1. Introduction
Decomposing tensor products of representations for a group
$G$
, for a Lie (super-) algebra
$\mathfrak {g}$
or for similar structures into indecomposable summands is key for understanding the representations of the given structure. In the world of complex semisimple Lie algebras, the prime example is
$\mathfrak {sl}_2$
, whose tensor products are described in terms of the Clebsch–Gordan rule. This description lifts to the quantum setting when replacing
$\mathfrak {sl}_2$
with
$\textrm{U}_q(\mathfrak {sl}_2)$
. In the world of quantum symmetric pairs (QSPs), the prime example is
$\textrm{U}_q^{\prime}(\mathfrak {sl}_2^{ \!\iota })\subset \textrm{U}_q(\mathfrak {sl}_2)$
. Loosely speaking, a QSP is a quantization of the canonical inclusion
$\textrm{U}(\mathfrak {g}^{ \iota })\subset \textrm{U}(\mathfrak {g})$
of universal enveloping algebras induced from an embedding of Lie algebras
$\mathfrak {g}^{\iota }\subset \mathfrak {g}$
given by a classical symmetric pair (i.e., an embedding of the Lie algebra of fixed points
$\mathfrak {g}^{\iota }$
for a Lie algebra involution
$\iota \colon \mathfrak {g}\to \mathfrak {g}$
). The resulting subalgebra
$\textrm{U}_q^{\prime}(\mathfrak {g}^{\iota })\subset \textrm{U}_q(\mathfrak {g})$
is not a Hopf subalgebra in general, but just a one-sided coideal. This deficiency has consequences for the representations of
$\textrm{U}_q^{\prime}(\mathfrak {g}^{\iota })$
and their morphisms: they cannot be tensored together; instead, representations of
$\textrm{U}_q(\mathfrak {g})$
act on this category. More precisely, the category of representations of
$\textrm{U}_q^{\prime}(\mathfrak {g}^{\iota })$
is a module category (even braided) in the sense of [Reference Brochier1], §5.1], [Reference Häring-Oldenburg11], §3 Def.7] over the (braided) monoidal category of representations of
$\textrm{U}_q(\mathfrak {g})$
. The purpose of this paper is to make this totally explicit in case of
$\textrm{U}_q^{\prime}(\mathfrak {sl}_2^{ \!\iota })\subset \textrm{U}_q(\mathfrak {sl}_2)$
using diagrammatic methods.
We extend a well-established connection between representation theory and diagrammatics from the familiar type
$A$
to other Lie types by developing a generalization of the Temperley–Lieb recoupling theory, [Reference Carter, Flath and Saito5], [Reference Kauffman and Lins14]. Our paper partially presents results obtained in the thesis [Reference Wojciechowski31], to which we also refer for more background material. Since the original theory in type
$A$
is ultimately based on quantum Schur–Weyl duality or quantum Skew-Howe duality [Reference Cautis, Kamnitzer and Morrison6], the starting point of this paper is the Schur–Weyl duality ([[Reference Bao and Wang3], Theorem 5.4], [[Reference Ehrig and Stroppel8], Theorem 10.4]) and Skew-Howe duality [[Reference Ehrig and Stroppel8], Theorem 10.28]). These are dualities between the type AIII quantum symmetric pairs and specialized 2-parameter type
$B$
Hecke algebras (which contain the type
$D$
Hecke algebras). These results and the invention and categorification of
$i$
-canonical bases in [Reference Bao and Wang3], [Reference Ehrig and Stroppel8], suggest that the representation theory of the smallest coideal
$\textrm{U}_q^{\prime}(\mathfrak {gl}_1)\subset \textrm{U}_q(\mathfrak {sl}_2)$
can be studied using Temperley–Lieb categories of type
$D$
and be described by the corresponding blob diagrammatics from [Reference Green10], [Reference tom Dieck26], [Reference tom Dieck27]. Such generalized Temperley–Lieb categories appeared in connection with
$\textrm{U}_q(\mathfrak {sl}_2)$
already in [Reference Iohara, Lehrer and Zhang12] to describe intertwiners between a dominant Verma tensored with copies of the standard representation. Despite the fact that the coideal
$\textrm{U}_q^{\prime}(\mathfrak {gl}_1)$
was already a prominent example in the theory of quantum homogeneous spaces, [Reference Müller and Schneider20], even before quantum symmetric pairs were classified in [Reference Letzter16], its representation category has not been studied in detail yet. For remarkable first steps see however for example [[Reference Watanabe29], Proposition 3.14] (for a study of weight modules and for classification results) and [Reference Watanabe30] (for crystal bases).
Overview and results. We start our study of tensor products and fusion rules in Section 2 in terms of deformed weight vectors. These are interesting eigenvectors for a commuting subalgebra which (in contrast to the quantum group case) does not consist of primitive elements. This means in particular that tensor products of weight vectors are not necessarily weight vectors. Various normalization problems of the Jones polynomial and other invariants related to
$\textrm{U}_q(\mathfrak {sl}_2)$
encode subtle sign issues, see for example [Reference Tingley28], [Reference Stroppel25] which can be explained by the interplay of type
$1$
and type
$-1$
representations. In Section 4, we therefore introduce simultaneous diagrammatics for type
$1$
and
$-1$
representations of
$\textrm{U}_q(\mathfrak {sl}_2)$
and describe diagrammatically the module category of representations of
$\textrm{U}_q^{\prime}(\mathfrak {gl}_1)$
which arises by restricting tensor products of representations of
$\textrm{U}_q(\mathfrak {sl}_2)$
to
$\textrm{U}_q^{\prime}(\mathfrak {gl}_1)$
. In Section 5, we relate this to the decompositions from Section 2. We define analogues of Jones–Wenzl projectors, prove uniqueness properties, and give a new combinatorial method to count dimensions of morphisms spaces for the coideal. We end the paper in Section 6 by introducing and then calculating certain
$\Theta$
-networks of type
$D$
.
All results of this paper which are formulated in terms of Temperley–Lieb categories, in particular the Temperley–Lieb algebra of type
$D$
, can be categorified easily (and straight-forwardly) in terms of (geometric bi)modules over the type
$D$
Khovanov arc algebras [Reference Ehrig and Stroppel7], that is a type
$D$
version of Khovanov’s famous arc algebra which he used to categorify the Reshethikhin–Turaev
$\textrm{U}_q(\mathfrak {sl}_2)$
-invariant of tangles. We however do not pursue this here. Instead we explicitly write out the decategorified picture in a way such that every step can be categorified. In type
$A$
, the Khovanov arc algebra can be realized, see [Reference Brundan and Stroppel2], as a cyclotomic quotient of a KLR-algebra [Reference Mathas19]. These algebras, introduced in [Reference Khovanov and Lauda15], [Reference Rouquier23], are generalizations of Nil-Hecke algebras which can be used to category the positive part of
$\textrm{U}_q(\mathfrak {sl}_2)$
. In parallel, the coideal
$\textrm{U}_q^{\prime}(\mathfrak {gl}_1)$
has been categorified recently in [Reference Brundan, Wang and Webster4] via a KLR approach using projective modules over the Nil–Brauer category. A direct connection between the two possible categorifications is not known at the moment and will be subject of a further study.
2. Fusion rules and decompositions for the coideal
Fix as ground field
$k=\mathbb {C}(q)$
. For
$n\in \mathbb {Z}$
set
$[n]=\frac {q^n-q^{-n}}{q-q^{-1}}\in k$
.
Definition 2.1. The quantum group
$\textrm{U}_q(\mathfrak {sl}_2)$
is the
$\mathbb {C}(q)$
-algebra generated by
$E,F,K,K^{-1}$
subject to the relations
\begin{equation*} KE=q^2EK, \, KF=q^{-2}KF, \, EF-FE=\frac {K-K^{-1}}{q-q^{-1}}, \, KK^{-1}=1=K^{-1}K. \end{equation*}
It is a Hopf algebra with comultiplication given by
\begin{equation*} \Delta (E)=E\otimes 1 + K \otimes E, \quad \Delta (F)=F\otimes K^{-1} + 1 \otimes F, \quad \Delta (K)=K\otimes K. \end{equation*}
Definition 2.2.
Let
$m\geq 0$
,
$\varepsilon \in \{{1},{-1}\}$
. The
$m+1$
-dimensional irreducible representation
$V_{m,\varepsilon }$
of type
$\varepsilon$
is the
$m+1$
-dimensional
$k$
-vector space with basis
$x_{0,\varepsilon },\ldots, x_{m,\varepsilon }$
and
$\textrm{U}_q(\mathfrak {sl}_2)$
-action given by
\begin{gather*} Ex_{i,\varepsilon }=\begin {cases} [m-i+1]x_{i-1,\varepsilon } & i\gt 0, \\ 0 & i=0, \end {cases} \quad Fx_{i,\varepsilon }=\begin {cases} \varepsilon [i+1]x_{i+1,\varepsilon } & i\lt m, \\ 0 & i=m, \end {cases} \\ Kx_{i,\varepsilon }=\varepsilon q^{m-2i}x_{i,\varepsilon }. \end{gather*}
Set
$V_{\varepsilon }\;:\!= V_{1,\varepsilon }$
the type
$\varepsilon$
standard representation, and
$k_{\varepsilon }\;:\!= V_{0,\varepsilon }$
the type
$\varepsilon$
trivial representation. We write
$x_{\varepsilon }\;:\!= x_{0,\varepsilon }, y_{\varepsilon }\;:\!= x_{1,\varepsilon }$
and
$1_{\varepsilon }\;:\!= x_{0,\varepsilon }$
for their bases.
