
= ρ(σ_{i}^{−1}, (g_{1}, …, g_{n}))x_{σ_{i}^{−1}(1)} ⊗ … ⊗ x_{σ_{i}^{−1}(n)}
= σ_{i}(x_{1} ⊗ … ⊗ x_{n}).

In particular we have

τ_{i}^{−1}τ_{i+1}^{−1} … τ_{j−1}^{−1}τ_{j}^{2}τ_{j−1} … τ_{i+1}τ_{i}(x_{1} ⊗ … ⊗ x_{n}) =
= ζ^{2}σ_{i}^{−1}σ_{i+1}^{−1} … σ_{j−1}^{−1}σ_{j}^{2}σ_{j−1} … σ_{i+1}σ_{i}(x_{1} ⊗ … ⊗ x_{n})
= ζ^{2}(x_{1} ⊗ … ⊗ x_{n}),

so that x_{1} ⊗ … ⊗ x_{n} ∈ P^{n}(ζ) by Lemma 2.4. Thus we have

⨁_{{(g_{1},... ,g_{n}) ζ-family}} P_{g_{1}} ⊗ … ⊗ P_{g_{n}} ⊆ P^{n}(ζ).

Conversely let ∑ x_{1} ⊗ … ⊗ x_{n} ∈ P^{n} = ⨁_{{(g_{1},... ,g_{n})}} P_{g_{1}} ⊗ … ⊗ P_{g_{n}} with homogeneous summands and assume that one of the summands is nonzero in P_{g_{1}} ⊗ … ⊗ P_{g_{n}} where (g_{1}, …, g_{n}) is not a ζ-family for example by χ(g_{i}, g_{i+1})χ(g_{i+1}, g_{i}) ≠ ζ^{2}. Then (τ_{i}^{2} − ζ^{2})(∑ x_{1} ⊗ … ⊗ x_{n}) has a non-zero component in P_{g_{1}} ⊗ … ⊗ P_{g_{n}}, hence ∑ x_{1} ⊗ … ⊗ x_{n} cannot be in P^{n}(ζ). This proves Proposition 7.2. Let ζ ∈ k^{∗} be given. Then

P^{n}(ζ) = ⨁_{{(g_{1},... ,g_{n}) ζ-family}} P_{g_{1}} ⊗ … ⊗ P_{g_{n}}.

By Lemma 7.1 and (12) the bracket multiplication of [5] is a special case of the bracket multiplication of this paper and (G, χ)-Lie algebras are Lie algebras over Yetter-Drinfeld modules. Example 7.3. As a new example of Lie algebras we give one family of examples of (G, χ)-Lie algebras. Let G = C_{3} = {0, 1, 2} be the cyclic group with 3 elements. Define the structure of a right kG-module on a kG-comodule V (i.e. on a C_{3}-graded vector space V = V_{0} ⊕ V_{1} ⊕ V_{2}) using the bicharacter χ : C_{3} ⊗_{ℤ}C_{3} ≅ C_{3} ⟶ k^{∗}, χ(1⊗1) = ξ a primitive 3-rd root of unity, by v · g := χ(deg(v) ⊗ g)v = χ(deg(v), g)v for g ∈ G and homogeneous elements v ∈ V. Then V is a Yetter-Drinfeld module.

Let A := end(V) be the inner endomorphism object of V in kG-comod. By Corollary 4.2 A is a Lie algebra. One verifies easily (see [5]) that the only non-zero components A^{n}(ζ) for the partial Lie multiplication are

A^{2}(−1) = A_{0} ⊗ (A_{1} ⊕ A_{2}) ⊕ (A_{0} ⊕ A_{1} ⊕ A_{2}) ⊗ A_{0}

and

A^{3}(ξ) = A_{1} ⊗ A_{1} ⊗ A_{1} ⊕ A_{2} ⊗ A_{2} ⊗ A_{2}.

Now let ⟨., .⟩ : V ⊗ V ⟶ k be a bilinear form on V in C. We define

𝔤(V)_{i} := {f ∈ A_{i}|∀v, w ∈ V, deg(v) = j : ⟨f(v), w⟩ = −χ(i, j)⟨v, f(w)⟩}.

This space is the homogeneous component of 𝔤(V) ⊆ A that becomes a Yetter-Drinfeld module.

