SKEW-PRIMITIVE ELEMENTS OF QUANTUM GROUPS

AND BRAIDED LIE ALGEBRAS

BODO PAREIGIS

Herrn Professor Dr. H. Röhrl zum 70. Geburtstag gewidmet.

Contents

1. Quantum groups, Yetter-Drinfel’d algebras, and G-graded algebras 3 2. Skew primitive elements 5 3. Symmetrization of B_{n}-modules 8 4. Lie algebras 11 5. Properties of Lie algebras 13 6. Derivations and skew-symmetric endomorphisms 14 7. Lie structures on C_{p^{n}}-graded modules 18 References 20

In the study of Lie groups, of algebraic groups or of formal groups, the concept of Lie algebras plays a central role. These Lie algebras consist of the primitive elements. It is difficult to introduce a similar concept for quantum groups. Many important quantum groups have braided Hopf algebras as building blocks. As we will see most primitive elements live in these braided Hopf algebras. In [P1] and [P2] we introduced the concept of braided Lie algebras for this type of Hopf algebras. In this paper we will give a survey of and a motivation for this concept together with some interesting examples.

By the work of Yetter [Y] we know that the category of Yetter-Drinfel’d modules is in a sense the most general category of modules carrying a natural braiding on the tensor power of each module (instead of a symmetric structure). The study of algebraic structures in such a category is a generalization of the study of group graded algebraic structures. We will describe the braid structure in the category of Yetter-Drinfel’d modules, the concept of a Hopf algebra in this category, and explain the reason why we want to study such braided Hopf algebras.

Date: April 15, 1997.

1991 Mathematics Subject Classification. Primary 16A10.

One of the big obstacles in this theory is the fact, that the set of primitive elements P (H) of a braided Hopf algebra H does not form a Lie algebra in the ordinary or slightly generalized sense. We will show, however, that there is still an algebraic structure on P (H) consisting of partially defined n-ary bracket operations, satisfying certain generalizations of the anti-symmetry and Jacobi relations. We call this structure a braided Lie algebra. This will generalize ordinary Lie algebras, Lie super algebras, and Lie color algebras. Furthermore we will show that the universal enveloping algebra of a braided Lie algebra is again a braided Hopf algebra leading us back to quantum groups.

Primitive elements of an ordinary Hopf algebra L are elements x ∈ L satisfying Δ(x) = x ⊗ 1 + 1 ⊗ x. The set of primitive elements P (L) of L forms a Lie algebra induced by the Lie algebra structure [x, y] := xy − yx on L. In fact one verifies that Δ([x, y]) = [x, y] ⊗ 1 + 1 ⊗ [x, y] if x, y ∈ P (L).

Since primitive elements are cocommutative they can only generate a cocommutative Hopf subalgebra of L. More general elements, skew-primitive elements with Δ(x) = x ⊗ g + g′ ⊗ x, are needed to generate quantum groups or general (noncommutative noncocommutative) Hopf algebras. But the skew-primitive elements do not form a Lie algebra anymore.

Many quantum groups are Hopf algebras of the special form L = kG⋆H = kG⊗H where H is a braided graded Hopf algebra over a commutative group G [L, Ma, R, S]. In this situation the primitive elements of H are skew-primitive elements of L. So the structure of a braided Lie algebra on the set of primitive elements in H induces a similar structure on a subset of the skew-primitive elements of L.

The central idea leading to the structure of braided Lie algebras is the concept of symmetrization. For any module P in the category of Yetter-Drinfel’d modules the n-th tensor power P^{n} of P has a natural braid structure. We construct submodules P^{n}(ζ) ⊆ P^{n} for any nonzero ζ in the base field k, that carry a (symmetric) S_{n}- structure. This is essentially an eigenspace construction for a family of operators. The Lie algebra multiplications will be defined on these S_{n}-modules P^{n}(ζ) for primitive n-th roots of unity ζ.

In the group graded case, there is a fairly explicit construction of these symmetrizations. In the last section of this paper we describe them in the C_{p^{n}}-graded case for a cyclic group of prime power order.

Apart from the explicit examples of braided Lie algebras we gave in [P1] we showed in [P2] that the set of derivations Der(A) of an algebra A in YD_{K}^{K} forms a braided Lie algebra. This is based on the existence of inner hom-objects in YD_{K}^{K}. In Theorem 6.3 we will show that the category YD_{K}^{K} is a closed monoidal category. We also construct another large family of braided Lie algebras consisting of skew-symmetric endomorphisms of a Yetter-Drinfel’d module with a bilinear form. This generalizes the construction of Lie algebras of classical groups.

I wish to thank Peter Schauenburg for valuable conversations especially on Theorems 2.2 and 6.3.

1. Quantum groups, Yetter-Drinfel’d algebras, and G-graded algebras

Quantum groups arise from deformations of universal enveloping algebras of Lie algebras. They often have the following form.

Let K = kG be the group algebra of a commutative group. Let H be a Hopf algebra in the category of Yetter-Drinfel’d modules over K. Then the biproduct K ⋆ H is a Hopf algebra [R, Ma, FM], which in general is neither commutative nor cocommutative. More generally quantum groups are Hopf algebras of the form H_{−} ⋆ K ⋆ H_{+} where H_{+} and H_{−} are dual to each other [L, S].

We investigate the question what the primitive (Lie) elements of these quantum groups are and whether they carry a specific structure (of a Lie algebra).

Let us first introduce the concept of Yetter-Drinfel’d modules over a Hopf algebra K with bijective antipode (see [M] 10.6.10). A Yetter-Drinfel’d module or crossed module over a Hopf algebra K is a vector space M which is a right K-module and a right K-comodule such that

(1)  ∑(x · c)_{[0]} ⊗ (x · c)_{[1]} = ∑ x_{[0]} · c_{(2)} ⊗ S(c_{(1)})x_{[1]}c_{(3)}

for all x ∈ M and all c ∈ K. Here we use the Sweedler notation Δ(c) = ∑ c_{(1)} ⊗ c_{(2)} with Δ : K ⟶ K ⊗ K and δ(x) = ∑ x_{[0]} ⊗ x_{[1]} with δ : M ⟶ M ⊗ K. The Yetter-Drinfel’d modules form a category YD_{K}^{K} in the obvious way (morphisms are the K-module homomorphisms which are also K-comodule homomorphisms).

The most interesting structure on YD_{K}^{K} is given by its tensor products. It is well known that the tensor product M ⊗ N of two vector spaces which are K-modules is again a K-module (via the comultiplication or diagonal of K). If M and N are K-comodules then their tensor product is also a K-comodule (via the multiplication of K). If M and N are Yetter-Drinfel’d modules over K, then their tensor product is a Yetter-Drinfel’d module, too. So with this tensor product YD_{K}^{K} is a monoidal category.

A monoidal or tensor product structure on an arbitrary category C allows to define the notion of an algebra A with a multiplication ∇ : A ⊗ A ⟶ A which is associative and unitary (by u : k ⟶ A). Similarly one can define coalgebras in C. There is, however, a problem with defining a bialgebra or Hopf algebra H in C. In the compatibility condition between multiplication and comultiplication of H

∑(h · h′)_{(1)} ⊗ (h · h′)_{(2)} = ∑ h_{(1)} · h′_{(1)} ⊗ h_{(2)} · h′_{(2)}

one uses in the formation of the right hand side ∑ h_{(1)} ·h′_{(1)} ⊗h_{(2)} ·h′_{(2)} = (∇⊗∇)(1⊗ τ ⊗ 1)(∑ h_{(1)} ⊗ h_{(2)} ⊗ h′_{(1)} ⊗ h′_{(2)}) = (∇ ⊗ ∇)(1 ⊗ τ ⊗ 1)(Δ ⊗ Δ)(h ⊗ h′) a switch or exchange function τ : H ⊗ H ⟶ H ⊗ H in the category C.

There exists such a nontrivial morphism τ : M ⊗ N ⟶ N ⊗ M in the category YD_{K}^{K} of Yetter-Drinfel’d modules. It is given by

(2)  τ : M ⊗ N ∋ m ⊗ n ↦ ∑ n_{[0]} ⊗ m · n_{[1]} ∈ N ⊗ M.

This is a natural transformation with the additional property of a braiding which we will discuss later. So we know now how to define a Hopf algebra H in YD_{K}^{K}. Observe that these Hopf algebras are not ordinary Hopf algebras since the condition for the compatibility between multiplication and comultiplication involves the new switch morphism.

