1 Introduction
In [Reference Rouquier, de la Peña and BautistaRou06], Rouquier introduced the 2-braid groups associated to the Artin braid groups using Rouquier complexes, which are certain complexes built from Soergel bimodules. As the category of Soergel bimodules is a categorification of the Hecke algebra [Reference SoergelSoe92, Reference SoergelSoe07], the 2-braid group can be viewed as a categorification of the Hecke algebra representation of the corresponding Artin braid group. Within the same paper [Reference Rouquier, de la Peña and BautistaRou06], Rouquier stated the faithfulness of the 2-braid group as a conjecture, where in type A, faithfulness is known due to the results in [Reference Khovanov and SeidelKS02]. This was extended to ADE types in [Reference Brav and ThomasBT10] and later on to all other finite types in [Reference JensenJen16].
In this paper, we provide an alternate proof of faithfulness of the 2-braid groups in the type
$B(=C)$
case. In contrast to the proofs in [Reference Brav and ThomasBT10, Reference JensenJen16], our proof is closer in spirit to the proof of the type A case in [Reference Rouquier, de la Peña and BautistaRou06], which we now briefly recall. In [Reference Khovanov and SeidelKS02], the authors provide a nil-categorification of the type A Temperley–Lieb algebra, using a certain monoidal, additive subcategory of
${\mathscr {A}}$
-bimodules for some quiver algebra
${\mathscr {A}}$
known as the type A zigzag algebra. Moreover, certain complexes over these
${\mathscr {A}}$
-bimodules (almost identical to Rouquier complexes) collectively define a braid group action on the homotopy category of projective modules over
${\mathscr {A}}$
. This action categorifies the Burau representation of the type A braid group and is shown to be faithful. Just as the Temperley–Lieb algebra is a quotient of the Hecke algebra, this subcategory of
${\mathscr {A}}$
-bimodules can be viewed as a quotient category of the category of Soergel bimodules; namely, one can construct an essentially surjective functor from the category of Soergel bimodules to this subcategory of
${\mathscr {A}}$
-bimodules. As such, the faithfulness of the type A 2-braid group follows from the faithfulness of the categorified Burau representation. (This is not to be confused with 2-faithfulness of a braid group involving cobordisms, see [Reference Khovanov and ThomasKT07].)
In our previous work [Reference Heng and NgeHN23], a type B analogue of [Reference Khovanov and SeidelKS02] was developed. In particular, we constructed a quiver algebra
${\mathscr {B}}$
which we called type B zigzag algebra (there is a different zigzag algebra that can be thought of as type C; see Remark 2.2), such that the type B Artin braid group acts faithfully on the homotopy category of projective modules over
${\mathscr {B}}$
using certain complexes of
${\mathscr {B}}$
-bimodules (which again are almost identical to the Rouquier complexes). As such, it is only natural to show that this whole story for type B also fits into the world of 2-braid groups and Soergel bimodules: such is the goal of this paper.
To be more precise, we describe a monoidal, additive subcategory of
${\mathscr {B}}_n$
-bimodules and construct an essentially surjective functor from the category of type
$B_n$
Soergel bimodules to the category of
$(\mathscr{B}, \mathscr{B})$
-bimodules
$\mathcal{B}$
. As in the case for type A, this produces an alternate proof of the faithfulness of the type B 2-braid groups. This result also partly serves as evidence that the zigzag algebra we introduced in [Reference Heng and NgeHN23] is indeed a reasonable definition of a type B zigzag algebra, at least from the point of view of braid group actions.
1.1 Outline of the paper
We follow closely the work of Jensen in [Reference JensenJen14], which spells out the details for type A. In particular, all of our results are formulated using the equivalent diagrammatic category of Soergel bimodules given in [Reference Elias and WilliamsonEW16].
Sections 2 and 3 are brief summaries of type B zigzag algebras
${\mathscr {B}}$
and 2-braid groups, respectively. In Section 2, we also define the relevant category of
$({\mathscr {B}}, {\mathscr {B}})$
-bimodules
${{\mathcal B}}$
(Definition 2.9) that categorifies a quotient of the type B Temperley–Lieb algebra (in the sense of [Reference GreenGre97], see Section 2.3). The main results of this paper and their proofs are all contained in Section 4, where we construct the quotient functor from the category of Soergel bimodules to the category of
$({\mathscr {B}},{\mathscr {B}})$
-bimodules
${{\mathcal B}}$
. The faithfulness of the 2-braid group is then a simple consequence.
2 Type B zigzag algebras
In this section, we recall the relevant objects and results from [Reference Heng and NgeHN23]: the construction of the type
$B_n$
zigzag algebra
${\mathscr {B}}_n$
and the fact that the type
$B_n$
Artin group
$\mathcal {A}(B_n)$
acts faithfully on the bounded homotopy category
${\text {Kom}}^b({\mathscr {B}}_n$
-
$\text {p}_{r}\text {g}_{r}\,\text {mod})$
of projective,
${{\mathbb Z}}$
-graded modules over
${\mathscr {B}}_n$
. We also describe the additive and monoidal category of
${\mathscr {B}}_n$
-bimodules that categorifies a quotient of the type B Temperley–Lieb algebra (in the sense of [Reference GreenGre97]).