For f ∈ 𝔤(V)_{0} and g ∈ 𝔤(V)_{i}, i ∈ C_{3}, v ∈ V_{j}, w ∈ V we have

⟨[f, g](v), w⟩= ⟨(fg − gf)(v), w⟩
= ⟨fg(v), w⟩ − ⟨gf(v), w⟩
= χ(i, j)⟨v, gf(w)⟩ − χ(i, j)⟨v, fg(w)⟩
= −χ(i, j)⟨v, [f, g](w)⟩,

hence [f, g] ∈ 𝔤(V)_{i}. Analogously one shows [g, f] ∈ 𝔤(V)_{i}.

For k = 1, 2, 3 let f_{k} ∈ 𝔤(V)_{i} (i = 1 or i = 2). Then

⟨[f_{1}, f_{2}, f_{3}](v),w⟩ = ∑_{σ∈S_{3}}⟨f_{σ(1)}f_{σ(2)}f_{σ(3)}(v), w⟩

= (−1) ∑_{σ∈S_{3}} χ(i, i + i + j)χ(i, i + j)χ(i, j)⟨v, f_{σ(3)}f_{σ(2)}f_{σ(1)}(w)⟩

= −⟨v, [f_{1}, f_{2}, f_{3}](w)⟩,

hence [f_{1}, f_{2}, f_{3}] ∈ 𝔤(V)_{0}. Thus we have a Lie algebra 𝔤(V). Depending on the choice of the bilinear form this is a generalization of the orthogonal or the symplectic Lie algebra.

8. Appendix

Proof. of Lemma 2.4:

Define actions π_{i,j} for 1 ≤ i < j ≤ n on M by

π_{i,j} := τ_{i}^{−1}τ_{i+1}^{−1} … τ_{j−2}^{−1}τ_{j−1}τ_{j−2} … τ_{i+1}τ_{i}  (13)

Observe that π_{i,i+1} = τ_{i}. Since τ_{i}τ_{j} = τ_{j}τ_{i} if |i − j| ≥ 2 a simple calculation gives

π_{i,j}τ_{k} = τ_{k}π_{i,j} for all k < i − 1 and all k > j,

π_{i,j}τ_{i−1} = τ_{i−1}π_{i−1,j},
π_{i,j}τ_{k} = τ_{k}π_{i,j} for all i < k < j − 1,
π_{i,j}τ_{j−1} = τ_{j−1}π_{i,j−1} if i < j − 1 and
π_{i,j}τ_{j−1} = τ_{i}τ_{i} = τ_{j−1}π_{i,j} if i = j − 1.

(14)

Let N ⊆ M be a kB_{n} submodule of M. Assume furthermore that τ_{i}^{2}τ_{i+1} = τ_{i+1}τ_{i}^{2} on N for all i = 1, …, n−2. Then τ_{i+1}^{2}τ_{i} = τ_{i}^{3} = τ_{i}τ_{i+1}^{2}. Consequently we have

τ_{j}^{2}τ_{i} = τ_{i}τ_{j}^{2}  (15)

on N for all i, j = 1, …, n − 1. Thus the τ_{j}^{2} commute with all φ ∈ B_{n} if they act on N.

We introduce the vector subspace M(ζ) ⊆ \overline{M(ζ)} ⊆ M by

\overline{M(ζ)} := \{z ∈ M|∀1 ≤ i < j ≤ n : π_{i,j}^{2}(z) = ζ^{2}z\}

and show that \overline{M(ζ)} is invariant under the action of the τ_{i} and τ_{i}^{2}τ_{i+1} = τ_{i+1}τ_{i}^{2} on M(ζ) for all i = 1, …, n − 2.

For z ∈ M(ζ) and i < j we have π_{i,j}^{2}τ_{k}(z) = τ_{k}π_{i,j}^{2}(z) = ζ^{2}τ_{k}(z) for all k with 1 ≤ k < i − 1 and j < k ≤ n by (14) and for all k with i < k < j−1 by (14). Furthermore we have π_{i,j}^{2}τ_{i−1}(z) = τ_{i−1}π_{i−1,j}^{2}(z) = ζ^{2}τ_{i−1}(z) by (14), π_{i,j}^{2}τ_{j−1}(z) = τ_{j−1}π_{i,j−1}^{2}(z) = ζ^{2}τ_{j−1}(z) (for i < j − 1) by (14), and π_{i,j}^{2}τ_{j−1}(z) = τ_{j−1}π_{i,j}^{2}(z) = ζ^{2}τ_{j−1}(z) (for i = j−1) by (14). So there remain two cases to investigate for which we use π_{i,j}^{2}(z) = ζ^{2}z and symmetrically π_{i,j}^{−2}(z) = ζ^{−2}z for all z ∈ M(ζ).