Given a Hopf algebra H in YD_{K}^{K} we can define the biproduct K ⋆ H [R] between K and H. The underlying vector space is the tensor product K ⊗ H. We denote the elements by ∑ c_{i} ⊗ h_{i} =: ∑ c_{i}#h_{i}. The (smash product) multiplication is given by

(3)  (c#h)(c′#h′) := ∑ cc′_{(1)}#(h · c′_{(2)})h′

and the (smash coproduct) comultiplication is given by

(4)  Δ(c#h) = ∑(c_{(1)}#(h_{(1)})_{[0]}) ⊗ (c_{(2)}(h_{(1)})_{[1]}#h_{(2)}).

If K is a Hopf algebra and H is a Hopf algebra in YD_{K}^{K} then K ⊗ H becomes a Hopf algebra with this multiplication and comultiplication, called the biproduct K ⋆ H (see [R, M, Ma]).

We will be mainly interested in the case where K = kG is the group ring of a commutative group. It is well known that the kG-comodules are precisely the Ggraded vector spaces ([M] Example 1.6.7). We denote this category by M^{kG}. From the comodule structure on a G-graded vector space M = ⊕_{h∈G}M_{h} we can construct a kGmodule structure such that M becomes a Yetter-Drinfel’d module. This construction depends on a bicharacter χ : G×G ⟶ k^{×} given by a group homomorphism χ : G⊗_{Z}G ⟶ k^{×}. Then it is easy to verify that M is in YD_{kG}^{kG} with the kG-module structure

m_{h} · g := χ(h, g)m_{h}

for homogeneous elements m_{h} ∈ M_{h}, h ∈ G and g ∈ G. So any bicharacter χ defines a functor M^{kG} ⟶ YD_{kG}^{kG}. This functor preserves tensor products. In particular any algebra or coalgebra in M^{kG} is also an algebra resp. coalgebra in YD_{kG}^{kG}. Since M^{kG} can be considered as a monoidal subcategory of YD_{kG}^{kG} via χ and thus has a switch map τ : M ⊗ N ⟶ N ⊗ M we can also define Hopf algebras in M^{kG} and they are also preserved by the functor induced by χ. In this situation τ turns out to be simply

τ(m_{h} ⊗ n_{g}) = χ(h, g)n_{g} ⊗ m_{h}.

Although one may define Yetter-Drinfel’d categories YD_{K}^{K} for arbitrary Hopf algebras K with bijective antipode hence in particular for arbitrary group rings kG (where G is not commutative) the above functor that induces Yetter-Drinfel’d modules from kG-comodules with a bicharacter can only be constructed for commutative groups G.

If A is an algebra in M^{kG} then kG#A carries the induced algebra structure

(5)  (g#a_{h})(g′#a_{h′}) = gg′#χ(h, g′)a_{h}a_{h′}.

If H is a Hopf algebra in M^{kG} then the Hopf algebra L = kG ⋆ H has the comultiplication

(6)  Δ_{L}(g#a_{h}) = ∑_{k∈G}(g#b_{k}) ⊗ (gk#b_{hk^{−1}})

if Δ_{H}(a_{h}) = ∑_{k∈G} b_{k} ⊗b_{hk^{−1}} where lower indices stand for the degree of homogeneous elements. Example 1.1. Let G = C_{3} = ⟨t⟩ be the cyclic group with three elements, generator t, and let χ(t, t) = ξ ∈ k be a primitive 3-rd root of unity. Then H = k[x]/(x^{3}) with Δ(x) = x ⊗ 1 + 1 ⊗ x is a Hopf algebra in YD_{kG}^{kG} where x has degree t, the generator of C_{3}.

To see that k[x]/(x^{3}) is a Hopf algebra in YD_{kG}^{kG} we check that Δ(x^{3}) = Δ(x)^{3}. We observe that (x ⊗ 1)(1 ⊗ x) = x ⊗ x but (1 ⊗ x)(x ⊗ 1) = ξx ⊗ x. So we have Δ(x)^{3} = (x⊗1+1⊗x)^{3} = x^{3} ⊗1+(1+ξ +ξ^{2})(x⊗x^{2})+(1+ξ +ξ^{2})(x^{2} ⊗x)+(1⊗x^{3}) = 0 = Δ(0) = Δ(x^{3}).

This example shows, that products and powers of primitive elements behave totally different in YD_{kG}^{kG} from how they behave in Vec, the category of vector spaces. This new behavior is central to the following observations on Lie algebras.

The biproduct kC_{3} ⋆ k[x]/(x^{3}) is isomorphic to the Hopf algebra k⟨t, x⟩/(t^{3} − 1, x^{3}, xt − ξtx) if we associate t#1 and t resp. 1#x and x.

2. Skew primitive elements

The last example shows the importance of elements x ∈ H (a Hopf algebra in YD_{K}^{K}) with Δ(x) = x⊗1+1⊗x. These elements are called primitive elements. They act like derivations. The primitive elements, homogeneous of degree g ∈ G, form a vector space P_{g}(H). In [P1] (after the proof of 3.2) and [P2] Lemma 5.1 we proved Lemma 2.1. The set of primitive elements of a Hopf algebra H in YD_{K}^{K} is a Yetter-Drinfel’d module P (H).

If K = kG and H ∈ M^{kG} then P (H) = ⨁_{g∈G} P_{g}(H) is also in M^{kG} (via the same bicharacter χ).

In general and especially in Hopf algebras of the form K ⋆ H we have to consider more general conditions for primitive elements. An element g ≠ 0 in a Hopf algebra L (in Vec) is called a group-like element if

Δ(g) = g ⊗ g.

This implies ε(g) = 1. By (4) a group-like element g ∈ K defines a group-like element g#1 ∈ L = K ⋆ H.

Let g, g′ ∈ L be group-like elements. An element x ∈ L is called a (g′, g)-primitive element or a skew primitive element if

Δ(x) = x ⊗ g + g′ ⊗ x.

This implies ε(x) = 0. Observe that a primitive element is (1, 1)-primitive, since 1 is a group-like element.

The (g′, g)-primitive elements form a vector space P_{(g′,g)}(L). For x ∈ P_{(g′,g)}(L) we have

Δ(g′^{−1}x) = (g′^{−1} ⊗ g′^{−1})(x ⊗ g + g′ ⊗ x) = g′^{−1}x ⊗ g′^{−1}g + 1 ⊗ g′^{−1}x,

so we get an isomorphism P_{(g′,g)}(L) ∋ x ↦ g′^{−1}x ∈ P_{(1,g′^{−1}g)}(L). Thus it suffices to study the spaces P_{(1,g)}(L).

If f : L ⟶ L′ is a Hopf algebra homomorphism then f obviously preserves group-like elements g ∈ L and (g′, g)-primitive elements x ∈ L are mapped into (f(g′), f(g))-primitive elements f(x) ∈ L′. So we get a homomorphism f : P_{(g′,g)}(L) ⟶ P_{(f(g′),f(g))}(L′) and in particular a homomorphism f : P_{(1,g)}(L) ⟶ P_{(1,f(g))}(L′).

Now let L = K ⋆ H and let h ∈ H be primitive and homogeneous of degree g ∈ K, a group-like element in K, i.e. Δ_{H}(h) = h ⊗ 1 + 1 ⊗ h and δ(h) = ∑ h_{[0]} ⊗ h_{[1]} = h ⊗ g with respect to the K-comodule structure δ : H ⟶ H ⊗ K of H. Then by (6)

Δ_{L}(1#h) = 1#h ⊗ g#1 + 1#1 ⊗ 1#h

hence 1#h ∈ P_{(1,g)}(L). So the following is a monomorphism

P_{g}(H) ∋ h ↦ 1#h ∈ P_{(1,g)}(L).

Theorem 2.2. Let K be a Hopf algebra with bijective antipode and H be a Hopf algebra in YD_{K}^{K}. Let L = K ⋆ H. For every group-like element g ∈ K we have

P_{(1,g)}(L) = P_{(1,g)}(K)#1 ⊕ 1#P_{g}(H).