Consider
$B\;:\!= q^{-1}EK^{-1}+F\in \textrm{U}_q(\mathfrak {sl}_2)$
. We directly obtain the following:
Lemma 2.3.
The element
$B\in \textrm{U}_q(\mathfrak {sl}_2)$
satisfies
$\Delta (B)=B\otimes K^{-1} + 1 \otimes B$
.
Definition 2.4.
The smallest type AIII right coideal
$\textrm{U}_q^{\prime}(\mathfrak {gl}_1)$
is the subalgebra of
$\textrm{U}_q(\mathfrak {sl}_2)$
generated by the element
$B\;:\!= q^{-1}EK^{-1}+F$
.
Remark 2.5.
By Lemma 2.3,
$\textrm{U}_q^{\prime}(\mathfrak {gl}_1)$
is a right coideal subalgebra of
$\textrm{U}_q(\mathfrak {sl}_2)$
, justifying the name. The coideal subalgebra
$\textrm{U}_q^{\prime}(\mathfrak {gl}_1)$
of
$\textrm{U}_q(\mathfrak {sl}_2)$
appears as one of the fundamental examples of quantum symmetric pairs [[Reference Letzter17], §7]. The left coideal version
${}^{\prime}\textrm{U}_q(\mathfrak {gl}_1)$
of
$\textrm{U}_q^{\prime}(\mathfrak {gl}_1)$
is generated by
${}^{\prime}\textrm {B}\;:\!= E+qKF$
.
The following describes tensor powers of
$V_{\varepsilon }$
over
$\textrm{U}_q^{\prime}(\mathfrak {gl}_1)$
.
Theorem 2.6 (Coideal tensor product decomposition). Let
$\varepsilon \in \{1,-1\}$
. Consider for
$n\in \mathbb {Z}$
the special elements in
$V_{\varepsilon }$
given by
\begin{equation*} v_{n, \varepsilon }= x_{\varepsilon } + \varepsilon q^{\varepsilon n}y_{\varepsilon }, \quad w_{n, \varepsilon }=x_{\varepsilon }-\varepsilon q^{-\varepsilon n}y_{\varepsilon }. \end{equation*}
-
i) Then
$V_{\varepsilon }$
decomposes over
$\textrm{U}_q^{\prime}(\mathfrak {gl}_1)$
into summands
which are eigenspaces for
\begin{equation*} V = L(1)\oplus L({-}1)\;:\!= \langle v_{0, \varepsilon }\rangle \oplus \langle w_{0, \varepsilon }\rangle, \end{equation*}
$B$
with eigenvalues
$1$
respectively
$-1$
.
-
ii) Let
$z$
be an eigenvector for
$B$
with eigenvalue
$[n]$
,
$n\in \mathbb {Z}$
. Then
$z\otimes v_{n, \varepsilon }$
and
$z\otimes w_{n,\varepsilon }$
are eigenvectors for
$B$
with eigenvalue
$[\varepsilon n+1]$
and
$[\varepsilon n-1]$
respectively.
-
iii) The
$n$
-th tensor power of
$V_{\varepsilon }$
decomposes into
$1$
-dimensional subrepresentations
$L([n-2k])$
which are eigenvectors for
$B$
with eigenvalue
$[n-2k]$
,(1)
\begin{equation} V_{\varepsilon }^{\otimes n} = \bigoplus _{k=0}^{n} \binom{n}{k} L([n-2k]). \end{equation}
Proof. Part i) is clear. With Lemma2.3, we calculate
\begin{align*} B (z\otimes (x_{\varepsilon }\pm \varepsilon q^{\pm \varepsilon n}y_{\varepsilon })) &=Bz \otimes K^{-1}(x_{\varepsilon }\pm \varepsilon q^{\pm \varepsilon n}y_{\varepsilon }) + z \otimes B(x_{\varepsilon }\pm \varepsilon q^{\pm \varepsilon n} y_{\varepsilon }) \\ &= [n] z \otimes (\varepsilon q^{-1} x_{\varepsilon } \pm q^{\pm \varepsilon n+1}y_{\varepsilon }) + z \otimes (\varepsilon y_{\varepsilon } \pm q^{\pm \varepsilon n} x_{\varepsilon }) \\ &=[\varepsilon n\pm 1] z \otimes (x_{\varepsilon }\pm \varepsilon q^{\pm \varepsilon n}y_{\varepsilon }), \end{align*}
\begin{equation*} \text {since } \varepsilon q^{-1}[n] \pm q^{\pm \varepsilon n} =[\varepsilon n\pm 1], \quad q^{\varepsilon n+1}[\pm n]+\varepsilon =\pm \varepsilon q^{\pm \varepsilon n} [\varepsilon n\pm 1]. \end{equation*}
This shows ii). Part iii) follows from i) and ii).
Corollary 2.7 (Clebsch–Gordan rule). Recall for
$m\geq 0$
the
$m+1$
-dimensional irreducible representation
$V_{m,\varepsilon }$
of
$\textrm{U}_q(\mathfrak {sl}_2)$
of type
$\varepsilon$
. Then
\begin{equation*} L([n])\otimes V_{m,\varepsilon }=\bigoplus _{k=0}^mL([n+m-2k]). \end{equation*}
Proof. This follows from Theorem2.6 using
$V_{n,\varepsilon }\otimes V_{1}\cong V_{n+1,\varepsilon }\otimes V_{n-1,\varepsilon }$
.
Remark 2.8.
For
$\varepsilon =1$
and
$n=0$
, Corollary
2.7
recovers the decomposition of [[Reference Müller and Schneider20], §4]. Concretely consider the element
$x=qEK^{-1}+F$
, obtained by choosing
$\alpha =0, \beta =q, \gamma =1$
in their setup. Then [[Reference Müller and Schneider20], Proposition4.2] states that the eigenvalues of the right action of
$x$
on
$V_{m,1}$
are of the form
$\{[m-2k]\mid k\in \{0,\ldots, m\}$
. Our element
$B$
is given by
$B=f(S'({-}x))$
, where
$S'$
is the antipode with respect to their comultiplication (call it
$\Delta '$
) and
$f$
is the
$\mathbb {C}$
-linear Hopf algebra isomorphism
\begin{equation*} f\colon \textrm{U}_q(\mathfrak {sl}_2)\to \textrm{U}_q(\mathfrak {sl}_2), \quad f(E)= EK^{-1}, f(F)=KF, f(K)=K^{-1}, F(q)= q^{-1}. \end{equation*}
Example 2.9.
By Corollary
2.7
,
$(L([2])\oplus L({-}[2])) \otimes V_{2,1}\otimes V_{2,1}$
decomposes as
\begin{equation*} L([6])\oplus L([4])^2\oplus L([2])^4 \oplus L(0)^4 \oplus L([{-}2])^4 \oplus L([{-}4])^2 \oplus L([{-}6]). \end{equation*}
In particular,
$L(0)$
appears with multiplicity
$4$
, which implies by adjunction
\begin{equation*} \dim \textrm {Hom}_{\textrm{U}_q^{\prime}(\mathfrak {gl}_1)}((L([2])\oplus L([{-}2]))\otimes V_{2,1}, V_{2,1})=4. \end{equation*}
Remark 2.10.
The coideal
$\textrm{U}_q^{\prime}(\mathfrak {gl}_1)\subset \textrm{U}_q(\mathfrak {sl}_2)$
can be extended to a coideal
$\textrm{U}_q^{\prime}(\mathfrak {gl}_1\times \mathfrak {gl}_1)\subset \textrm{U}_q(\mathfrak {gl}_2)$
. Concretely if
$D_1, D_{-1}$
are the additional group-like generators of
$\textrm{U}_q(\mathfrak {gl}_2)$
subject to the additional relations
\begin{equation*} D_iE_i=q^{i}E_iD_i, \, D_iF_i=q^{-i}F_iD_i, \, D_1D_{-1}=D_{-1}D_1, \, K=D_1D_{-1}^{-1} \end{equation*}
we let
$\textrm{U}_q^{\prime}(\mathfrak {gl}_1\times \mathfrak {gl}_1)$
be the subalgebra generated by
$B$
and
$D\;:\!= D_1D_2$
. This coideal has the advantage that
$D$
distinguishes different tensor powers of
$V_{\varepsilon }$
.