2.1 Type
$B_n$
Artin group
For
$n \geq 2,$
the type
$B_n$
Artin group
${\mathcal A}({B_n})$
is the group generated by n generators:
$$ \begin{align*} \sigma_1^B,\sigma_2^B,\ldots, \sigma_{n}^B \end{align*} $$
subject to the relations
$$ \begin{align} \sigma_1^B \sigma_2^B \sigma_1^B \sigma_2^B &= \sigma_2^B \sigma_1^B \sigma_2^B \sigma_1^B; \qquad\qquad\qquad\qquad\qquad\ \end{align} $$
$$ \begin{align} \sigma_j^B \sigma_k^B &= \sigma_k^B \sigma_j^B, \quad\text{for } |j-k|> 1; \qquad\quad\ \ \end{align} $$
$$ \begin{align} \sigma_j^B \sigma_{j+1}^B \sigma_j^B &= \sigma_{j+1}^B \sigma_{j}^B \sigma_{j+1}^B, \quad \text{for } j= 2,3, \ldots, n-1. \end{align} $$
Its corresponding Coxeter group
$W(B_n)$
is generated by
$s_1,s_2,...,s_n$
subject to Equations (2-1), (2-2), (2-3) (with s in place of
$\sigma ^B$
) and
$s_j^2 = 1$
for all
$j \geq 1$
.
2.2 Type
$B_n$
zigzag algebra
${\mathscr {B}}_n$
Consider the quiver
$Q_n$
in Figure 1.
Take its path algebra
${\mathbb R} Q_n$
over
${\mathbb R}$
and consider the path length grading on
${\mathbb R} Q_n$
, where exceptionally the ‘imaginary’ path
$(ie_j)$
has grading
$0.$
Note that
$(ie_j)$
is the loop in Figure 1 and is not to be confused with the constant path
$e_j$
(also length 0). In this paper, we use the notation
$(1)$
to denote a grading shift down by
$1.$
Figure 1 The quiver
$Q_n$
. Note that paths are read from left to right. All arrows have length 1, except the loops
$(ie_j)$
, which have length 0.
We are now ready to define the zigzag algebra of type
$B_n$
.
Definition 2.1. The zigzag path algebra of
$B_n$
, denoted by
${\mathscr {B}}_n$
, is the quotient algebra of the path algebra
${\mathbb R} Q_n$
modulo the usual zigzag relations given by
$$ \begin{align*} (j|j-1)(j-1|j) &= (j|j+1)(j+1|j) \quad (=: X_j);\nonumber\\ (j-1|j)(j|j+1) &= 0 = (j+1|j)(j|j-1); \end{align*} $$
for
$2\leq j \leq n-1$
, in addition to the relations
$$ \begin{align} (ie_j)(ie_j) &= -e_j, \quad \text{for } j \geq 2 ;\\(ie_{j-1})(j-1|j) &= (j-1|j)(ie_j), \quad\ \ \ \text{for } j\geq 3; \nonumber\\(ie_{j})(j|j-1) &= (j|j-1)(ie_{j-1}), \quad \text{for } j\geq 3; \nonumber\\(1|2)(ie_2)(2|1) &= 0; \nonumber\\(ie_2) X_2 &= X_2 (ie_2).\nonumber \end{align} $$
Remark 2.2. One can define a ‘type C’ zigzag algebra by switching the positions of the (length zero) loops
$(ie_j)$
, so that we only have
$(ie_1)$
on vertex 1 and nowhere else. The relations are defined similarly.
Since the relations are all homogeneous with respect to the path length grading,
${\mathscr {B}}_n$
is a
${{\mathbb Z}}$
-graded algebra. As a
${\mathbb R}$
-vector space,
${\mathscr {B}}_n$
has dimension
$8n-6$
, with the following basis:
$$ \begin{align*} \{ &e_1 , \ldots, e_{n}, ie_2 , \ldots, ie_{n}, \\ & (1|2), \ldots, (n-1 | n), (2|1), \ldots, (n | n-1), (ie_2)(2|1), (1|2)(ie_2), (ie_2)(2|3), \ldots, (ie_{n-1})(n-1|n), \\ & (3|2)(ie_2), \ldots, (n|n-1)(ie_{n-1}), (1|2|1), \ldots, (2n-1|2n-2|2n-1), (ie_2)(2|1|2), \ldots, (ie_n)(n|n-1|n) \}. \end{align*} $$
The indecomposable (left) projective
${\mathscr {B}}_n$
-modules are given by
$P^B_j := {\mathscr {B}}_n e_j$
. For
$j=1$
,
$P^B_j$
is naturally a
$({\mathscr {B}}_n, {\mathbb R})$
-bimodule; there is a natural left
${\mathscr {B}}_n$
-action given by the multiplication of the algebra and the right
${\mathbb R}$
-action induced by the natural
${\mathbb R}$
-vector space structure. Nonetheless, for
$j\geq 2$
, we endow
$P^B_j$
with a right
${\mathbb C}$
-action. To this end, note that Equation (2-4) is analogous to the relation satisfied by the complex imaginary number i. We define a right
${\mathbb C}$
-action on
$P^B_j$
by
$p * (a+ib) = ap + bp(ie_j)$
for
$p \in P^B_j, a+ib\in {\mathbb C}$
. Further note that this right action restricted to
${\mathbb R}$
agrees with both the natural right and left
${\mathbb R}$
-action. This makes
$P^B_j$
into a
$({\mathscr {B}}_n,{\mathbb C})$
-bimodule for
$j\geq 2$
. Dually, we define
${}_jP^B := e_j{\mathscr {B}}_n$
, where we similarly consider it as a (
${\mathbb R},{\mathscr {B}}_n)$
-bimodule for
$j=1$
and as a
$({\mathbb C},{\mathscr {B}}_n)$
-bimodule for
$j\geq 2$
.