Proof. Let π : K ⋆ H ⟶ K be defined by π(c#h) = cε_{H}(h) and ι : K ⟶ K ⋆ H by ι(c) = c#1. Then one checks easily that π and ι are Hopf algebra homomorphisms and that πι = id_{K}. Thus π and ι preserve group-like elements and skew-primitive elements. In particular we have for any group-like element g ∈ K that ι(g) = g#1 ∈ L is group-like. We identify the group-like elements g in K with the group-like elements ι(g) in L. For any (1, g)-primitive element x ∈ L the element π(x) ∈ K is also (1, g)-primitive. Furthermore π and ι define a direct sum decomposition K ⋆ H = Im(ι) ⊕ Ker(π).

We have already seen 1#P_{g}(H) ⊆ P_{(1,g)}(L). Furthermore if c ∈ P_{(1,g)}(K) then ι(c) = c#1 ∈ P_{(1,g)}(L) so that P_{(1,g)}(L) ⊇ P_{(1,g)}(K)#1 + 1#P_{g}(H).

Given a (1, g)-primitive element x ∈ L = K ⋆ H for g ∈ K. We study how x decomposes with respect to the direct sum decomposition x = ιπ(x) + (x − ιπ(x)). The element π(x) is (1, g)-primitive since x is (1, g)-primitive. So ιπ(x) ∈ P_{(1,g)}(L)#1.

Furthermore ιπ(x) ∈ P_{(1,g)}(L) implies y := x − ιπ(x) ∈ P_{(1,g)}(L) ∩ Ker(π). We have Δ_{L}(y) = y ⊗ g + 1 ⊗ y and (π ⊗ 1)Δ_{L}(y) = 1 ⊗ y, since y ∈ Ker(π). Let y = ∑ c#h ∈ L = K ⋆ H then

\begin{aligned}1 ⊗ y = (π ⊗ 1)Δ_{L}(y) & = (π ⊗ 1)(∑(c_{(1)}#(h_{(1)})_{[0]}) ⊗ (c_{(2)}(h_{(1)})_{[1]}#h_{(2)})) \\ & = ∑ c_{(1)}ε_{H}((h_{(1)})_{[0]}) ⊗ c_{(2)}(h_{(1)})_{[1]}#h_{(2)}. \end{aligned}

We apply 1 ⊗ ε_{K}#1 and get

\begin{aligned}∑ 1#ε_{K}(c)h & = (1 ⊗ ε_{K}#1)(∑ 1 ⊗ c#h) \\ & = ∑ c_{(1)}ε_{K}(c_{(2)})#ε_{H}((h_{(1)})_{[0]})ε_{K}((h_{(1)})_{[1]})h_{(2)} = ∑ c#h, \end{aligned}

so we know

y = x − ιπ(x) = 1#h

for some h ∈ H.

Since y is skew-primitive we get 0 = ε_{L}(y) = ε_{H}(h). Furthermore Δ_{L}(y) = y ⊗ g + 1 ⊗ y = 1#h ⊗ g#1 + 1#1 ⊗ 1#h = ∑(1#(h_{(1)})_{[0]}) ⊗ ((h_{(1)})_{[1]}#h_{(2)}) implies

h ⊗ g#1 + 1_{H} ⊗ 1_{K}#h = ∑(h_{(1)})_{[0]} ⊗ (h_{(1)})_{[1]}#h_{(2)}

so that by applying 1 ⊗ ε#1 we get h ⊗ 1 + 1 ⊗ h = ∑ h_{(1)} ⊗ h_{(2)} = Δ_{H}(h). i.e. h is primitive. Furthermore with 1 ⊗ 1#ε we get h ⊗ g = ∑ h_{[0]} ⊗ h_{[1]} = δ(h) where δ : H ⟶ H ⊗ K is the given comodule struktur of H. Thus h is homogeneous of degree g so that

x − ιπ(x) ∈ 1#P_{g}(H).

So we have shown P_{(1,g)}(L) ⊆ P_{(1,g)}(K)#1 ⊕ 1#P_{g}(H). □

Corollary 2.3. Let G be a commutative group, χ be a bicharacter of G. Let H be a Hopf algebra in M^{kG}. Let L = kG ⋆ H. For every g ∈ G we have

P_{(1,g)}(L) = k(g − 1)#1 ⊕ 1#P_{g}(H).

Proof. The only thing to check is P_{(1,g)}(kG) = k(g − 1). We have Δ(g − 1) = g ⊗ g − 1 ⊗ 1 = (g − 1) ⊗ g + 1 ⊗ (g − 1). Conversely if x = ∑ α_{i}g_{i} is in P_{(1,g_{0})}(kG) then by comparing coefficients one obtains x = α_{0}(g_{0} − 1). □

In particular we have P_{(1,1)}(kG ⋆ H) = 1#P_{1}(H).

In order to study the (g′, g)-primitive elements in L = kG ⊗ H it suffices now to study P (H). We are interested in obtaining an algebraic structure on P (H) similar to the Lie algebra structure on the primitive elements of an ordinary Hopf algebra in Vec. The usual Lie multiplication on P (H) induced by the multiplication of the Hopf algebra H in YD_{kG}^{kG} cannot be used as the following example shows. Example 2.4. Let x, y ∈ P (H) and H a Hopf algebra in YD_{K}^{K}. If we define [x, y] := xy − ∇τ(x ⊗ y) then

Δ_{H}[x, y] = [x, y] ⊗ 1 + 1 ⊗ [x, y] + x ⊗ y − τ^{2}(x ⊗ y).

So in general the element [x, y] ∈ H will not be a primitive element unless τ^{2}(x⊗y) = x ⊗ y.

However, in Example 1.1 we found the fact that x^{3} may be primitive if x is primitive.

3. Symmetrization of B_{n}-modules

We want to find a reasonable algebraic structure (of a generalized Lie algebra) on the set of primitive elements of a Hopf algebra in the category YD_{K}^{K} of Yetter-Drinfel’d modules. We also want to get a generalized Lie algebra from every (noncommutative) algebra A in YD_{K}^{K} by suitable definition of Lie multiplications with the help of the algebra multiplication. We expect that the Lie products are (skew-)commutative and satisfy some kind of Jacobi identity. The (skew-)commutativity of an ordinary Lie algebra P results from the action of S_{2} (the symmetric group) on P ⊗ P. In the case of an algebra A made into a Lie algebra this skew-commutativity results from the following composition of maps

[., .] = ∇ ∘ SkSymm : A ⊗ A ⟶ SkSymm(A ⊗ A) ⟶ A

where SkSymm denotes the set of anti-symmetric tensors in A ⊗ A and the antisymmetrization process itself. In general the Lie multiplication must only be defined on SkSymm(P ⊗ P) since

[x, y] = [x ⊗ y] = \frac{1}{2}[x ⊗ y − y ⊗ x]

where \frac{1}{2}(x ⊗ y − y ⊗ x) ∈ SkSymm(P ⊗ P).

This is a special case of the following more general observation. If a finite group G acts on a module M then the map M ∋ m ↦ ∑_{g∈G} gm ∈ G-Inv(M) sends any m ∈ M into the set of G-invariant elements G-Inv(M) = {m ∈ M|∀g ∈ G : gm = m}. This process is only possible for finite groups G. In the above case S_{2} acts on P ⊗P by σ(x⊗y) = −y⊗x. We want to use this process to define a generalized Lie algebra. We will not restrict ourselves to binary multiplications, since the Jacobi identity indicates that higher order multiplications might be of interest, too. Furthermore generalized Lie multiplications will only be partially defined, on subspaces of P ⊗ … ⊗ P.

The reason for the fact that the Lie bracket [x, y] of two primitive elements x, y ∈ P (H) is not primitive in Example 2.4 results from the following observation. The operation of the switch map τ : P ⊗ P ⟶ P ⊗ P induces only an operation of the group ℤ rather than ℤ/(2) on P ⊗ P. Observe that ℤ = B_{2} is the 2-nd braid group, whereas ℤ/(2) = S_{2} is the 2-nd symmetric group.

The switch morphism τ : P ⊗ P ⟶ P ⊗ P satisfies the (quantum-) Yang-Baxter equation

(τ ⊗ 1)(1 ⊗ τ)(τ ⊗ 1) = (1 ⊗ τ)(τ ⊗ 1)(1 ⊗ τ)

hence it induces the action of the n-th braid group B_{n} on P^{n}. The braid group B_{n} is generated by elements τ_{1}, …, τ_{n−1} (τ_{i} acting on P^{n} by switching the i-th and (i+1)-st component) and has relations

\begin{matrix}τ_{i}τ_{j} = τ_{j}τ_{i} for |i − j| ≥ 2, \\ τ_{i}τ_{i+1}τ_{i} = τ_{i+1}τ_{i}τ_{i+1}. \end{matrix}

So the n-th symmetric group S_{n} with generators σ_{1}, …, σ_{n−1} is a canonical quotient of B_{n} (by τ_{i} ↦ σ_{i}) by τ_{i}^{2} = id. This observation is the reason that τ is called a braiding for the category YD_{K}^{K}.