In the first case we get

π_{i,j}^{2}τ_{i}(z) = τ_{i}^{−1} … τ_{j−1}^{2} … τ_{i}τ_{i}(z) = τ_{i}^{−1} … τ_{j−1}^{2} … τ_{i+1}(ζ^{2}z)
= ζ^{2}τ_{i}^{−1} … τ_{j−1}^{2} … τ_{i+1}(z) = ζ^{2}τ_{i}^{−1}π_{i+1,j}^{2}(z)
= ζ^{2}τ_{i}^{−1}(ζ^{2}z) = ζ^{2}τ_{i}^{−1}τ_{i}^{2}(z) = ζ^{2}τ_{i}(z)

for i + 1 < j and π_{i,i+1}^{2}τ_{i}(z) = τ_{i}^{3}(z) = ζ^{2}τ_{i}(z).

In the second case we get

π_{i,j}^{2}τ_{j}(z) = τ_{i}^{−1} … τ_{j−1}^{2} … τ_{i}τ_{j}(z) = ζ^{2}τ_{i}^{−1} … τ_{j−1}^{2} … τ_{i}τ_{j}τ_{j}^{−2}(z)

= ζ^{2}τ_{j}^{−1}τ_{j}τ_{i}^{−1} … τ_{j−1}^{2} … τ_{i}τ_{j}^{−1}(z) = ζ^{2}τ_{j}^{−1}τ_{i}^{−1} … τ_{j}τ_{j−1}^{2}τ_{j}^{−1} … τ_{i}(z)

= ζ^{2}τ_{j}^{−1}τ_{i}^{−1} … τ_{j−1}^{−1}τ_{j}^{2}τ_{j−1} … τ_{i}(z) = ζ^{2}τ_{j}^{−1}πi, j + 1^{2}(z)
= ζ^{2}τ_{j}^{−1}(ζ^{2}z) = ζ^{2}τ_{j}(z).

Hence we have τ_{i}(z) ∈ M(ζ) for all z ∈ M(ζ) and all i = 1, …, n−1. The claim τ_{i}^{2}τ_{i+1} = τ_{i+1}τ_{i}^{2} is clear from the invariance and the fact,

that τ_{i}^{2} on M(ζ) is multiplication by ζ^{2}.

Since the τ_{i}^{2} commute in their action on \overline{M(ζ)} with all φ ∈ B_{n} it is clear that \overline{M(ζ)} ⊆ M(ζ). □

We now study specific braids. The following identity

...  ... ...  ... =

implies

τ_{1}^{−1} … τ_{i−1}^{−1}τ_{i}^{2}τ_{i−1} … τ_{1} = τ_{i} … τ_{2}τ_{1}^{2}τ_{2}^{−1} … τ_{i}^{−1}  (16)

and similarly τ_{1} … τ_{i−1}τ_{i}^{2}τ_{i−1}^{−1} … τ_{1}^{−1} = τ_{i}^{−1} … τ_{2}^{−1}τ_{1}^{2}τ_{2} … τ_{i} for all i = 1, …, n.

Let B_{n} ∋ φ ↦ φ̃ ∈ S_{n} denote the canonical epimorphism.

For each braid φ ∈ B_{n} there exists a braid φ_{(i)} ∈ B_{n+1} such that the diagram

[figure: P ⊗ … ⊗ P^{2} ⊗ … ⊗ P; P ⊗ … ⊗ P; 1 ⊗ … ⊗ f ⊗ … ⊗ 1; i; P ⊗ … ⊗ P^{2} ⊗ … ⊗ P; P ⊗ … ⊗ P; 1 ⊗ … ⊗ f ⊗ … ⊗ 1; j; φ_{(i)}; φ]

commutes for all f : P^{2} ⟶ P in C (where j = φ̃(i)). The braid φ_{(i)} can be given explicitly, but we are only interested in the following special forms

\begin{aligned}τ_{j(i)} & = τ_{j+1} & if j > i; & τ_{i−1(i)} & = τ_{i}τ_{i−1}; \\ τ_{j(i)} & = τ_{j} & if j < i − 1; & τ_{i(i)} & = τ_{i}τ_{i+1} \end{aligned}

which can be easily verified.