3. Different types of Temperley–Lieb algebras
We describe the endomorphism algebra of
$V_{\varepsilon }^{\otimes n}$
using a specialized type
$B$
Temperley–Lieb algebra
$\textrm{TL}(B_n)$
. We fix
$\varepsilon \in \{1,-1\}$
.
Definition 3.1.
Let
$n\geq 1$
. The (specialized) type
$B$
Temperley–Lieb algebra
$\textrm{TL}(B_n)_{\varepsilon }$
is the
$\mathbb {C}(q)$
-algebra with generators
$s_0, U_1, \ldots, U_{n-1}$
subject to the following relations
(2)
and
(3)
, which are the type
$A$
Temperley–Lieb relations and the (specialized) type
$B$
Temperley–Lieb relations involving the generator
$s_0$
respectively:
\begin{equation} U_i^2=-\varepsilon (q+q^{-1})U_i,\quad U_iU_j=U_jU_i,\\ U_i U_{i+1} U_i=U_i, U_{i+1} U_{i} U_{i+1}=U_{i+1}, \end{equation}
where
$|i-j| \geq 2$
and
$i,j\in \{1,\ldots, n-1\}$
such that expressions are defined, and
\begin{equation} s_0^2=1,\quad U_1 s_0 U_1=0,\quad s_0 U_i = U_i s_0 \end{equation}
for
$i\in \{2,\ldots, n-1\}$
. The subalgebra of
$\textrm{TL}(B_n)_{\varepsilon }$
generated by
$U_i$
for
$1\leq i\leq n-1$
, respectively by
$U_i$
for
$0\leq i \leq n$
with
$U_0\;:\!= s_0U_1s_0$
, is the Temperley–Lieb algebra
$\textrm{TL}(A_{n-1})_\varepsilon$
of type
$A_{n-1}$
and
$\textrm{TL}(D_{n})_\varepsilon$
of type
$D_n$
respectively.
Remark 3.2.
The algebra
$\textrm{TL}(B_n)_{\varepsilon }$
appears as
$T'D_n$
in [[Reference tom Dieck27], Definition 2.3], our
$s_0$
is
$\varepsilon _0-1$
in their notation. The algebra is closely related to the algebra
$TB_n$
introduced in [[Reference tom Dieck26], §4], where a basis theorem was established. A blob diagrammatics description for
$\textrm{TL}(B_n)$
and
$\textrm{TL}(D_n)$
was established in [[Reference Green10], Theorems 4.1, 4.2]. Our algebra
$\textrm{TL}(D_n)_{\varepsilon }$
is the quotient of Green’s type
$D_n$
Temperley–Lieb algebra by the relation
$U_0U_1=0=U_1U_0$
, which means that the closed dotted loop is zero. For the relationship between the Temperley–Lieb algebras see [[Reference tom Dieck27], Theorem 2.7]. Our algebra
$\textrm{TL}(D_n)$
is
$T''(D_n)$
in the notation from there.
The following lemma allows to define an involution
$\sigma$
on
$\textrm{TL}(B_n)$
and gives a realization of
$\textrm{TL}(D_n)$
as its fixed-point subalgebra.
Lemma 3.3.
There is a well-defined
$k$
-linear algebra involution
\begin{equation*} \sigma \colon \operatorname {TL}(B_n) \to \operatorname {TL}(B_n), \quad s_0 \mapsto -s_0\quad U_i \mapsto U_i \quad \text { where } i=1,\ldots, n-1. \end{equation*}
The fixed-point subalgebra
$\textrm{TL}(B_n)^{\sigma }$
for this involution is precisely
$\textrm{TL}(D_n)$
.
Proof. One checks the assignments yield an algebra involution. The second part follows from the diagrammatic description of
$\textrm{TL}(D_n)$
in [Reference Green10], Theorem 4.2] as the span of all type
$B$
diagrams with an even number of dots and Remark 3.2.
Next we recall the connection between
$\textrm{TL}(B_n)$
and a type
$B$
Hecke algebra.
Definition 3.4.
The (specialized) Hecke algebra
$\mathcal {H}_{1,q}(B_n)$
of type
$B_n$
is the
$\mathbb {C}(q)$
-algebra with generators
$s_0, H_1, \ldots, H_{n-1}$
, subject to the type
$A$
Hecke relations
-
(H1)
$H_i^2=1 + (q^{-1}-q)H_i$
for all
$1\leq i\leq n-1$
, -
(H2)
$H_iH_{i+1}H_i=H_{i+1}H_iH_{i+1}$
for all
$1\leq i \leq n-2$
, -
(H3)
$H_iH_j=H_j H_i$
for all
$0\leq i,j \leq {n-1}$
such that
$|i-j|\geq 2$
,
and the additional, specialized type
$B$
Hecke relations
-
(BH1)
$s_0^2=1$
, -
(BH2)
$s_0H_1s_0H_1=H_1s_0H_1s_0$
, -
(BH3)
$s_0H_i=H_is_0$
for all
$2\leq i \leq n-1$
.
Lemma 3.5.
There are surjective
$\mathbb {C}(q)$
-algebra morphisms
$\varphi _{\varepsilon }\colon \mathcal {H}_{1,q}(B_n)\rightarrow \textrm{TL}(B_n)_{\varepsilon }$
defined on the generators as
$\varphi _{\varepsilon }(s_0)= s_0, \varphi _{\varepsilon }(H_i)= U_i + \varepsilon q^{-\varepsilon }$
, where
$i=1,\ldots, n-1$
.
Proof. This is a reformulation of [[Reference tom Dieck27], Proposition 2.12]. In the notation from [[Reference tom Dieck27], Proposition 2.12]
$\mathcal {H}_{1,q}(B_n)$
is
$H'(D_n)$
, and the map from there is our map precomposed with the
$q$
-linear automorphism sending
$s_0$
to
$s_0$
and
$H_i$
to
$-H_i^{-1}$
.
We next show that the action of
$\mathcal {H}_{1,q}(B_n)$
on
$V_{\varepsilon }^{\otimes n}$
factors through
$\textrm{TL}(B_n)$
.
Definition 3.6.
We define
$\mathbb {C}(q)$
-linear endomorphisms
\begin{equation*} \kappa _{V_{\varepsilon }}\colon V_{\varepsilon }\to V_{\varepsilon }, \quad x_{\varepsilon }\mapsto y_{\varepsilon }, \quad y_{\varepsilon }\mapsto x_{\varepsilon }, \quad \text {and}\quad R_{V_{\varepsilon },V_{\varepsilon }} \colon V_{\varepsilon } \otimes V_{\varepsilon } \rightarrow V_{\varepsilon } \otimes V_{\varepsilon } \text { by} \end{equation*}
\begin{align*} \begin {array}{l@{\quad}c@{\quad}l@{\quad}l@{\quad}c@{\quad}l} x_{\varepsilon }\otimes x_{\varepsilon } &\mapsto & \varepsilon q^{-\varepsilon } x_{\varepsilon } \otimes x_{\varepsilon }, & x_{\varepsilon } \otimes y_{\varepsilon } &\mapsto & y_{\varepsilon } \otimes x_{\varepsilon } + \delta _{\varepsilon, -1}(q^{-1}-q)x_{\varepsilon }\otimes y_{\varepsilon },\\ y_{\varepsilon }\otimes y_{\varepsilon } &\mapsto & \varepsilon q^{-\varepsilon } y_{\varepsilon }\otimes y_{\varepsilon },& y_{\varepsilon } \otimes x_{\varepsilon } &\mapsto & x_{\varepsilon } \otimes y_{\varepsilon } + \delta _{\varepsilon, 1}(q^{-1}-q)y_{\varepsilon }\otimes x_{\varepsilon }. \end {array} \end{align*}
Lemma 3.7. There is a well-defined algebra homomorphism such that
Proof. This (and well-definedness in Definition3.6) follows from the definitions.
Definition 3.8.
We define an endomorphism
$\underline {H}_{V_\varepsilon, V_\varepsilon }$
of
$V_{\varepsilon }\otimes V_{\varepsilon }$
by
\begin{equation*} \underline {H}_{V_\varepsilon, V_\varepsilon }\;:\!= R_{V_{\varepsilon },V_{\varepsilon }} - \varepsilon q^{-\varepsilon } \textrm {id}_{V_{\varepsilon }\otimes V_{\varepsilon }.} \end{equation*}
One easily verifies that this endomorphism commutes with the action of
$\textrm{U}_q(\mathfrak {sl}_2)$
and that moreover the following holds:
Lemma 3.9.