Any S_{n}-module is a B_{n}-module by the residue homomorphism B_{n} ⟶ S_{n}. Conversely, is there a way to construct an S_{n}-module from a given B_{n}-module in a canonical way? Why do we want to consider S_{n}-modules rather than B_{n}-modules? The main reason is, as we saw above, that B_{n} is an infinite group and S_{n} is a finite group.

The first question leads to the process of symmetrization of a B_{n}-module or braid module as follows. Any element ζ ∈ k^{×} induces an algebra automorphism ζ : kB_{n} ⟶ kB_{n} by ζ(τ_{i}) := ζ · τ_{i}, due to the fact that the relations for the braid group are homogeneous. The algebra homomorphism νζ : kB_{n} ⟶ kB_{n} ⟶ kS_{n} induces a forgetful functor_{kS_{n}}M ⟶ _{kB_{n}}M which has the right adjoint Hom_{kB_{n}}(kS_{n}, −) : _{kB_{n}}M ⟶ _{kS_{n}}M. So any braid module M and any ζ ∈ k^{×} induces a module Hom_{kB_{n}}(kS_{n}, M) over the symmetric group. Since the algebra homomorphism νζ is surjective we get a submodule

M(ζ) := Hom_{kB_{n}}(kS_{n}, M) ⊆ Hom_{kB_{n}}(kB_{n}, M) = M.

In [P2] following Definition 2.3 we proved

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

and computed the action of S_{n} on M(ζ) as

(7)  σ_{i}(m) = ζ^{−1}τ_{i}(m).

So we have kS_{n}-submodules M(ζ) ⊆ M for every ζ ∈ k^{×}. Since they are constructed similar to eigenspaces for the eigenvalues ζ^{2} they form direct sums in M.

If P ∈ YD_{K}^{K} then P^{n} = P ⊗ … ⊗ P is in YD_{K}^{K} and B_{n} acts on P^{n}. The symmetrization with respect to ζ ∈ k^{×} gives a module P^{n}(ζ) ∈ YD_{K}^{K} ([P2] Theorem 2.5).

Now we consider the special case K = kG and a G-graded vector space M ∈ M^{kG} for a commutative group G with a bicharacter χ : G ⊗_{Z} G ⟶ k^{×}. As we saw in the first section M can be considered as a Yetter-Drinfel’d module in YD_{kG}^{kG}. The space M decomposes into homogeneous components M = ⊕_{g∈G}M_{g}. The components themselves are again Yetter-Drinfel’d modules, since M_{g} is G-graded.

Assume that M is in YD_{kG}^{kG} and that B_{n} operates on M by morphisms in the category YD_{kG}^{kG}. Then B_{n} operates also on the homogeneous components M_{g}. Since M(ζ) is a Yetter-Drinfel’d module it decomposes into homogeneous components M(ζ)_{g} = M_{g}(ζ).

Let P ∈ M^{kG} and ∑ x_{g_{1}} ⊗ … ⊗ x_{g_{n}} ∈ P^{n} be an element with the x_{g_{i}} homogeneous of degree g_{i} ∈ G. Then the action of τ_{i}^{2} ∈ B_{n} looks particularly simple

τ_{i}^{2}(∑ x_{g_{1}} ⊗ … ⊗ x_{g_{n}}) = χ(g_{i+1}, g_{i})χ(g_{i}, g_{i+1})(∑ x_{g_{1}} ⊗ … ⊗ x_{g_{n}}).

One shows that the element ∑ x_{g_{1}} ⊗ … ⊗ x_{g_{n}} is in P^{n}(ζ) iff χ(g_{j}, g_{i})χ(g_{i}, g_{j})(∑ x_{g_{1}} ⊗ … ⊗ x_{g_{n}}) = ζ^{2}(∑ x_{g_{1}} ⊗ … ⊗ x_{g_{n}}) for all i ≠ j. This implies χ(g_{j}, g_{i})χ(g_{i}, g_{j}) = ζ^{2} for all i ≠ j, if the given element is not zero. In [P2] Proposition 7.2 we proved Proposition 3.1. Let ζ ∈ k^{∗} be given. Then

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

Here (g_{1}, …, g_{n}) is called a ζ-family if χ(g_{i}, g_{j})χ(g_{j}, g_{i}) = ζ^{2} for all i ≠ j. Example 3.2. We close this section with some important examples.

1. If G = {0} then χ(0, 0) = 1 and YD_{kG}^{kG} ≅ Vec. The braiding in Vec is the usual switch map τ(x ⊗ y) = y ⊗ x hence τ^{2} = id. If M is a B_{n}-module then the equation ϕ^{−1}τ_{i}^{2}ϕ(m) = ζ^{2}m has a non-trivial solution only if ζ = ±1. Hence P^{n}(ζ) = 0 for all ζ ≠ ±1. So the only nontrivial ”symmetrization” is P^{n}(1) = P^{n}(−1) = P^{n}. Since τ^{2} = id the braid group B_{n} acts on P^{n} by ordinary permutations generated by the action of the canonical switch map. If ϕ ∈ B_{n} and σ its canonical image in S_{n} then the action (7) given by the symmetrization on z ∈ P^{n} is σ(z) = sgn(σ)ϕ(z), where ϕ is acting as ordinary switch permutation.

2. Let G = ℤ/2ℤ = {0, 1} the cyclic group of order two with the (only) nontrivial bicharacter χ(i, j) = (−1)^{ij}. Then M^{kG} is the category of (2-graded) super vector spaces with the braiding τ(x_{i} ⊗ y_{j}) = (−1)^{ij}y_{j} ⊗ x_{i}. Again τ^{2} = id so that P^{n}(ζ) = 0 for all ζ ≠ ±1. The only symmetrization is P^{n}(1) = P^{n}(−1) = P^{n}.

3. Let G be an arbitrary finite abelian group with bicharacter χ such that χ(h, g) = χ(g, h)^{−1}. Then M^{kG} is the category of (G-graded) color vector spaces with the braiding τ(x_{h} ⊗ y_{g}) = χ(h, g)y_{g} ⊗ x_{h}. Again τ^{2} = id so that P^{n}(ζ) = 0 for all ζ ≠ ±1. The only symmetrization is P^{n}(1) = P^{n}(−1) = P^{n}.

4. The first interesting example is G = ℤ/3ℤ = {0, 1, 2} with the bicharacter χ(i, j) = ξ^{ij} where ξ is a primitive 3-rd root of unity. Then M^{kG} is the category of 3-graded vector spaces with the braiding τ(x_{i} ⊗ y_{j}) = ξ^{ij}y_{j} ⊗ x_{i}. The homogeneous elements m ∈ P^{n}(ζ) = ⨁_{{(g_{1},...,g_{n}) ζ -family}} P_{g_{1}} ⊗ … ⊗ P_{g_{n}} for P ∈ M^{kG} have to satisfy ϕ^{−1}τ^{2}ϕ(m) = ξ^{2ij}m = ζ^{2}m. The possibilities for ζ are 1, −1, ξ, −ξ, ξ^{2}, −ξ^{2}. By computing all possible ζ-families we get for example

(P ⊗ P)(−1) = (P ⊗ P_{0}) + (P_{0} ⊗ P),
(P ⊗ P ⊗ P)(ξ) = (P_{1} ⊗ P_{1} ⊗ P_{1}) ⊕ (P_{2} ⊗ P_{2} ⊗ P_{2}),
P^{6}(−ξ) = P_{1}^{6} ⊕ P_{2}^{6},
(P ⊗ P ⊗ P)(ξ^{2}) = 0,
P^{6}(−ξ^{2}) = 0.