By (16) we have for all z ∈ P^{n+1}(−1, ζ)

τ_{i}^{2}τ_{i−1} … τ_{1}(z) = τ_{i−1} … τ_{1}(z).  (17)

Lemma 8.1. For z ∈ P^{n+1}(−1, ζ), φ ∈ B_{n} and j := φ̃(i) we have

\begin{aligned}φ_{(i)}τ_{i−1} … τ_{1}(z) & = τ_{j−1} … τ_{1}(1 ⊗ φ)(z); \\ φ_{(i)}τ_{i} … τ_{1}(z) & = τ_{j} … τ_{1}(1 ⊗ φ)(z). \end{aligned}

Proof. To prove this we first observe that these two relations are compatible with the group structure of B_{n}. For φ̃ψ̃(i) = φ̃(j) = k we have

\begin{aligned}φ_{(j)}ψ_{(i)}τ_{i−1} … τ_{1}(z) & = φ_{(j)}τ_{j−1} … τ_{1}(1 ⊗ ψ)(z) = τ_{k−1} … τ_{1}(1 ⊗ φψ)(z); \\ φ_{(j)}ψ_{(i)}τ_{i} … τ_{1}(z) & = φ_{(j)}τ_{j} … τ_{1}(1 ⊗ ψ)(z) = τ_{k} … τ_{1}(1 ⊗ φψ)(z) \end{aligned}

so we only have to show these relations for the generators φ = τ_{j}, j = 1, …, n − 1. In these cases we have

τ_{j(i)}τ_{i−1} … τ_{1}(z) = τ_{j+1}τ_{i−1} … τ_{1}(z)

= τ_{i−1} … τ_{1}τ_{j+1}(z)

= τ_{i−1} … τ_{1}(1 ⊗ τ_{j})(z)  for j > i;
τ_{j(i)}τ_{i−1} … τ_{1}(z) = τ_{j}τ_{i−1} … τ_{1}(z)

= τ_{i−1} … τ_{j}τ_{j+1}τ_{j} … τ_{1}(z) = τ_{i−1} … τ_{j+1}τ_{j}τ_{j+1} … τ_{1}(z) = τ_{i−1} … τ_{1}τ_{j+1}(z)

= τ_{i−1} … τ_{1}(1 ⊗ τ_{j})(z)  for j < i − 1;
τ_{i−1(i)}τ_{i−1} … τ_{1}(z) = τ_{i}τ_{i−1}τ_{i−1} … τ_{1}(z)

= τ_{i}τ_{i−2} … τ_{1}(z) = τ_{i−2} … τ_{1}τ_{i}(z) = τ_{i−2} … τ_{1}(1 ⊗ τ_{i−1})(z);

τ_{i(i)}τ_{i−1} … τ_{1}(z) = τ_{i}τ_{i+1}τ_{i−1} … τ_{1}(z)

= τ_{i} … τ_{1}τ_{i+1}(z) = τ_{i} … τ_{1}(1 ⊗ τ_{i})(z);

τ_{j(i)}τ_{i} … τ_{1}(z) = τ_{j+1}τ_{i} … τ_{1}(z)

= τ_{i} … τ_{1}τ_{j+1}(z)

= τ_{i} … τ_{1}(1 ⊗ τ_{j})(z)  for j > i;
τ_{j(i)}τ_{i} … τ_{1}(z) = τ_{j}τ_{i} … τ_{1}(z)

= τ_{i} … τ_{j}τ_{j+1}τ_{j} … τ_{1}(z) = τ_{i} … τ_{j+1}τ_{j}τ_{j+1} … τ_{1}(z) = τ_{i} … τ_{1}τ_{j+1}(z)

= τ_{i} … τ_{1}(1 ⊗ τ_{j})(z)  for j < i − 1;
τ_{i−1(i)}τ_{i−1} … τ_{1}(z) = τ_{i}τ_{i−1}τ_{i}τ_{i−1}τ_{i−2} … τ_{1}(z)

= τ_{i−1}τ_{i}τ_{i−1}^{2}τ_{i−2} … τ_{1}(z) = τ_{i−1}τ_{i}τ_{i−2} … τ_{1}(z) = τ_{i−1}τ_{i−2} … τ_{1}τ_{i}(z) = τ_{i−1} … τ_{1}(1 ⊗ τ_{i−1})(z);