The morphism
$\Phi _{\varepsilon }$
from Lemma
3.7
factors through
$\textrm{TL}(B_n)$
where
$\Psi _{\varepsilon }\colon \textrm{TL}(B_n)_{\varepsilon } \rightarrow {\textrm {End}}_{\mathbb {C}(q)}(V_{\varepsilon }^{\otimes n})$
is given by
\begin{equation*} s_0 \mapsto \kappa _{V_{\varepsilon }} \otimes \textrm {id}_{V^{\otimes (n-1)}}, \quad U_i \mapsto \textrm {id}_{V^{\otimes (i-1)}} \otimes \underline {H}_{V_\varepsilon, V_\varepsilon } \otimes \textrm {id}_{V^{\otimes (n-i-1)}}\ \ \text { where } i=1,\ldots, n-1. \end{equation*}
In particular, the image of
$\Phi _{\varepsilon }$
is contained in the centralizer of the
$\textrm{U}_q^{\prime}(\mathfrak {gl}_1)$
-action.
Theorem 3.10 (Schur–Weyl duality). The map
$ \Psi _{\varepsilon }\colon \textrm{TL}(B_n)_{\varepsilon }\rightarrow {\textrm {End}}_{\textrm{U}_q^{\prime}(\mathfrak {gl}_1)}(V_{\varepsilon }^{\otimes n})$
is an isomorphism of
$\mathbb {C}(q)$
-algebras.
Proof. First assume
$\varepsilon =1$
. Using Schur–Weyl duality [Reference Ehrig and Stroppel8], Theorem 10.4] between
$\textrm{U}_q^{\prime}(\mathfrak {gl}_1)$
and
$\mathcal {H}_{1,q}(B_n)$
, the morphism
$\Phi _{1}$
is surjective and hence, by the commutativity of (4), the map
$\Psi _{1}$
is also surjective. In order to show injectivity we compare dimensions. By the explicit decomposition into isotypical components in Theorem2.6, the dimension of
${\textrm {End}}_{\textrm{U}_q^{\prime}(\mathfrak {gl}_1)}(V_{1}^{\otimes n})$
is
$\sum _{k=0}^{n}\binom {n}{k}^2$
which agrees with
$\dim _k \textrm{TL}(B_n)_{\varepsilon }=\binom {2n}{n}$
from [Reference Green10]. For
$\varepsilon =-1$
, note first that
$V_{1}\cong V_{-1}\otimes k_{-1}$
. Then use
$V_{1}^{\otimes n}\cong V_{-1}^{\otimes n}\otimes k_{-1}^{\otimes n}$
and the fact that
${-} \otimes k_{-1}$
is a self-inverse functor.
4. Presentations via generalized string calculus
In this section, we consider a monoidal category
$\mathcal {V}=(\mathcal {V},{\otimes })$
of
$\textrm{U}_q(\mathfrak {sl}_2)$
-representations and a module category
$\mathcal {M}=\mathcal {M}_{(\mathcal {V}, \otimes )}$
of
$\textrm{U}_q^{\prime}(\mathfrak {gl}_1)$
-representations. For each of them, we provide a diagrammatic presentation,
$\mathscr {V}=(\mathscr {V}, \otimes )$
respectively
$\mathscr {M}=\mathscr {M}_{(\mathscr {V}, \otimes )}$
.
4.1. Diagrammatics for
$\textrm{U}_q(\mathfrak {sl}_2)$
We first summarize (without proofs) easy facts.
Definition 4.1.
Let
$\mathcal {V}=\langle V_1,k_{-1}\rangle$
be the full monoidal subcategory of
$\textbf{Rep}(\textrm{U}_q(\mathfrak {sl}_2))$
generated by
$V_1$
and
$k_{-1}$
.
Proposition 4.2. The following hold:
-
i) If
$L$
is an irreducible representation of
$\textrm{U}_q(\mathfrak {sl}_2)$
of type
$1$
, then
$L\otimes k_{-1}$
is an irreducible representation of type
$-1$
. -
ii) Let
$n\geq 1$
,
$\varepsilon, \varepsilon '\in \{1,-1\}$
. The Clebsch–Gordan rule holds:
\begin{equation*} V_{n,\varepsilon }\otimes V_{\varepsilon^{\prime}}\cong V_{n+1,\varepsilon \varepsilon^{\prime}}\oplus V_{n-1,\varepsilon \varepsilon^{\prime}}. \end{equation*}
-
iii) The following map defines an isomorphism of
$\textrm{U}_q(\mathfrak {sl}_2)$
-representations:
-
iv) Let
$\varepsilon _1,\ldots, \varepsilon _r, \varepsilon^{\prime}_{1},\ldots, \varepsilon^{\prime}_{s}\in \{-1,1\}$
. We have vector spaces
\begin{equation*} \textrm {Hom}(V_{\varepsilon _1}\otimes \cdots \otimes V_{\varepsilon _r}, V_{\varepsilon _{1}^{\prime}}\otimes \cdots \otimes V_{\varepsilon^{\prime}_{s}})\!\cong \!\begin {cases} \textrm {Hom}(V_1^{\otimes r}, V_1^{\otimes s})& \text {if } \varepsilon _1\cdots \varepsilon _{r}=\varepsilon^{\prime}_{1}\cdots \varepsilon _{s}^{\prime}, \\ 0 & \text {else.} \end {cases} \end{equation*}
Definition 4.3.
Let
$\mathscr {V}$
be the
$k$
-linear monoidal category generated by two objects 
and
$-$
, and morphisms
subject to the following relations:
-
(
$\mathscr {V}$
1) invertibility of braiding:
-
(
$\mathscr {V}$
2) snake relations:
-
(
$\mathscr {V}$
3) bubble relations:
-
(
$\mathscr {V}$
4)
${-}{-}\cong{1}$
-relation:
-
(
$\mathscr {V}$
5) Brauer relations:
Proposition 4.4. The following statements about
$\mathscr {V}$
hold:
-
i) The vertically reflected images of the Brauer relations hold:
-
ii) A compatibility between the snake and braid holds:
-
iii) If
$x,y$
are words in such that
$\#\{{-}\text { in }x\}$
and
$\#\{ {-}\text { in }y\}$
are different modulo
$2$
, then
$\textrm {Hom}_{\mathscr {V}}(x,y)=0$
. -
iv) For a word
$z$
in , let
$\bar {z}$
be
$z$
with all
$-$
removed. Set
$z'\;:\!= \bar {z}$
, if this number of
$-$
removed was even and
$z'\;:\!= {-} \bar {z}$
if the number of
$-$
removed was odd. Then for all
$x,y$
words in we have
$\textrm {Hom}_{\mathscr {V}}(x,y)\cong \textrm {Hom}_{\mathscr {V}}(x',y')$
. -
v) For
$n\in \mathbb {N}_0$
Let
$r,s\in \mathbb {N}_0$
. Then (respectively is spanned by crossingless completely red matchings (respectively crossingless red matchings with a blue strand on the left).
such that 
, let 
we have 
Let 
(respectively 
is spanned by crossingless completely red matchings (respectively crossingless red matchings with a blue strand on the left).
Proof. For i) use first the Brauer relation (
$\mathscr {V}5$
) and to it cups at the bottom and apply the snake relation (
$\mathscr {V}2$
) on both sides. ii) follows by applying relation (
$\mathscr {V}5$
) to a subdiagram and then applying a snake relation. iii) is clear by definition. To prove iv) one can use the isomorphisms 
, 
, 
, 
to obtain
$z\cong z'$
in
$\mathscr {V}$
. Hence
$\textrm {Hom}_{\mathscr {V}}(x,y)\cong \textrm {Hom}(x',y')$
. We divide the proof of v) into two claims:
Claim 1: We can assume without loss of generality that the diagram we consider contains no closed components
$C$
. To see this take a diagram
$d$
which contains such a closed component
$C$
(when writing
$d$
as a composition of generators tensored with identities). Thus
$C$
must contain a cup and a cap. We do a case by case distinction: Case 1:
$C$
is a small circle
$\bigcirc$
. Using bubble removal,
$C$
can be replaced by a scalar. Case 2:
$C$
contains a kink 
or 
. Then replace
$C$
by
$\vert$
using the snake relations. Case 3: The only remaining possibility is that the closed component gets crossed by some number of strands of a different color. Using height moves, the invertibility of the braiding and the Brauer relation repeatedly we can assume the strands that cross
$C$
, start on one side of
$C$
and end on the different side of
$C$
. For instance:
Now the claim follows by induction on the number of strands crossing
$C$
(in the top right picture this number is two) using the Brauer relation and the invertibility of the braiding.
Claim 2: Assume
$d$
is a diagram in this
$\textrm {Hom}$
-space. Then one of the following holds.
-
(a) The diagram is entirely red, that is there is no blue component at all. (In this case the statement is just about the usual Temperley–Lieb category.)
-
(b) We can rewrite
$d$
as
In case d 
part (a) follows by Claim 1. If d 
part (a) we have only one blue component by Claim 1. This component is a strand
$S$
connecting the two
$-$
. Let
$m$
be the maximum of strands which appear at some level to the left of
$S$
. If
$m=0$
, we are done. If
$m\gt 0$
, look at the first time this appears (reading from top to bottom). After applying a height move, we end up in one of the following two situations:
In any case, we reduce
$m$
or at least reduce the number of times
$m$
is achieved. Repeating this argument gives finally
$m=0$
, which shows (b).