The particular choice of the number of tensor factors in this example will become clear in the next section. The action of the symmetric group on these symmetrizations is

σ(x_{i} ⊗ y_{j}) = −y_{j} ⊗ x_{i},
σ_{1}(x_{1} ⊗ y_{1} ⊗ z_{1}) = y_{1} ⊗ x_{1} ⊗ z_{1},

i.e. the ordinary permutation – and this holds for all elements in S_{3} and also for elements in P_{2} ⊗ P_{2} ⊗ P_{2},

σ_{1}(x_{1} ⊗ y_{1} ⊗ z_{1} ⊗ u_{1} ⊗ v_{1} ⊗ w_{1}) = −y_{1} ⊗ x_{1} ⊗ z_{1} ⊗ u_{1} ⊗ v_{1} ⊗ w_{1}.

4. Lie algebras

Let P be a Yetter-Drinfel’d module in YD_{K}^{K}. Then P^{n}(ζ) is an S_{n}-module. We will have to consider morphisms [ ] : P^{n}(ζ) ⟶ P in YD_{K}^{K}. If we suppress the summation index and the summation sign then we may write the bracket operation on elements z = x_{1} ⊗ … ⊗ x_{n} ∈ P^{n}(ζ) as [x_{1}, …, x_{n}] := [z]. Furthermore we define

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

Observe that the components x_{1}, …, x_{n} in these expressions are interchanged and changed according to the action of the braid group resp. the symmetric group on P^{n}(ζ), so x_{σ(1)} ⊗ … ⊗ x_{σ(n)} is only a symbolic expression, not the usual permutation of the tensor factors given by the permutation of the indices.

We need another submodule of P^{n} whose special properties will not be investigated. Define

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 YD_{K}^{K}, P^{n+1}(−1, ζ) is again an object in YD_{K}^{K}.

For z = x⊗y_{1} ⊗…⊗y_{n} ∈ P^{n+1}(−1, ζ) we write y_{1} ⊗…⊗y_{i−1} ⊗x⊗y_{i} ⊗…⊗y_{n} := τ_{i−1} … τ_{1}(z). If the morphisms [ ]_{n} : P^{n} ⟶ P and [ ]_{2} : P^{2} ⟶ P are suitably defined then we write

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

Now we have the tools to give the definition of a braided Lie algebra. Definition 4.1. A Yetter-Drinfel’d 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 ≠ 1 is called a braided Lie algebra or a Lie algebra in YD_{K}^{K} if the following identities hold:

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

[z] = [σ(z)],

(2) (1. Jacobi identity) for all n ∈ ℕ, for all primitive n-th roots of unity ζ ≠ 1, and for all z = x_{1} ⊗ … ⊗ x_{n+1} ∈ 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 notation (8) (and where (1 … i) is a cycle in S_{n}),

(3) (2. Jacobi identity) for all n ∈ ℕ, for all primitive n-th roots of unity ζ ≠ 1, 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 notation (9).

Observe that the bracket operations are only partially defined and should not be considered as multilinear operations, since P^{n}(ζ) ⊆ P^{n} is just a submodule in YD_{K}^{K} and does not necessarily decompose into an n-fold tensor product. The elements in P^{n}(ζ) are, however, of the form z = ∑_{k} x_{k,1} ⊗ … ⊗ x_{k,n}.

Clearly the braided Lie algebras in YD_{K}^{K} form a category LYD_{K}^{K}. Before we investigate its properties we discuss some examples.

Example 4.2. 1. If G = {0} as in Example 3.2.1. then the only required morphism for a braided Lie algebra P is [ ] : P^{2}(−1) ⟶ P since −1 is a primitive 2-nd root of unity. This is the usual bracket operation of Lie algebras. The action of B_{2} on P ⊗ P is given by the canonical switch map τ(x ⊗ y) = y ⊗ x. The induced action of S_{2} with respect to ζ = −1 is then

σ(x ⊗ y) = −y ⊗ x

by (7). Thus axiom 1. gives [x, y] = [σ(x ⊗ y)] = −[y, x], the usual anti-symmetry relation. With this action of S_{2} on P^{2}(−1) = P ⊗P one gets the usual Jacobi identity from both braided Jacobi identities.

2. Let G = ℤ/2ℤ = {0, 1} with the nontrivial bicharacter χ(i, j) = (−1)^{ij}. In Example 3.2.2. we saw that the only non-trivial symmetrization occurs with respect to ζ = −1, a primitive 2-nd root of unity. So the only bracket is defined on (P ⊗ P)(−1) = P ⊗ P. The operation of B_{2} on P ⊗ P is the braid action and σ(x_{i} ⊗ y_{j}) = (−1)(−1)^{ij}y_{j} ⊗ x_{i}. So we get [x_{i}, y_{j}] = [σ(x_{i} ⊗ y_{j})] = −[y_{j}, x_{i}] if at least one of the degrees i or j is zero and we get [x_{1}, y_{1}] = [σ(x_{1} ⊗ y_{1})] = [y_{1}, x_{1}]. In this case we get the notion of Lie super algebras since the braided Jacobi identities translate to the Jacobi identity for Lie super algebras.

3. Let G be an arbitrary finite abelian group with bicharacter χ such that χ(h, g) = χ(g, h)^{−1}. Again we get only one bracket operation [ ] : P ⊗P ⟶ P and anti-symmetry and Jacobi identities translate to those for Lie color algebras.

4. The example G = ℤ/3ℤ = {0, 1, 2} with bicharacter χ(i, j) = ξ^{ij} where ξ is a primitive 3-rd root of unity has three bracket operations

[ ] : P^{2}(−1) = (P ⊗ P_{0}) + (P_{0} ⊗ P) ⟶ P,

[ ] : P^{3}(ξ) = (P_{1} ⊗ P_{1} ⊗ P_{1}) ⊕ (P_{2} ⊗ P_{2} ⊗ P_{2}) ⟶ P,
[ ] : P^{6}(−ξ) = P_{1}^{6} ⊕ P_{2}^{6} ⟶ P.

Here the 1. Jacobi identity means for example

[x_{1}, [x_{2}, x_{3}, x_{4}]] + [x_{2}, [x_{1}, x_{3}, x_{4}]] + [x_{3}, [x_{1}, x_{2}, x_{4}]] + [x_{4}, [x_{1}, x_{2}, x_{3}]] = 0,

and the 2. Jacobi identity

[x, [y_{1}, y_{2}, y_{3}]] = [[x, y_{1}], y_{2}, y_{3}] + [y_{1}, [x, y_{2}], y_{3}] + [y_{1}, y_{2}, [x, y_{3}]].

Further explicit examples of braided Lie algebras can be found in [P1].

5. Properties of Lie algebras

The definition of braided Lie algebras, although it generalizes the notion of the known Lie algebras, Lie super algebras, and Lie color algebras, gains its interest from the properties that these Lie algebras have. We cite some of these properties in brief. Theorem 5.1. ([P2] 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  defined by  [z] := ∑_{σ∈S_{n}} ∇^{n}σ(z).

for all n ∈ ℕ and all roots of unity ζ ≠ 1 in k^{×}.

This defines a functor –^{L} : AYD_{K}^{K} ⟶ LYD_{K}^{K} from the category AYD_{K}^{K} of algebras in YD_{K}^{K} to LYD_{K}^{K}.

In [P2] Theorem 5.3 we proved Theorem 5.2. For any algebra A the morphism p : A ∋ a ↦ a ⊗ 1 + 1 ⊗ a ∈ A ⊗ A is a Lie algebra homomorphism in YD_{K}^{K}.

An easy consequence of this theorem is Theorem 5.3. ([P2] Corollary 5.4) Let H be a Hopf algebra in YD_{K}^{K}. Then the set of primitive elements P (H) forms a Lie algebra in YD_{K}^{K}.

This defines a functor P : HYD_{K}^{K} ⟶ LYD_{K}^{K} from the category HYD_{K}^{K} of Hopf algebras in YD_{K}^{K} to LYD_{K}^{K}. This is the most interesting result which solves the question for the algebraic structure of the primitive elements of a Hopf algebra in YD_{K}^{K}. In particular the braided Lie brackets live also on the set of skew primitive elements K ⋆ H as partially defined operations. Theorem 5.4. The functor –^{L} : AYD_{K}^{K} ⟶ LYD_{K}^{K} has a left adjoint U : LYD_{K}^{K} ⟶ AYD_{K}^{K}, called the universal enveloping algebra. Theorem 5.5. ([P2] Theorem 6.1) The universal enveloping algebra U(P) of a braided Lie algebra P is a Hopf algebra in YD_{K}^{K}.