τ_{i(i)}τ_{i} … τ_{1}(z) = τ_{i}τ_{i+1}τ_{i} … τ_{1}(z)

= τ_{i+1}τ_{i}τ_{i+1} … τ_{1}(z) = τ_{i+1}τ_{i} … τ_{1}τ_{i+1}(z) = τ_{i+1} … τ_{1}(1 ⊗ τ_{i})(z)

where we used (17) in the 3. and 7. equations. □

Lemma 8.2. For all z ∈ P^{n+1}(−1, ζ) and all f : P^{2} ⟶ P we have

(P^{i−1} ⊗ f ⊗ P^{n−i})τ_{i−1} … τ_{1}(z) ∈ P^{n}(ζ).

Proof. For all φ ∈ B_{n} and all k = 1, …, n we have

τ_{k}^{2}φ (P^{i−1} ⊗ f ⊗ P^{n−i})τ_{i−1} … τ_{1}(z) =

= τ_{k}^{2}(P^{j−1} ⊗ f ⊗ P^{n−j})φ_{(i)}τ_{j−1} … τ_{1}(z) = (P^{j−1} ⊗ f ⊗ P^{n−j})τ_{k(j)}^{2}τ_{j−1} … τ_{1}(1 ⊗ φ)(z) = (P^{j−1} ⊗ f ⊗ P^{n−j})τ_{j−1} … τ_{1}(1 ⊗ τ_{k}^{2}φ)(z) = φ(P^{i−1} ⊗ f ⊗ P^{n−i})τ_{i−1} … τ_{1}(z)

hence (P^{i−1} ⊗ f ⊗ P^{n−i})τ_{i−1} … τ_{1}(z) ∈ P^{n}(ζ). □

Now we can give the

Proof. of Proposition 3.1:

We first show that P^{n+1}(ζ) ⊆ P ⊗(P^{n}(ζ)) ⊆ P^{n+1}. Let z = ∑_{k} z_{k,1}⊗ …⊗z_{k,n+1} be in P^{n+1}(ζ) with linearly independent z_{k,1}. Let φ, τ_{i} ∈ B_{n} be given. Define 1 ⊗ φ ∈ B_{n+1} resp. 1 ⊗ τ_{i} ∈ B_{n+1} by the operation of φ resp. τ_{i} on the factors z_{k,2} ⊗ … ⊗ z_{k,n+1}, e.g. 1 ⊗ τ_{i}^{(n)} = τ_{i+1}^{(n+1)}. Then

\begin{aligned}∑ z_{k,1} ⊗ φ^{−1}τ_{i}^{2}φ(∑ z_{k,2} ⊗ … ⊗ z_{k,n+1}) & = (1 ⊗ φ^{−1}τ_{i}^{2}φ)(z) \\ & = ∑_{k} z_{k,1} ⊗ ζ^{2} ∑ z_{k,2} ⊗ … ⊗ z_{k,n+1}. \end{aligned}

Since the z_{k,1} are linearly independent, the terms ∑ z_{k,2} ⊗ … ⊗ z_{k,n+1} are in P^{n}(ζ) hence z ∈ P ⊗ P^{n}(ζ).

Now we show that a factorization as given in the following diagram exists

[figure: P^{n+1}(ζ); P ⊗ P^{n}(ζ); ι; P^{2}(−1); P ⊗ P.; ι; 1 ⊗ [.,.]; 1 ⊗ [.,.]]

The morphism 1⊗[., .] : P ⊗P^{n}(ζ) ⟶ P ⊗P is in C. Consider the braiding τ : P ⊗ P^{n}(ζ) ⟶ P^{n}(ζ) ⊗ P. Since it is a natural transformation the diagram

[figure: P ⊗ P^{n}(ζ); P^{n}(ζ) ⊗ P; τ; P ⊗ (P ⊗ … ⊗ P); (P ⊗ … ⊗ P) ⊗ P; φ]

commutes with φ = τ_{n} … τ_{1} = τ_{(P,P^{n})}, so τ(∑_{k} z_{k,1} ⊗ (∑ z_{k,2} ⊗ … ⊗ z_{k,n+1})) = τ_{n} … τ_{1}(∑_{k} z_{k,1} ⊗ z_{k,2} ⊗ … ⊗ z_{k,n+1}). Hence we get

