Logarithmic Knot Invariants Arising From Restricted Quantum Groups
Abstract.
We construct knot invariants from the radical part of projective modules of the restricted quantum group at , and we also show a relation between these invariants and the colored Alexander invariants. These projective modules are related to logarithmic conformal field theories.
1. Introduction
Various knot invaraiants are constructed from the quantum matrix of the quantum group. However, most of them are constructed from semisimple algebras. Our concern in this note is constructing knot invariant arising from nonsemisimple representations. We focus on the restricted quantum group and construct knot invariant which is understood as a derivative of the colored Alexander invariant [1], [9].
Let and be the center and the Jacobson radical of respectively. Then is a direct sum of and , where is the subalgebra of generated by the primitive idempotents and . Let be a knot in . By using the idea of the universal invariant in [10], we can associate an element with . Then is expressed as
The first term is a linear combination of the primitive idempotents and the coefficient of each idempotent corresponds to the colored Jones invariant. In this paper, we study about the knot invariants coming from . The space has a natural basis corresponding to the indecomposable modules, and the coefficients of with respect to this basis are also knot invariants.
In the construction of , we assume that is a single component knot. For a multicomponent link, the construction in this paper does not work well and we need another idea to extend for link case.
A threemanifold invariant is constructed from the ‘integral’ of [8], [4], [7], [11], which is defined for a finite dimensional Hopf algebra. Meanwhile, an action of on the center of is given in [6] and [3]. By using our invariants constructed here, these two theories can be combined as the usual topological quantum field theory, e.g. [12], [2], related to . The detail will be given elsewhere.
We review the definition of and its representations in Section 2. The construction of is given in Section 3. In Sections 4 and 5, we show some property of the invariants coming from , especially the relation to the colored Alexander invariant in [9].
2. Restricted quantum groups
2.1. Definition
Let be a positive integer and . The semirestricted quantum group is the quotient of the usual quantum group defined by the following generators and relations as an algebra.
The restricted quantum group is obtained from by inquiring one more relation . The coproduct, counit and antipode of and are defined as follows.
2.2. matrix
By introducing a symbol such that , We can define an matrix of satisfying
where if . The explicit form of is given as follows.
(2.1) 
where and , and is an element such that .
2.3. Irreducible mudules
The irreducible modules of are labeled by and , and is spanned by weight vectors , with the action of given by
where .
2.4. Projective modules
Projective modules of which are fundamental to investigate the structure of the center of are labeled by . Note that a module is also naturally a module.
Let be any integer . The projective module has the basis
and the action of is given by
and
Let be an integer . The projective module has the basis
and the action of is given by
and
Note that the diagonal part is a direct sum of irreducible modules.
2.5. Center
The dimension of the center of is . The basis of is given by the canonical central elements in [3] as follows: Two special primitive idempotents and , other primitive idempotents , , and elements corresponding to the radical part. These basis satisfy the following relations.
(2.2)  
3. Logarithmic invariants of knots
3.1. Knots and (1,1)tangles
In this paper, knots and tangles are oriented and framed. For a connected (1,1)tangle , let be the knot obtained by joining the two open ends as in Figure 1. For two tangles and , it is known that and are isotopic as framed knots if and only if and are isotopic as framed tangles. So, in the following, we sometimes mix up invariants of connected (1,1)tangles and invariants of knots.
3.2. Framed braid
Framed braid group on strings is defined by the following generators and relations.
The generators correspond to the positive and negative crossings, and represent the blackboard framing corresponding to the twist as in Figure 2. Let be the symmetric group of letters , 2, , , and be the group homomorphism from to sending to the transposition and to the identity for , 2, , .
3.3. Alexander’s and Markov’s theorems
Then we have the framed versions of Alexander’s and Markov’s theorem as follows.
Theorem 3.3.1.
Any framed link is isotopic to the closure of some framed braid .
Theorem 3.3.2.
Two framed braids , have isotopic closures if and only if can be transformed to by a finite sequence of moves of the following two types.

for , .

for .
3.4. Representation of on
The universal matrix satisfies the YangBaxter equation
where and acts on the th and th components of the tensor product. Let
and
Then is a ribbon Hopf algebra with the ribbon element . Therefore, we can define a homomorphism from to by
3.5. Universal invariant
Let be a knot and let be a framed braid whose closure is equivalent to . Then is expressed as follows.
Let be a (1,1)tangle corresponding to . The element corresponding to is defined by
The element commutes with any elements of and is in the center . Therefore, we have
(3.1) 
where , are scalars and are invariants of the closure of . Hence we can also denote them by , .
4. Properties of and
We show some property of and .
Theorem 4.0.1.

For the connected sum of two knots and ,

For a knot , the invariant is equal to the colored Jones invariant corresponding to the dimensional irreducible module normalized as
Proof.
For two tangles , , let be the tangle obtained by joining below . Then, for two knots and , is a tangle representing the connected sum . Therefore, by using (2.2) and (3.1), we have
and we obtain (1).
The center acts on as identity, and the other basis , and acts on as zero. Hence corresponds to the scalar representing the action of on , which is equal to the colored Jones invariant and we get (2). ∎
5. Relation to the colored Alexander invariant
5.1. Relation
Let be a framed knot and be the corresponding framed tangle. Let be the scalar multiple of the colored Alexander invariant defined in [9]. Then we have the following.
Theorem 5.1.1.
The invariants , , are given by the colored Alexander invariants as follows.
Before proving the above, we introduce some representations of .
5.2. Nonintegral representations
We introduce highest weight representations of for nonintegral weights and obtain the projective modules as a specialization of cetrain nonirreducible module.
First, we define the irreducible module for noninteger number as follows. Let be the module spanned by weight vectors , . The action of to is given by
where .
Next, we define a nonirreducible module which is isomorphic to direct sum of two nonintegral highest modules. Let be an integer with and be the module which is spanned by weight vectors and for . The action of is given by
5.3. Colored Alexander invariant
Let be a framed knot, be the corresponding framed tangle, and be the corresponding central element in the semirestricted quantum group which is defined as by using the universal Rmatrix of given by (2.1). Let be the representation matrix of on with repsect to the above basis . Then the diagonal element corresponding to and are equal to and respectively, where is given in [9] as the scalar corresponding to the tangle . Note that is a scalar multiple of the colored alexander invariant and itself is also an invariant of if is a single component knot.
5.4. Proof of Theorem 4
The matrix has offdiagonal elements at components for . Let be the component of . Then
(5.1)  
Let
Then and so is a highest weight vector of weight . Therefore, on the one hand,
On the other hand, from (5.1),
Thus we have
(5.2) 
and then
Hence, we get
(5.3) 