Now we can apply the usual type
$A$
Temperley–Lieb category diagrammatics to the completely red part, and v) follows.
Definition 4.5.
We define morphisms of
$\textrm{U}_q(\mathfrak {sl}_2)$
-modules
by the formulas
Theorem 4.6. The assignments
extend to an equivalence of
$k$
-linear monoidal categories
$\Phi \colon \mathscr {V}\to \mathcal {V}$
.
Proof. Well-defined
$k$
-linear monoidal functor: Relation (
${\mathscr {V}1}$
) is clear by definition, relation (
${\mathscr {V}2}$
) is a standard type
$1$
relation, for blue this is clear by definition, (
${\mathscr {V}3}$
) and (
${\mathscr {V}4}$
) are obvious, for (
$\mathscr {V}5$
) this is an explicit calculation.
Fullness: Let
$x,y\in \mathcal {V}$
. By Proposition4.2 (
$\mathscr {V}4$
) we may assume
${\textrm {sgn}}\ {x}={\textrm {sgn}}\ {y}$
. By the proof of (
$\mathscr {V}4$
), any morphism can be written as a composition as shown in Figure 1 i).
Compositions with
$f\in \textrm {Hom}_{\textrm{U}_q(\mathfrak {sl}_2)}(V_1^{\otimes r}, V_1^{\otimes s})$
for some
$r,s$
.
Hence, fullness follows from the analogous statement for type
$1$
, which gives that
$f$
lies in the image of the usual Temperley–Lieb category.
Faithfulness: Since the vector space generators from Proposition4.4 v) are sent bijectively to the type
$A$
Temperley–Lieb basis of
${\textrm {End}}_{\textrm{U}_q(\mathfrak {sl}_2)}(V_1^{\otimes n})$
, they are linearly independent and
$\Phi$
is faithful.
Remark 4.7.
The relations imply that there is an action of the (ordinary type
$A$
braid group) on
$\mathscr {V}$
via the morphisms
That is, these morphisms satisfy the braid relations
$\beta _1\beta _2\beta _1=\beta _2\beta _1\beta _2$
involving three strands (of arbitrary color), where
$\beta _1$
and
$\beta _2$
denote the crossing of the first respectively last two strands (with the color induced from the chosen color of the strands). Naturality holds however only up to sign. Thus, these morphisms do not provide the structure of a braided monoidal category, see also [[Reference Jantzen13].
4.2. Diagrammatics for
$\textrm{U}_q^{\prime}(\mathfrak {gl}_1)$
Recall notation from Definition2.2.
Definition 4.8.
Let
$\mathcal {M}=\langle k_1\rangle \mathcal {V}\subseteq \textbf{Rep}(\textrm{U}_q^{\prime}(\mathfrak {gl}_1))$
be the right module full subcategory generated by
$k_{1}={1}$
over the monoidal category
$\mathcal {V}\subseteq \textbf{Rep}(\textrm{U}_q(\mathfrak {sl}_2))$
, see [Reference Häring-Oldenburg11]. By definition, this means we have a commutative diagram:
Definition 4.9.
Let
$(\mathscr {M}, \otimes \colon \mathscr {M}\times \mathscr {V} \to \mathscr {M})$
be the
$k$
-linear right module category over
$\mathscr {V}$
generated by one object
${1}'$
and morphisms
subject to the relations
-
(
$\mathscr {M}$
1) quadratic relations:
-
(
$\mathscr {M}$
2) left unit and counit:
-
(
$\mathscr {M}$
3)
${1}'\cong {1}'\otimes {-}$
-relations:
-
(
$\mathscr {M}$
4) wrecking ball-relation:
Remark 4.10.
Recall the definition of the red crossing from Remark
4.7
. Then (
${\mathscr {M}1}$
) implies the following reflection equation:
The involved dotted braid morphism is the type
$D_2$
full twist. Similarly to type
$A$
[[Reference Rozansky24], Theorem 2.2], powers of this full twist, after renormalization with
$q^2$
, converge in
$q^{-1}$
-adic norm to the type
$D_2$
Jones–Wenzl projector
$d_{2,1}$
, see Theorem
5.6
for the explicit formula. For a precise statement and proof see [Reference Wojciechowski31].
Theorem 4.11.
There is a
$k$
-linear equivalence of categories
$\Phi '\colon \mathscr {M}\to \mathcal {M}$
, which is a bijection on objects such that the following diagram commutes
It defines an equivalence of module categories via the equivalence from Theorem 4.6 .
Proof of Theorem 4.11. We define
$\Phi '\colon \mathscr {M} \to \mathcal {M}$
by mapping
${1}'\mapsto k_{1}$
and
using notation from Definition3.6.
Functor of module categories: All the images define morphisms of
$\textrm{U}_q^{\prime}(\mathfrak {gl}_1)$
-modules and checking the relations is an easy calculation.
Fullness: Let
$x,y\in \mathcal {M}$
, that is tensor products of
$V_{1}$
and
$k_{\pm 1}$
tensored with
$k_{1}={1}$
on the left. We use the braiding from Remark4.7 and blue cups and caps to replace
$x, y$
with
$\textrm{U}_q(\mathfrak {sl}_2)$
-isomorphic objects
$x'\cong x$
,
$y'\cong y$
with at most one tensor factor
$k_{-1}$
on the very left. Concretely
$x'=V_1^{\otimes r}$
(or
$x'=k_{-1}\otimes V_1^{\otimes r}$
) and
$y'=V_1^{\otimes s}$
(or
$y'=k_{-1}\otimes V_1^{\otimes s}$
) for some
$r,s\geq 0$
. Then use the
$\textrm{U}_q^{\prime}(\mathfrak {gl}_1)$
-equivariant isomorphisms 
to replace, if necessary, the left tensor factor
$k_{-1}$
with
$k_1$
. This allows to write any
$\textrm{U}_q^{\prime}(\mathfrak {gl}_1)$
-equivariant linear map
$x\to y$
as a composition of the form displayed in Figure 1 ii). The space
$\textrm {Hom}_{\textrm{U}_q^{\prime}(\mathfrak {gl}_1)}(V_1^{\otimes r}, V_1^{\otimes s})$
becomes zero, if
$r+s$
is odd, by Theorem2.6. Hence we can assume
$r+s=2n$
for some
$n\geq 0$
. After using that
$V_1$
is self-dual via the cup and cap morphisms we get from an adjunction
\begin{equation*} \textrm {Hom}_{\textrm{U}_q^{\prime}(\mathfrak {gl}_1)}(V_1^{\otimes r}, V_1^{\otimes s})\cong \textrm {Hom}_{\textrm{U}_q^{\prime}(\mathfrak {gl}_1)}(V_1^{\otimes n}, V_1^{\otimes n})\cong \textrm{TL}(B_n)_1. \end{equation*}
Since all the generators of
$\textrm{TL}(B_n)_1$
lie in the image of the functor and all the isomorphisms we used are in the image, we obtain fullness.
Faithfulness: First notice that every object in
$\mathscr {M}$
is isomorphic to some tensor power of 
by diagrammatic analogues of the isomorphisms as above used for
$\mathcal {M}$
. Hence, it is enough to show for every
$r,s\geq 0$
the faithfulness of
$\Phi '$
on the
$\textrm {Hom}$
-space 
. If
$r+s$
is odd, we have by choice of generators 
, so faithfulness is clear. If
$r+s=2n$
is even, use the self-duality of 
to replace 
with 
. In order to conclude the theorem, one shows analogously to the proof of Proposition4.4 v) that that every blue closed component involving 
and 
can be removed, the crucial relation being the wrecking ball (
${\mathscr {M}4}$
). Hence, 
is spanned by completely red type
$B$
Temperley–Lieb diagrams, which form a basis of the image under the functor by Theorem3.10.
5. Different types of Jones–Wenzl projectors
We next connect the direct sum decomposition from Section 2 with the diagrammatics from Section 4. This means we study the Karoubian envelope of
$\mathscr {M}\simeq \mathcal {M}$
, or in more elementary terms, idempotents in 
Let
$\varepsilon \in \{1,-1\}, n\geq 1$
. Recall the Jones–Wenzl projectors, see for example [[Reference Carter, Flath and Saito5], §3.5]:
Definition 5.1.
The type
$A_{n-1}$
Jones–Wenzl projector
$a_{n-1,\varepsilon }$
is visualized as an uncolored (white) box on
$n$
strands and determined by the recursive description
Remark 5.2. Definition 5.1 only deals with the ordinary Jones–Wenzl projector. We omit the higher versions (projecting onto other irreducible representations) here, but refer to [[Reference Wojciechowski31], §3, in particular Definition 3.17] for an analogous treatment.
Definition 5.3.