This defines a left adjoint functor U : LYD_{K}^{K} ⟶ HYD_{K}^{K} to P : HYD_{K}^{K} ⟶ LYD_{K}^{K}. Example 5.6. The one dimensional vector space kx considered as a ℤ/3ℤ-graded space with x of degree 1 ∈ G = ℤ/3ℤ = {0, 1, 2} is a braided Lie algebra in YD_{kG}^{kG} with [x ⊗ x ⊗ x] = 0 and [x ⊗ x ⊗ x ⊗ x ⊗ x ⊗ x] = 0. The universal enveloping algebra of kx is k[x]/(x^{3}), the Hopf algebra discussed in Example 1.1.

6. Derivations and skew-symmetric endomorphisms

We shall give two examples which show how to construct large families of Lie algebras from Yetter-Drinfel’d algebras and from Yetter-Drinfel’d modules with a bilinear form in a similar way as one does for classical Lie algebras.

For this purpose we need inner hom-objects in YD_{K}^{K}. Let V, W be Yetter-Drinfel’d modules in YD_{K}^{K}. Then Hom(V, W) is a right K-module by

(10)  (fh)(v) = f(vS^{−1}(h_{(2)}))h_{(1)}.

This is equivalent to

(11)  (fh_{(1)})(vh_{(2)}) = f(vh_{(3)}S^{−1}(h_{(2)}))h_{(1)} = f(v)h,

i.e. the evaluation Hom(V, W) ⊗ V ⟶ W is a K-module homomorphism. We define a map δ_{0} : Hom(V, W) ⟶ Hom(V, W ⊗ K) by

(12)  δ_{0}(f)(v) := f(v_{[0]})_{[0]} ⊗ f(v_{[0]})_{[1]}S(v_{[1]})

that “dualizes” the right module structure on Hom(V, W).

Let hom(V, W) be the pullback (in Vec) in the diagram

[figure: hom(V, W); Hom(V, W) ⊗ K; δ_{1}; Hom(V, W); Hom(V, W ⊗ K); δ_{0}; ι; j]

hom(V, W) thus can be written as

(13) hom(V, W) = {f ∈Hom(V, W)|∃ ∑ f_{0} ⊗ f_{1} ∈ Hom(V, W) ⊗ K∀v ∈ V : ∑ f_{0}(v) ⊗ f_{1} = ∑ f(v_{[0]})_{[0]} ⊗ f(v_{[0]})_{[1]}S(v_{[1]}) = δ_{0}(f)(v)}

Lemma 6.1. hom(V, W) is a K-comodule.

Proof. Since K is faithfully flat we get that again is a pullback. So we get a uniquely determined homomorphism δ : hom(V, W) ⟶ hom(V, W) ⊗ K such that

[figure: hom(V, W) ⊗ K; Hom(V, W) ⊗ K ⊗ K; δ_{1} ⊗ K; Hom(V, W) ⊗ K; Hom(V, W ⊗ K) ⊗ K; δ_{0} ⊗ K; ι ⊗ K; j ⊗ K]

[figure: hom(V, W); δ; (1 ⊗ Δ)δ_{1}; δ_{1}; hom(V, W) ⊗ K; Hom(V, W) ⊗ K ⊗ K; δ_{1} ⊗ K; Hom(V, W) ⊗ K; Hom(V, W ⊗ K) ⊗ K; δ_{0} ⊗ K; ι ⊗ K; j ⊗ K]

commutes. To show that the outer diagram commutes we compute

((δ_{0} ⊗ 1)δ_{1}(f))(v) = (δ_{0} ⊗ 1)(f_{0} ⊗ f_{1})(v)
= (δ_{0}(f_{0}))(v) ⊗ f_{1}
= f_{0}(v_{[0]})_{[0]} ⊗ f_{0}(v_{[0]})_{[1]}S(v_{[1]}) ⊗ f_{1}
= f(v_{[0]})_{[0]} ⊗ f(v_{[0]})_{[1]}S(v_{[2]}) ⊗ f(v_{[0]})_{[2]}S(v_{[1]})
= f(v_{[0]})_{[0]} ⊗ Δ(f(v_{[0]})_{[1]}S(v_{[1]}))
= ((1 ⊗ Δ)δ_{1}(f))(v)

hence (δ_{0} ⊗ 1)δ_{1}(f) = (1 ⊗ Δ)δ_{1}(f). Since δ_{1} = (ι ⊗ K)δ and (δ_{1} ⊗ K)δ = (1 ⊗ Δ)δ_{1} we get for the induced map δ the equality (ι ⊗ K ⊗ K)(δ ⊗ K)δ = (1 ⊗ Δ)(ι ⊗ K)δ = (ι ⊗ K ⊗ K)(1 ⊗ Δ)δ and thus (δ ⊗ K)δ = (1 ⊗ Δ)δ. Consequently hom(V, W) is a K-comodule. □

Lemma 6.2. hom(V, W) is a Yetter-Drinfel’d module.

Proof. We first show that hom(V, W) is a K-module. Let f ∈ hom(V, W) and h ∈ K. Then

(14)  δ_{0}(fh) = j(f_{0}h_{(2)} ⊗ S(h_{(1)})f_{1}h_{(3)})

since δ_{0}(fh)(v) = (fh)(v_{[0]})_{[0]} ⊗ (fh)(v_{[0]})_{[1]}S(v_{[1]}) = (f(v_{[0]}S^{−1}(h_{(2)}))h_{(1)})_{[0]} ⊗ (f(v_{[0]}S^{−1}(h_{(2)}))h_{(1)})_{(1)}S(v_{[1]}) = f(v_{[0]}S^{−1}(h_{(4)}))_{[0]}h_{(2)} ⊗ S(h_{(1)})f(v_{[0]}S^{−1}(h_{(4)}))_{[1]}h_{(3)}S(v_{[1]}) = f(v_{[0]}S^{−1}(h_{(4)}))_{[0]}h_{(2)} ⊗ S(h_{(1)})f(v_{[0]}S^{−1}(h_{(4)}))_{[1]}h_{(3)}S(v_{[1]})S(h_{(5)})h_{(6)} = f(v_{[0]}S^{−1}(h_{(4)}))_{[0]}h_{(2)} ⊗ S(h_{(1)})f(v_{[0]}S^{−1}(h_{(4)}))_{[1]}S(S(S^{−1}(h_{(5)}))v_{[1]}S^{−1}(h_{(3)}))h_{(6)} = f((vS^{−1}(h_{(3)}))_{[0]})_{[0]}h_{(2)} ⊗ S(h_{(1)})f((vS^{−1}(h_{(3)}))_{[0]})_{[1]}S((vS^{−1}(h_{(3)}))_{[1]})h_{(4)} = f_{0}(vS^{−1}(h_{(3)}))h_{(2)} ⊗ S(h_{(1)})f_{1}h_{(4)} = (f_{0}h_{(2)})(v) ⊗ S(h_{(1)})f_{1}h_{(3)}

Thus fh ∈ hom(V, W) by definition of hom(V, W). So hom(V, W) is a K-submodule of Hom(V, W). Furthermore (14) shows also that hom(V, W) is a Yetter-Drinfel’d module. □

Theorem 6.3. The category of Yetter-Drinfel’d modules YD_{K}^{K} is a closed monoidal category.

Proof. It suffices to show for a homomorphism g : X ⊗ V ⟶ W in YD_{K}^{K} that the induced map g̃ : X ⟶ Hom(V, W) factors through a Yetter-Drinfel’d homomorphism ḡ : X ⟶ hom(V, W). We have

δ_{0}(g̃(x))(v) = g̃(x)(v_{[0]})_{[0]} ⊗ g̃(x)(v_{[0]})_{[1]}S(v_{[1]})
= g(x ⊗ v_{[0]})_{[0]} ⊗ g(x ⊗ v_{[0]})_{[1]}S(v_{[1]})
= g(x_{[0]} ⊗ v_{[0]}) ⊗ x_{[1]}v_{[1]}S(v_{[2]})
= g̃(x_{[0]})(v) ⊗ x_{[1]}
= j(g̃(x_{[0]}) ⊗ x_{[1]})(v)

or δ_{0}(g̃(x)) = j(g̃(x_{[0]}) ⊗ x_{[1]}) so that g̃(x) ∈ hom(V, W) which defines a homomorphism ḡ : X ⟶ hom(V, W). Furthermore this proves δ(ḡ(x)) = ḡ(x_{[0]}) ⊗ x_{[1]} = (ḡ ⊗ 1)δ(x) which shows that ḡ : X ⟶ hom(V, W) is a comodule homomorphism. Finally we have

ḡ(xh)(v) = g(xh ⊗ v)
= g(xh_{(1)} ⊗ vS^{−1}(h_{(3)})h_{(2)})
= g((x ⊗ vS^{−1}(h_{(2)}))h_{(1)})
= g(x ⊗ vS^{−1}(h_{(2)}))h_{(1)}
= ḡ(x)(vS^{−1}(h_{(2)}))h_{(1)}
= (ḡ(x)h)(v)

so that ḡ(xh) = ḡ(x)h, i.e. ḡ is a Yetter-Drinfel’d homomorphism. □

We denote the evaluation map corresponding to id ∈ Hom(hom(V, W), hom(V, W)) by ϑ : hom(V, W) ⊗ V ⟶ W.