τ(1 ⊗ [., .])(z) = ([., .] ⊗ 1)τ_{n} … τ_{1}(z)  (18)

and similarly

τ([., .] ⊗ 1)(z) = (1 ⊗ [., .])τ_{1} … τ_{n}(z)  (19)

for z ∈ P^{n+1}(ζ). This implies τ^{2}(1 ⊗ [., .])(z) = τ([., .] ⊗ 1)τ_{n} … τ_{1}(z) = (1 ⊗ [., .])τ_{1} … τ_{n}τ_{n} … τ_{1}(z) = (1 ⊗ [., .])ζ^{2n}(z) = (−1)^{2}(1 ⊗ [., .])(z), so that (1 ⊗ [., .])(z) is in P^{2}(−1) and thus ∑[z_{k,1}, [z_{k,2}, …, z_{k,n+1}]] is defined.

The second claim of the Proposition is proved in a symmetric way.

□

We continue with the

Proof. of Proposition 3.2:

We use (18), (19), and (16) to get

τ^{2}(1⊗ [., .]_{n})(z) =
= (1 ⊗ [., .]_{n})τ_{1} … τ_{n}τ_{n} … τ_{1}(z)
= (1 ⊗ [., .]_{n})τ_{1} … τ_{n−1}τ_{n}^{2}τ_{n−1}^{−1} … τ_{1}^{−1}τ_{1} … τ_{n−1}τ_{n−1} … τ_{1}(z)

= (1 ⊗ [., .]_{n})τ_{n} … τ_{2}τ_{1}^{2}τ_{2}^{−1} … τ_{n}^{−1}τ_{1} … τ_{n−2}τ_{n−1}^{2}τ_{n−2} … τ_{1}(z)

= (1 ⊗ [., .]_{n})(τ_{n} … τ_{2}τ_{1}^{2}τ_{2}^{−1} … τ_{n}^{−1})(τ_{n−1} … τ_{2}τ_{1}^{2}τ_{2}^{−1} … τ_{n−1}^{−1}) … (τ_{1}^{2})(z)

= (1 ⊗ [., .]_{n})(z)

for all z ∈ P^{n+1}(−1, ζ) hence (1⊗[., .]_{n})(z) is in P^{2}(−1) and [., [., .]_{n}]_{2}(z) is defined.

Now we prove that [., [., .]_{2}, .]_{n}τ_{i−1} … τ_{1} : P^{n+1}(−1, ζ) ∋ x⊗y_{1} ⊗…⊗ y_{n} ↦ [y_{1}, …, [x, y_{i}], …, y_{n}] ∈ P is well defined. Let z ∈ P^{n+1}(−1, ζ). Then we have τ_{1}^{2}τ_{2}^{−1} … τ_{i}^{−1}(z) = τ_{2}^{−1} … τ_{i}^{−1}(z) since τ_{2}^{−1} … τ_{i}^{−1} = 1 ⊗ τ_{1}^{−1} … τ_{i−1}^{−1}. If we represent y = τ_{2}^{−1} … τ_{i}^{−1}(z) = ∑ a_{i} ⊗ b_{i} ∈ P^{2} ⊗ P^{n−1} in shortest form, then the set {b_{i}} is linearly independent, so ∑ a_{i}⊗b_{i} = τ_{1}^{2}(∑ a_{i} ⊗b_{i}) = ∑ τ_{1}^{2}(a_{i})⊗b_{i}, hence τ_{1}^{2}(a_{i}) = a_{i} and y ∈ P^{2}(−1)⊗P^{n−1}. So we get τ_{i−1} … τ_{1}τ_{i} … τ_{2}(y) = τ_{i−1} … τ_{1}(z) ∈ P^{i−1} ⊗ P^{2}(−1) ⊗ P^{n−i} and (1 ⊗ … ⊗ [., .] ⊗ … ⊗ 1)τ_{i−1} … τ_{1}(z) ∈ P^{n} is defined.

By Lemma 8.2 we have (1 ⊗ … ⊗ [., .]_{2} ⊗ … ⊗ 1)(z) ∈ P^{n}(ζ), so that [., [., .]_{2}, .]_{n}τ_{i−1} … τ_{1}(z) is well defined. □

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[6] Bodo Pareigis: Symmetric Yetter-Drinfeld Categories are Trivial.

www.mathematik.uni-muenchen.de /personen/professoren.html.

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Mathematisches Institut der Universität, Theresienstr.39, 80333 München, Germany

 E-mail address: pareigis@rz.mathematik.uni-muenchen.de