We define elements
$b_{n,{+}, \varepsilon }, b_{n,{-}, \varepsilon }\in \textrm{TL}(B_n)_{\varepsilon }$
recursively and visualize them as red respectively blue boxes on
$n$
strands:
First we define for
$n=1$
Assuming that
$b_{n, {+}, \varepsilon }$
and
$b_{n, {-}, \varepsilon }$
are already defined, set
We call
$b_{n,{+},\varepsilon }$
(and
$b_{n,{-},\varepsilon }$
) the dominant (respectively antidominant) type
$B_n$
Jones–Wenzl projector of type
$\varepsilon$
. Additionally, we define the type
$D_n$
Jones–Wenzl projector
$d_n$
of type
$\varepsilon$
as the sum
The red and blue projectors appeared abstractly in [[Reference tom Dieck27], §3], in their notation we have
$b_{n,{+},\varepsilon }=g_n$
and
$b_{n,{-},\varepsilon }=f_n$
, however no diagrammatics were involved and the type
$D$
projectors not defined. Our ’new’ diagrammatical depiction is an attempt at generalizing the type
$A$
Jones–Wenzl diagrammatics. The green projectors appeared also in non-quantized form in [Reference Queffelec and Wedrich22], where quotients of affine Temperley–Lieb algebras were studied, in a context at first glance independent of type
$B/D$
.
Remark 5.4.
The involution
$\sigma$
from Lemma
3.3
swaps
$b_{1,{+},\varepsilon }$
and
$b_{1,{-},\varepsilon }$
, thus
$\sigma (b_{n,{+},\varepsilon })=b_{n,{-},\varepsilon }$
for all
$n$
. As a consequence, the type
$D_n$
Jones–Wenzl projector
$d_{n, \varepsilon }$
is contained in the fixed-point subalgebra
$\textrm{TL}(D_n)_{\varepsilon }^\sigma$
, and thus indeed of type
$D$
.
Example 5.5.
Explicitly, the dominant Jones–Wenzl projector of type
$B_2$
equals
whereas the antidominant Jones–Wenzl projector of type
$B_2$
looks like
We will prove recursion formulas for type
$D$
independent of the type
$B$
result:
Theorem 5.6 (Type
$D$
recursion). The type
$D$
projectors
$d_{n,\varepsilon }$
are described by the following recursive formulas:
Before the proof, we establish diagrammatic properties of type
$B$
projectors.
5.1. Diagrammatic properties
The following theorem formulates an important property of Jones–Wenzl projectors: The annihilation of Temperley–Lieb generators.
Theorem 5.7 (Type
$B$
uniqueness). Let
$\eta \in \{{+},{-}\}$
. The element
$b_{n,\eta, \varepsilon }$
is the unique non-zero idempotent in
$\textrm{TL}(B_n)_{\varepsilon }$
which satisfies
\begin{equation} s_0b_{n,\eta, \varepsilon }=\eta b_{n,\eta, \varepsilon }=b_{n,\eta, \varepsilon }s_0, \quad U_ib_{n,\eta, \varepsilon }=0=b_{n,\eta, \varepsilon }U_i \, \end{equation}
for all
$i=1,\ldots, n-1$
.
Proof. The proof is analogous to the type
$A$
proof, see for instance [Reference Lickorish18], Lem. 2]. For uniqueness one uses the decomposition
\begin{equation} \textrm{TL}(B_n)_{\varepsilon }={\langle 1, s_0\rangle } \oplus (U_1,\ldots, U_{n-1}), \end{equation}
The rest of the proof follows by simultaneous induction over the three statements:
-
(I
$_{n}$
)
$b_{n \eta, \varepsilon }$
is an idempotent. -
(II
$_{n}$
)
$b_{n, \eta, \varepsilon }$
annihilates
$U_j$
, if
$1\leq j \leq n-1$
. -
(III
$_{n}$
)
$U_{n}b_{n, \eta, \varepsilon }U_n=-\varepsilon \frac {q^{n}+ q^{{-}n}}{q^{n-1}+ q^{{-}(n-1)}}U_{n}b_{n-1, \eta, \varepsilon }=-\varepsilon \frac {q^{n}+ q^{{-}n}}{q^{n-1}+ q^{{-}(n-1)}}b_{n-1, \eta, \varepsilon }U_{n}$
.
As a consequence of Definition5.3, Theorem5.7 implies two stacking rules.
Corollary 5.8 (Box absorption). Let
$n\geq m\geq 1$
,
$\eta \in \{ {+}, {-}\}$
. We have
\begin{equation*} b_{n,\eta, \varepsilon } b_{m,\eta, \varepsilon } = b_{n,\eta, \varepsilon } = b_{m,\eta, \varepsilon } b_{n,\eta, \varepsilon } \end{equation*}
which we indicate diagrammatically by
Corollary 5.9 (Red-blue orthogonality). Let
$n,m\geq 1$
. The dominant and antidominant type
$B$
projectors are orthogonal, that is we have
\begin{equation*} b_{n,{+}, \varepsilon } b_{m,{-},\varepsilon }=0=b_{m,{-},\varepsilon } b_{n,{+}, \varepsilon } \end{equation*}
which we display diagrammatically as
These rules and Theorem5.7 imply:
Corollary 5.10.
The element
$d_{n,\varepsilon }$
is non-zero, idempotent and satisfies
\begin{equation*} s_0d_{n,\varepsilon }s_0=d_{n,\varepsilon }, \quad d_{n,\varepsilon }U_i= 0 = U_i d_{n,\varepsilon } \, \textit {\ for\ all\ } i=0,1,\ldots, n-1. \end{equation*}
We conclude the following symmetry relating type
$B$
and
$D$
projectors.
Corollary 5.11 (Symmetry). Let
$\eta \in \{{+},{-}\}$
. Then
$b_{n,\eta, \varepsilon }=\frac {1}{2}(d_{n,\varepsilon }+\eta s_0d_{n,\varepsilon })$
.
Proof. Corollary5.10 implies that the element
$y_{\eta }\;:\!=\frac {1}{2}(d_{n,\varepsilon }+\eta s_0d_{n,\varepsilon })$
is idempotent, kills
$U_1,\ldots, U_{n-1}$
and satisfies
$s_0y_{\eta }=\eta y_{\eta }=y_{\eta }s_0$
. Moreover it is non-zero, since
$0\neq d_{n,\varepsilon }\in \textrm{TL}(D_n)_{\varepsilon }$
, which implies
$d_{n,\varepsilon }$
and
$s_0d_{n,\varepsilon }$
are linearly independent. Hence
$y_{\eta }=b_{n,\eta, \varepsilon }$
by the uniqueness in Theorem5.7.
Corollary 5.12 (Type
$D_n$
uniqueness). The projector
$d_{n,\varepsilon }$
is the unique element in
$\textrm{TL}(D_n)_{\varepsilon }$
, which satisfies the properties from Corollary
5.10
.
Proof. Let
$x\in \textrm{TL}(D_n)_{\varepsilon }$
and assume it satisfies the properties from Corollary5.10. By the argument as for Corollary5.11 we get
$b_{n,\eta, \varepsilon }=\frac {1}{2}(x+\eta s_0x)$
for
$\eta \in \{{+},{-}\}$
. Hence by definition,
$d_{n,\varepsilon }=b_{n,{+},\varepsilon }+b_{n,{-},\varepsilon }=\frac {1}{2}(x+s_0x)+\frac {1}{2}(x- s_0x)=x.$
We can now also prove the recursive formula for the type
$D$
projectors.
Proof of Theorem 5.6. Assume
$n\geq 2$
. The recursive formula follows from
where
$\lambda _n\;:\!=\varepsilon \frac {q^{n-1}+q^{-(n-1)}}{q^n+q^{-n}}$
. We used in the non-trivial step that
\begin{equation*} b_{n,{-}, \varepsilon } U_nb_{n,{+}, \varepsilon } = 0 = b_{n,{+}, \varepsilon } U_n b_{n,{-}, \varepsilon }. \end{equation*}
Indeed, Corollary5.8 implies that
$b_{n,\eta, \varepsilon }=b_{1,\eta, \varepsilon }b_{n,\eta, \varepsilon }b_{1,\eta, \varepsilon }$
which we can apply to both
$+,-$
to factor out a copy of
$b_{1,{+}, \varepsilon }$
and
$b_{1,{-}, \varepsilon }$
each. Since
$b_{1,\eta, \varepsilon }$
and
$U_n$
commute as
$n\geq 2$
, the statement follows now from Corollary5.9 for
$b_{1,+,\varepsilon }$
and
$b_{1,-,\varepsilon }$
.
The recursive formulas for the type
$D$
projectors in Definition5.3 appeared in non-quantized form in [Reference Queffelec and Wedrich22] as idempotents projecting onto the maximal and lowest weight space of the standard representation of
$\mathfrak {sl}_2$
. The next theorem lifts this interpretation to the quantum coideal setting.