Now we consider derivations on algebras A in YD_{K}^{K}. A derivation from A to A is a linear map (d : A ⟶ A) ∈ hom(A, A) such that

d(ab) = d(a)b + ∇(1 ⊗ ϑ)(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). Lemma 6.4. Let A be an algebra in YD_{K}^{K}. Then the set

Der(A) := {d ∈ hom(A, A)|d(ab) = d(a)b + ∇(1 ⊗ ϑ)(τ ⊗ 1)(d ⊗ a ⊗ b)∀a, b ∈ A}

is a Yetter-Drinfel’d module in YD_{K}^{K}.

In fact Der(A) is the kernel in YD_{K}^{K} of the morphism in Hom(end(A), hom(A ⊗ A, A)) ≅ Hom(end(A)⊗A⊗A, A) given by ϑ(1⊗∇)−∇(ϑ⊗1)−∇(1⊗ϑ)(τ ⊗1)). It is easily checked that Der(A) together with its operation on A is the universal derivation module on A, i.e. module M in YD_{K}^{K} together with an operation ϑ : M ⊗ A ⟶ A such that (1 ⊗ ∇)ϑ = ∇(ϑ ⊗ 1) + ∇(1 ⊗ ϑ)(τ ⊗ 1). Theorem 6.5. ([P2] Corollary 5.6) The Yetter-Drinfel’d module of derivations Der(A) of an algebra A is a Lie algebra.

Now let V be in YD_{K}^{K} with inner endomorphism object A := end(V). Given a bilinear form ⟨., .⟩ : V ⊗V ⟶ k in YD_{K}^{K}. We collect the set 𝔤 of skew-symmetric endomorphisms f ∈ end(V) for which in principle the following hold: ⟨f(v), w⟩ = −⟨v, f(w)⟩ for all v, w ∈ V. Since there is a switch between f and v in these expressions the correct condition is

⟨f(v), w⟩ = − ∑⟨v_{i}, f_{i}(w)⟩

where τ(f ⊗ v) = ∑ v_{i} ⊗ f_{i} resp.

μ(ϑ ⊗ 1)(f ⊗ v ⊗ w) = −μ(1 ⊗ ϑ)(τ ⊗ 1)(f ⊗ v ⊗ w.

Thus 𝔤 is the difference kernel of two morphisms in YD_{K}^{K}. Hence 𝔤 is an (universal) object satisfying the diagrammatic condition

[figure: 𝔤; 𝔤; V; V; V; V; =; −]

Theorem 6.6. For a Yetter-Drinfel’d module V ∈ YD_{K}^{K} with bilinear form μ = ⟨., .⟩ : V ⊗V ⟶ k the Yetter-Drinfel’d module 𝔤 of skew-symmetric endomorphisms is a Lie subalgebra of end(V)^{L} in YD_{K}^{K}.

Proof. We give a diagrammatic proof where we use the multiplication ∇ : 𝔤 ⊗ 𝔤 ⟶ A = end(V) and the fact that the evaluation ϑ : 𝔤 ⊗ V ⟶ V of 𝔤 on V is associative with respect to this multiplication. For the n-fold multiplication we write ∇^{n} : 𝔤^{n} ⟶ A. Let π_{n} : 𝔤^{n} ⟶ 𝔤^{n} be given by π_{2} := τ_{𝔤⊗𝔤}, π_{n+1} = (π_{n} ⊗ 1)τ_{𝔤⊗𝔤^{n}}. Then

μ(ϑ ⊗ 1)(∇^{n} ⊗ 1 ⊗ 1) = (−1)^{n}μ(1 ⊗ ϑ)(τ ⊗ 1)(∇^{n}π_{n} ⊗ 1 ⊗ 1).

We prove this by induction. For n = 1 this is the defining condition for 𝔤. The induction step is

[figure: ∇^{n}; A; 𝔤; 𝔤^{n}; V; V; A; 𝔤; 𝔤^{n}; V; V; A; 𝔤; 𝔤^{n}; V; V; A; 𝔤; 𝔤^{n}; V; V; =; =; =; =; (−1); (−1); =; (−1)^{n+1}; A; 𝔤; 𝔤^{n}; V; V; =]

[figure: =; (−1)^{n+1}; π_{n}; A; 𝔤; 𝔤^{n}; V; V; =; (−1)^{n+1}; π_{n}; A; 𝔤; 𝔤^{n}; V; V; =; (−1)^{n+1}; ∇^{n+1}π_{n+1}; A; 𝔤^{n+1}; V; V]

In the first and the last term we indicated the multiplication ∇^{n} : 𝔤^{n} ⟶ A resp. ∇^{n+1} : 𝔤^{n+1} ⟶ A. Furthermore we indicated the use of π_{n} where appropriate.

Now let ζ be a primitive n-th root of unity and z ∈ 𝔤^{n}(ζ). Then ζ^{\frac{n(n−1)}{2}} = (−1)^{n−1} and π_{n}(z) = ζ^{\frac{n(n−1)}{2}} ρ_{n}(z) by (7) and the definition of π_{n}, where ρ_{n} is the image of π_{n} under the canonical map B_{n} ⟶ S_{n}. So we get

μ(ϑ ⊗ 1)([z] ⊗ v ⊗ w) =
= ∑_{σ} μ(ϑ ⊗ 1)(∇^{n}σ(z) ⊗ v ⊗ w)
= ∑_{σ} μ(ϑ ⊗ 1)(∇^{n} ⊗ 1 ⊗ 1)(σ(z) ⊗ v ⊗ w)
= (−1)^{n} ∑_{σ} μ(1 ⊗ ϑ)(τ ⊗ 1)(∇^{n}π_{n} ⊗ 1 ⊗ 1)(σ(z) ⊗ v ⊗ w)
= (−1) ∑_{σ} μ(1 ⊗ ϑ)(τ ⊗ 1)(∇^{n} ⊗ 1 ⊗ 1)(ρ_{n}σ(z) ⊗ v ⊗ w)
= (−1)μ(1 ⊗ ϑ)(τ ⊗ 1)([z] ⊗ v ⊗ w),

hence [z] ∈ 𝔤. Thus 𝔤 is a Lie subalgebra of end(V). □

7. Lie structures on C_{p^{n}}-graded modules

In this section we assume that G = C_{p^{t}} = ℤ/(p^{t}) is the cyclic group with p^{t} elements where p ≠ 2 is prime and that the field k has characteristic ≠ 2 and contains a p^{t}-th primitive root of unity ξ. We want to get information on the nontrivial symmetrizations of G-comodules.

A bicharacter χ : G ⊗_{ℤ} G ⟶ k^{×} is uniquely defined by the value of χ(1, 1) as χ(i, j) = χ(1, 1)^{ij} ∈ k. Since ℤ/(p^{t}) ⊗_{ℤ} ℤ/(p^{t}) ≅ ℤ/(p^{t}) this amounts to a homomorphism φ : ℤ/(p^{t}) ∋ g ↦ ξ^{g} ∈ k^{×} for some element ξ ∈ k^{×}. Such a homomorphism has a unique image factorization ℤ/(p^{t}) ⟶ ℤ/(p^{s}) ⟶ k^{×} with g ↦ g ↦ ξ^{g} with the second homomorphism injective. Then ξ has order p^{s} so it is a primitive p^{s}-th root of unity. Without loss of generality we may assume s = t and ξ a primitive p^{t}-th root of unity.

We wish to compute the ζ-symmetrization (P ⊗ … ⊗ P)(ζ) = P^{n}(ζ) where P is a G = ℤ/(p^{t})-graded vector space and ζ is an n-th primitive root of unity (with n > 1) so that we can determine the domain of the possible Lie multiplications.