Consider the tensor product decomposition of
$V_{\varepsilon }^{\otimes n}$
over
$\textrm{U}_q^{\prime}(\mathfrak {gl}_1)$
in Theorem2.6. We write
${+}\;:\!= 1$
and
${-}\;:\!= -1$
to make notation compatible, so that we can consider the elements
$b_{n,\varepsilon, \varepsilon }\;:\!= b_{n,\varepsilon }$
and
$ b_{n,-\varepsilon, \varepsilon }\;:\!= b_{n,-\varepsilon } \in \textrm{TL}(B_n)_{\varepsilon }$
.
Theorem 5.13.
Under the isomorphism
$\Psi _{\varepsilon }\colon \textrm{TL}(B_n)_{\varepsilon }\xrightarrow {\cong } {\textrm {End}}_{\textrm{U}_q^{\prime}(\mathfrak {gl}_1)}(V_\varepsilon ^{\otimes n})$
from Theorem
3.10
, the projector
$b_{n,\varepsilon, \varepsilon }$
becomes the idempotent projection
\begin{equation*} \pi _{n,\varepsilon, \varepsilon }\colon V_{\varepsilon }^{\otimes n}\rightarrow L(\varepsilon ^{n-1}[n])\hookrightarrow V_{\varepsilon }^{\otimes n}, \end{equation*}
and the projector
$b_{n,-\varepsilon, \varepsilon }$
becomes the idempotent projection
\begin{equation*} \pi _{n,-{\varepsilon },\varepsilon }\colon V_{\varepsilon }^{\otimes n}\rightarrow L({-}\varepsilon ^{n-1}[n])\hookrightarrow V_{\varepsilon }^{\otimes n}. \end{equation*}
The type
$D_n$
Jones–Wenzl idempotent
$d_{n,\varepsilon }$
projects onto
$L([n])\oplus L({-}[n])$
.
Proof. The strategy of the proof is to verify that the images
$\Psi _{\varepsilon }^{-1}(\pi _{n,\varepsilon, \varepsilon })$
,
$\Psi _{\varepsilon }^{-1}(\pi _{n,-\varepsilon, \varepsilon })$
satisfy the characterizing properties of the projectors
$b_{n,\varepsilon, \varepsilon }$
and
$b_{n,-\varepsilon, \varepsilon }$
using the explicit vectors in the decomposition in Theorem2.6. First observe that
$\Psi _{\varepsilon }(s_0)(v_{0,\varepsilon })=\varepsilon v_{0,\varepsilon }$
and similarly
$\Psi _{\varepsilon }(s_0)(w_{0,\varepsilon })=-\varepsilon w_{0,\varepsilon }$
. The subrepresentation
$L(\varepsilon ^{n-1}[n])$
is spanned by
\begin{equation*} v_{0,\varepsilon }\otimes v_{1,\varepsilon } \otimes \cdots \otimes v_{n-1,\varepsilon } \text { for } \varepsilon =1 \quad \text {and}\quad v_{0,\varepsilon } \otimes w_{1, \varepsilon } \otimes v_{-2,\varepsilon } \otimes w_{3,\varepsilon } \otimes \cdots \text { for } \varepsilon =-1 \end{equation*}
by Theorem2.6 ii), where in the
$\varepsilon =-1$
case the tensor factors alternate between
$v$
’s and
$w$
’s with negative subscripts for the
$v$
’s. In particular,
$s_0$
acts on
$L(\varepsilon ^{n-1}[n])$
by
$\varepsilon$
by the first observation. Similarly, the subrepresentation
$L({-}\varepsilon ^{n-1}[n])$
is spanned by
\begin{equation*} w_{0,\varepsilon }\otimes w_{-1,\varepsilon }\otimes \cdots \otimes w_{-(n-1),\varepsilon } \text { if } \varepsilon =1, \, w_{0,\varepsilon } \otimes v_{-1, \varepsilon } \otimes w_{2, \varepsilon } \otimes v_{-3, \varepsilon } \otimes \cdots \text { if } \varepsilon =-1. \end{equation*}
Hence
$s_0$
acts by
$-\varepsilon$
on
$L({-}\varepsilon ^{n-1}[n])$
. We conclude that
$\Psi ^{-1}(\pi _{n,\varepsilon, \varepsilon })$
satisfies
\begin{equation*} s_0\Psi ^{-1}(\pi _{n,\varepsilon, \varepsilon })=\varepsilon \Psi ^{-1}(\pi _{n,\varepsilon, \varepsilon })=\Psi ^{-1}(\pi _{n,\varepsilon, \varepsilon })s_0, \end{equation*}
and similarly that
$\Psi ^{-1}(\pi _{n,-{\varepsilon },\varepsilon })$
satisfies
\begin{equation*} s_0\Psi ^{-1}(\pi _{n,-\varepsilon, \varepsilon })=-\varepsilon \Psi ^{-1}(\pi _{n,-\varepsilon, \varepsilon })=\Psi ^{-1}(\pi _{n,\varepsilon, \varepsilon })s_0. \end{equation*}
Next in order to conclude that
$U_i\Psi ^{-1}(\pi _{n,\varepsilon, \varepsilon })=0$
and
$U_i\Psi ^{-1}(\pi _{n,-\varepsilon, \varepsilon })=0$
, observe
$[n]$
and
$-[n]$
do not appear as eigenvalues of
$B$
acting on
$V_{\varepsilon }^{\otimes l}$
by Theorem2.6 iv) whenever
$l\lt n$
. This shows that all morphisms
$V_{\varepsilon }^{\otimes n}\rightarrow V_{\varepsilon }^{\otimes n}$
which factor through such
$V_{\varepsilon }^{\otimes l}$
become zero, when composed with the projection onto
$L([n])$
respectively
$L({-}[n])$
. In particular, this applies to the morphisms
This shows that
$\Psi ^{-1}(\pi _{n,\varepsilon, \varepsilon })$
(and
$\Psi ^{-1}(\pi _{n,-\varepsilon, \varepsilon })$
) satisfies the characterizing properties of
$b_{n,\varepsilon, \varepsilon }$
(respectively
$b_{n,-\varepsilon, \varepsilon }$
), and hence they agree with these elements by Theorem5.7. The claim for
$d_{n,\varepsilon }$
follows from
$d_{n,\varepsilon }=b_{n,{+}, \varepsilon }+b_{n,{-}, \varepsilon }=b_{n,\varepsilon, \varepsilon }+b_{n,-\varepsilon, \varepsilon }.$
We finally illustrate how the diagrammatics can be used to compute
$\textrm {Hom}$
-spaces.
Example 5.14.
Let
$a,b,c\geq 0$
. Assume everything is type
$1$
for simplicity. There are four kinds of
$\textrm {Hom}$
-spaces over the coideal we are interested in:
In order to determine these
$\textrm {Hom}$
-spaces one can either use Corollary
2.7
or count all diagrams connecting three boxes of appropriate colors and numbers of strands. For example, for
$a=b=c=2$
one can see by Corollary
2.7
\begin{equation*} (L([2])\oplus L([{-}2]))\otimes V_2\cong L([4])\oplus L([2]) \oplus L([0])^2 \oplus L({-}[2]) \oplus L({-}[4]), \end{equation*}
which implies
$\dim \textrm {Hom}((L([2])\oplus L([{-}2]))\otimes V_2, L([2])\oplus L([{-}2]))=2$
. The corresponding diagram basis consists of the two only possible connections
between
$d_2$
,
$a_1$
on the bottom and
$d_2$
on the top. If one replaces the target
$L([2])\oplus L([{-}2])$
with
$V_2 \cong L([2])\oplus L(0)\oplus L([{-}2])$
over the coideal, the dimension of the
$\textrm {Hom}$
-space increases by
$2$
. This corresponds to the two additional diagrams
which become possible since the type
$A$
projector doesn’t kill a dotted cup. This is compatible with Example
2.9
. Thinking further, if one replaces the first tensor factor
$L([2])\otimes L({-}[2])$
in the source with
$V_2=L([2])\oplus L(0)\oplus L({-}[2])$
one obtains
\begin{equation*} V_2 \otimes V_2\cong L([4])\oplus L([2])^2 \oplus L([0])^3 \oplus L({-}[2])^2 \oplus L({-}[4]) \end{equation*}
which increases the dimension of
$\textrm {Hom}$
by
$3$
. The additional three diagrams are:
6. Application:
$\Theta$
–networks
We finish the paper by an outlook how our tools allow to define and compute
$\Theta$
-networks generalizing the networks arising in the Turaev–Viro invariants, [Reference Carter, Flath and Saito5] and in the context of
$3j$
-symbols, see also [Reference Frenkel, Stroppel and Sussan9]. First recall:
Definition 6.1.
A triple
$(a,b,c)\in {\mathbb {N}_0}^{3}$
is called admissible if one (all) of the following equivalent conditions is satisfied:
-
i)
$a\in \{b+c,b+c-2,b+c-4,\ldots, |b-c|\}$
-
ii)
$b\in \{a+c,a+c-2,a+c-4,\ldots, |a-c|\}$
-
iii)
$c\in \{a+b,a+b-2,a+b-4,\ldots, |a-b|\}$
To any admissible triple
$(a,b,c)$
we associate a triple
$(i,j,k)\in \mathbb {N}_0^{3}$
by
\begin{equation*} i\;:\!= \frac {b+c-a}{2}, \, j\;:\!= \frac {a+c-b}{2}, \, k\;:\!= \frac {a+b-c}{2}. \end{equation*}
In particular, we can recover
$(a,b,c)$
from
$(i,j,k)$
as
$a=j+k, \quad b=i+k, \quad c=i+j$
.
Lemma 6.2.
The following are equivalent for a triple
$(a,b,c)\in \mathbb {N}_0^{3}$
:
-
•
$(a,b,c)$
is admissible,
-
•
$\textrm {Hom}_{\textrm{U}_q(\mathfrak {sl}_2)}(V_a\otimes V_b, V_c)\neq 0$
, -
•
$\textrm {Hom}_{\textrm{U}_q^{\prime}(\mathfrak {gl}_1)}(V_a\otimes V_b, V_c)\neq 0$
, -
•
$\textrm {Hom}_{\textrm{U}_q^{\prime}(\mathfrak {gl}_1)}(V_a\otimes V_b, L([c]))\neq 0$
, -
•
$\textrm {Hom}_{\textrm{U}_q^{\prime}(\mathfrak {gl}_1)}(V_a\otimes V_b, L([{-}c]))\neq 0$
,
-
•
$\textrm {Hom}_{\textrm{U}_q^{\prime}(\mathfrak {gl}_1)}(L([a])\otimes V_b, V_c)\neq 0$
, -
•
$\textrm {Hom}_{\textrm{U}_q^{\prime}(\mathfrak {gl}_1)}(L([{-}a])\otimes V_b, V_c)\neq 0$
, -
•
$\textrm {Hom}_{\textrm{U}_q^{\prime}(\mathfrak {gl}_1)}(L([a])\otimes V_b, L([c]))\neq 0$
, -
•
$\textrm {Hom}_{\textrm{U}_q^{\prime}(\mathfrak {gl}_1)}(L([{-}a])\otimes V_b, L([{-}c]))\neq 0$
.
Proof. This follows from Corollary2.7.
Remark 6.3.
By the Clebsch–Gordan rule, the first
$\textrm {Hom}$
-space in Lemma
6.2
is at most one dimensional. The up-to-scalar unique non-zero morphism is a unique type
$A$
planar diagram connecting the type
$A$
projectors
$a_{a-1}, a_{b-1}, a_{c-1}$
, where
$i$
many strands connect
$a_{b-1}$
and
$a_{c-1}$
,
$j$
many connect
$a_{a-1}$
and
$a_{c-1}$
, and
$k$
many connect
$a_{a-1}$
and
$a_{b-1}$
. As we saw in Example
5.14
, the
$\textrm {Hom}$
-spaces in Lemma
6.2
are in general higher dimensional with bases labeled by all possible type
$B$
diagrams one can draw between them. Hence in order to define analogues
$\Theta$
-networks one has to not only fix an admissible triple but also fix a type
$B$
diagram connecting the projectors. For the rest of the section we work with type
$D$
projectors, instead of type
$B$
, and fix the standard type
$A$
connection. The first choice does not lose any information, since the type
$D$
projector carries all of the information about the type
$B$
projectors. A full treatment of all the choices is material of another paper, which also treats
$3j$
-symbols similarly to [[Reference Frenkel, Stroppel and Sussan9], Theorem 8] with respect to the bases we defined.
Definition 6.4.
Let
$\varepsilon \in \{1,-1\}$
. Let
$(a,b,c)$
be an admissible triple. A
$\Theta$
-network
$\Theta _{a,b,c,\varepsilon }^D$
of type
$D$
is the rational function
$f(1)$
, where
$f\colon \mathbb {C}(q)\rightarrow \mathbb {C}(q)$
is given by
The outer cup and cap should be read as
$j$
nested strands, the left small cup/cap as
$k$
nested strands and the right small cup/cap as
$i$
nested strands.
The following theorem gives a recursive description of type
$D$
$3j$
-symbols.
Theorem 6.5.
For all
$a\geq 1$
we have
-
i)
$\Theta _{a,0,a,\varepsilon }^D=({-}\varepsilon )^a \left (q^a+q^{-a}\right )$
-
ii)
$\Theta _{a,1,a-1,\varepsilon }^D=({-}\varepsilon )^a \left (q^a+q^{-a}\right )$
-
iii)
$\Theta _{a-1,1,a,\varepsilon }^D=({-}\varepsilon )^a \frac {[a]}{[a-1]}\left (q^{a-1}+q^{-(a-1)}\right )$
Moreover for any admissible triple
$(a,b,c)$
with
$a\geq 1, b\geq 2, c\geq 1$
the type
$D$
$3j$
-symbol
$\Theta _{a,b,c,\varepsilon }^D$
satisfies the following recursion:
\begin{equation*} \Theta _{a,b,c,\varepsilon }^D=-\varepsilon \frac {[c]}{[c-1]}\Theta _{a,b-1,c-1,\varepsilon }^D+\varepsilon \frac {[(a+b-c)/2]^2}{[b][b-1]}\Theta _{a,b-2,c,\varepsilon } \end{equation*}
Before we outline a proof of the theorem, we recall two useful formulas for computing
$\Theta$
-networks in type
$A$
from [[Reference Masbaum and Vogel21], Lemmas 1 and 2]. Both lemmas describe rules how to delete strands, which is helpful for recursion.
Lemma 6.6.
Let
$i,j,k\in \mathbb {N}^3_0$
such that
$i,j\geq 1$
. Then we have:
Lemma 6.7.
Let
$n\geq 1$
. Then we have
The characterizing property of the type
$D$
Jones–Wenzl projector directly implies:
Lemma 6.8.
We have
$d_{n,\varepsilon }a_{n-1,\varepsilon }=d_{n,\varepsilon }=a_{n-1,\varepsilon }d_{n,\varepsilon }$
. Diagrammatically speaking:
The next lemma is a type
$D$
version of Lemma6.7.
Lemma 6.9.
Let
$n\geq 2$
. We have the following formula:
\begin{equation*} U_nd_{n, \varepsilon }U_n=-\varepsilon \frac {q^{n}+q^{-n}}{q^{n-1}+q^{-(n-1)}} U_nd_{n-1,\varepsilon }. \end{equation*}
Diagramatically this becomes:
Proof. We already proved this statement for
$b_{n,{+}, \varepsilon }$
and
$b_{n,{-}, \varepsilon }$
as part III
$_n$
of the proof of Theorem5.7. Now use
$d_{n,\varepsilon }=b_{n,{+}, \varepsilon }+b_{n,{-}, \varepsilon }$
. The diagrammatic version follows from surjectivity of
$\cap \colon V_{\varepsilon }\otimes V_{\varepsilon }\to \mathbb {C}(q)$
and injectivity of
$\cup \colon \mathbb {C}(q) \to V_{\varepsilon }\otimes V_{\varepsilon }$
.
As direct consequence of the previous lemma, we obtain:
Corollary 6.10.
Let
$n\geq 1$
. We have the following trace formula:
Proof. This statement is clear for
$n=1,2$
by the explicit descriptions. For the induction step use Lemma6.9.
Now we can prove the main theorem:
Proof of Theorem 6.5. The formulas for
$\Theta _{a,0,a,\varepsilon }^D$
,
$\Theta _{a,0,a-1,\varepsilon }^D$
and
$\Theta _{a-1,0,a,\varepsilon }^D$
are direct consequences of Lemma6.7, Lemma6.8, Lemma6.9 and Corollary6.10. For the recursion formula we sketch the (diagrammatic) proof: First apply the type
$A$
Jones–Wenzl recursion formula (Definition5.1) to the middle Jones–Wenzl projector on
$b$
strands. This yields two summands. For the first one use Lemma6.7 to obtain
$-\varepsilon \frac {[c]}{[c-1]}\Theta _{a,b-1,c-1,\varepsilon }^D$
. For the second summand use Lemma6.8 twice to write the type
$D$
Jones–Wenzl projector
$d_{n,\varepsilon }$
as
$a_{n-1,\varepsilon }d_{n,\varepsilon }a_{n-1,\varepsilon }$
. Then bend the two
$A_{n-1}$
projectors over to the right. Afterwards apply Lemma6.6 to the right hand side twice. Then use that
$a_{n-1,\varepsilon }$
is idempotent twice and bend the type
$A$
Jones–Wenzl projectors to the left to obtain
$\varepsilon \frac {[(a+b-c)/2]^2}{[b][b-1]}\Theta _{a,b-2,c,\varepsilon }$
as the second summand.
Acknowledgements
We like to thank the referee for helpful comments. The second author thanks Gaetan Borot and the Humboldt-Universität zu Berlin for the CIMPA school in Puerto Madryn in 2023. Both authors were supported by the Hausdorff Center of Mathematics (EXC 2047) in Bonn.