To determine P^{n}(ζ) = ⨁_{{(g_{1},...,g_{n})ζ-family}} P_{g_{1}} ⊗ … ⊗ P_{g_{n}} (Proposition 3.1) we have to find the ζ-families (g_{1}, …, g_{n}) in G i.e. families with χ(g_{i}, g_{j})^{2} = ζ^{2} for all i ≠ j. Since χ(g_{i}, g_{j}) has order p^{r} for some r, there is only a restricted choice for the primitive root of unity ζ and for the arity of the possible Lie multiplication on P.

We have n > 1 and ζ ∈ k^{×} a primitive n-th root of unity. We wish to determine all ζ-families (g_{1}, …, g_{n}) of elements in G = ℤ/(p^{t}) satisfying χ(g_{i}, g_{j})^{2} = ζ^{2} or ξ^{2g_{i}g_{j}} = ζ^{2} for all i ≠ j. This amounts to ξ^{g_{i}g_{j}} = εζ with ε = ±1. In the case ε = +1 we get ζ = ξ^{g_{i}g_{j}} ∈ Im(φ). Hence the order of ζ is n = p^{t′} and ζ = ξ^{b} for b ∈ ℤ/(p^{t}). In the case ε = −1 we get −ζ = ξ^{g_{i}g_{j}} ∈ Im(φ) (and ζ ∉ Im(φ) since p is odd). Hence the order of ζ is n = 2p^{t′} and ζ = −ξ^{b} for b ∈ ℤ/(p^{t}). Together this says ζ = εξ^{b} has order n = (\frac{3}{2} − \frac{1}{2}ε)p^{t′} for ε = ±1.

If t′ = 0 then ζ = ±1. If ζ = 1 then we get the trivial case P^{1}(1) = P. If ζ = −1 the primitive 2-nd root of unity, then (g_{1}, g_{2}) is a ζ-family iff it satisfies χ(g_{1}, g_{2}) = 1 (the value −1 is not possible) iff g_{1}g_{2} ≡ 0(p^{t}).

Assume now that t′ > 0 hence b ≠ 0. Then there are n > 2 components g_{i} in a ζ-family (g_{1}, …, g_{n}). Choose representatives g_{i} ∈ ℕ with 0 < g_{i} < p^{t}. Observe that g_{i} ≠ 0 since b ≠ 0 and b ≡ g_{i}g_{j}(p^{t}). So we can write g_{i} = n_{i}p^{r_{i}} with (n_{i}, p) = 1 and 0 ≤ r_{i} < t and b = qp^{s} with (q, p) = 1 and 0 ≤ s < t.

We have g_{i}g_{j} ≡ b(p^{t}) for all i ≠ j hence n_{i}n_{j}p^{r_{i}}p^{r_{j}} ≡ qp^{s}(p^{t}). Since g_{i}g_{j} ≠ 0 in ℤ/(p^{t}) we get r_{i} +r_{j} < t hence r_{i} +r_{j} = s for all i ≠ j. Thus r_{i} = r for all i = 1, …, n and s = 2r < t. Since b = qp^{s} has order p^{t−s} in ℤ/(p^{t}) we get t′ = t − 2r and n = (\frac{3}{2} − \frac{1}{2}ε)p^{t−2r}.

Finally we have g_{i}(g_{j} − g_{k}) = n_{i}(n_{j} − n_{k})p^{2r} ≡ 0(p^{t}) hence n_{j} − n_{k} ≡ 0(p^{t−2r}) or

n_{i} ≡ n_{j}(p^{t−2r}), ∀i ≠ j

and b ≡ n_{i}n_{i}p^{2r}(p^{t}). It is easy to check that any family (g_{1}, …, g_{n}) satisfying these equations is a ζ-family. This proves the following Theorem 7.1. Let p ≠ 2 be a prime and t ≥ 1. Then the (−1)-families (g_{1}, g_{2}) are those with g_{1}g_{2} ≡ 0(p^{n}).

For any choice of • r such that 0 ≤ 2r < t,

• ε ∈ {+1, −1},

• m ∈ {1, …, p^{n} − 1} with (m, p) = 1, there are ζ-families (g_{1}, …, g_{n}) with ζ = εξ^{b}, b ≡ m^{2}p^{2r}(p^{n}), and n = (\frac{3}{2} − \frac{1}{2}ε)p^{t−2r}. The g_{i} can be chosen as g_{i} ≡ mp^{r} + a_{i}p^{t−r}(p^{n}) with a_{i} ∈ {0, …, p^{r} − 1}.

These are all families on which a Lie multiplication can be defined. Example 7.2. 1. Let p = 3 and t = 1. Then there is the (−1)-symmetrization

P^{2}(−1) = P_{0} ⊗ P + P ⊗ P_{0},

since g_{1}g_{2} ≡ 0(3) iff one of the factors g_{i} is zero. To get all other symmetrizations observe that r = 0 hence b ≡ g_{1}g_{1} ≡ 1(3). So there are 2 cases ζ = ξ and n = 3 or ζ = −ξ and n = 6. The corresponding possible ζ-families are (1, 1, 1) and (2, 2, 2) resp. (1, 1, 1, 1, 1, 1) and (2, 2, 2, 2, 2, 2) hence

P^{3}(ξ) = P_{1} ⊗ P_{1} ⊗ P_{1} ⊕ P_{2} ⊗ P_{2} ⊗ P_{2}

and

\begin{aligned}P^{6}(−ξ) = & P_{1} ⊗ P_{1} ⊗ P_{1} ⊗ P_{1} ⊗ P_{1} ⊗ P_{1} \\ & ⊕P_{2} ⊗ P_{2} ⊗ P_{2} ⊗ P_{2} ⊗ P_{2} ⊗ P_{2}. \end{aligned}

2. Let p = 3 and t = 2. Then P^{2}(−1) = P_{0} ⊗ P + P ⊗ P_{0} + (P_{3} + P_{6}) ⊗ (P_{3} + P_{6}). For larger ζ-families the only choice for r is r = 0. Since m ∈ {1, 2, 4, 5, 7, 8} we get b ∈ {1, 4, 7}. We get families with 9 or 18 elements g_{i} and these elements must all be equal, g_{i} ∈ {1, 2, 4, 5, 7, 8}.

3. Let p = 3 and t = 3. We consider the families of length > 2. There are two choices for r ∈ {0, 1}. So we get families with 3, 6, 27, and 54 elements. For the choice r = 1, m = 4, ε = +1 the ξ^{9}-families (g_{1}, g_{2}, g_{3}) are composed of g_{i} = 4·3+a_{i}·9 with a_{i} ∈ {0, 1, 2}. There is for example the ξ^{9}-family (3, 12, 21). So any braided Lie algebra P in M^{kG} with G = ℤ/(27) has a Lie operation [ ] : P_{3} ⊗ P_{12} ⊗ P_{21} ⟶ P_{9}.

References

[FM] Davida Fischman and Susan Montgomery, A Schur Double Centralizer Theorem for Cotriangular Hopf Algebras and Generalized Lie Algebras, J. Algebra 168 (1994), 594-614.

[L] Lusztig, George: On Quantum Groups. J. Alg. 131 (1990), 466-475

[Ma] Shahn Majid, Crossed Products by Braided Groups and Bosonization, J. Algebra 163 (1994), 165-190.

[M] Susan Montgomery: Hopf Algebras and Their Actions on Rings. CBMS 82, AMS-NSF, 1993.

[P1] Bodo Pareigis: On Lie Algebras in Braided Categories. Banach Center Publications Vol. 40 (1997). www.mathematik.uni-muenchen.de /personen/professoren.html.

[P2] Bodo Pareigis: On Lie Algebras in the Category of Yetter-Drinfel’d-Modules. To appear in Applied Categorical Structures – www.mathematik.uni-muenchen.de /personen/professoren.html.

[R] David E. Radford: The Structure of Hopf Algebras with a Projection. J. Alg. 92 (1985), 322-347.

[S] Yorck Sommerhäuser: Deformed Enveloping Algebras. New York Journal of Mathematics 2 (1996), 1-23.

[Y] Yetter, David N.: Quantum Groups and Representations of Monoidal Categories. Math. Proc. Camb. Phil. Soc. 108 No 2 (1990), 261-290.

Mathematisches Institut der Universität München, Germany

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