It is easy to check that we have the following isomorphisms of
${{\mathbb Z}}$
-graded bimodules.
Proposition 2.3 [Reference Heng and NgeHN23, Proposition 3.5]
Denote
${}_jP^B_k := {}_jP^B\otimes _{{\mathscr {B}}_n}P^B_k$
. We have that
Remark 2.4. Note that all the graded bimodules in Proposition 2.3 can be restricted to
$({\mathbb R}, {\mathbb R})$
-bimodules by identifying
${}_{\mathbb R} {\mathbb C}_{\mathbb R} \cong {\mathbb R}\oplus {\mathbb R}$
as
$({\mathbb R}, {\mathbb R})$
-bimodules. For example,
${}_1 P_2^B$
as an
$({\mathbb R}, {\mathbb R})$
-bimodule is generated by
$(1|2)$
and
$(1|2)i$
, and so it is isomorphic to the bimodule
${\mathbb R}(-1) \oplus {\mathbb R}(-1) \cong {}_{\mathbb R} {\mathbb C}_{\mathbb R} (-1)$
.
Lemma 2.5 [Reference Heng and NgeHN23, Lemma 3.7]
Denote
$\mathbb {K}_1 := {\mathbb R}$
and
$\mathbb {K}_j := {\mathbb C}$
when
$j \geq 2$
. The maps
$$ \begin{align*} \beta_j: P^B_j \otimes_{\mathbb{K}_j} {}_jP^B \to {\mathscr{B}}_n \quad\text{and}\quad \gamma_j: {\mathscr{B}}_n \to P^B_j \otimes_{\mathbb{K}_j} {}_jP^B (2), \end{align*} $$
defined by
$$ \begin{align*} &\beta_j(x\otimes y) := xy, \\ &\quad\gamma_j(1) := \begin{cases} X_j \otimes e_j + e_j \otimes X_j + (j+1|j) \otimes (j|j+1) \\ \quad + (-ie_{j+1})(j+1|j) \otimes (j|j+1)(ie_{j+1}) &\text{for } j=1;\\ X_j \otimes e_j + e_j \otimes X_j + (j-1|j) \otimes (j|j-1) + (j+1|j) \otimes (j|j+1) &\text{for } 1<j < n; \\ X_j \otimes e_j + e_j \otimes X_j + (j-1|j) \otimes (j|j-1) &\text{for } j = n, \end{cases} \end{align*} $$
are
$({\mathscr {B}}_n,{\mathscr {B}}_n)$
-bimodule maps.
Definition 2.6. Define the following complexes of graded
$({\mathscr {B}}_n,{\mathscr {B}}_n)$
-bimodules:
$$ \begin{align*} R_j &:= (0 \to P^B_j \otimes_{\mathbb{K}_j} {}_jP^B \xrightarrow{\beta_j} {\mathscr{B}}_n \to 0) \quad\text{and} \\ R_j' &:= (0 \to {\mathscr{B}}_n \xrightarrow{\gamma_j} P^B_j \otimes_{\mathbb{K}_j} {}_jP^B(2) \to 0) \end{align*} $$
for each
$j \in \{1,2, \ldots , n\},$
with both
${\mathscr {B}}_n$
in cohomological degree 0,
$\mathbb {K}_1 = {\mathbb R}$
and
$\mathbb {K}_j = {\mathbb C}$
for
$j \geq 2$
.
Theorem 2.7 [Reference Heng and NgeHN23, Theorem 3.13]
Let
${\text {Kom}}^b({\mathscr {B}}_n$
-
$p_r g_r \,\mathrm{mod})$
denote the homotopy category of complexes of projective, graded left
${\mathscr {B}}_n$
-modules. We have a (weak)
$\mathcal {A}(B_n)$
-action on
${\text {Kom}}^b({\mathscr {B}}_n$
-
$p_r g_r \,\mathrm{mod})$
, where each standard generator
$\sigma ^B_j$
for
$j \geq 1$
of
$\mathcal {A}(B_n)$
acts on a complex
$M \in {\text {Kom}}^b({\mathscr {B}}_n$
-
$p_rg_r\,\mathrm{mod})$
via
$R_j$
:
$$ \begin{align*}\sigma^B_j(M):= R_j \otimes_{{\mathscr{B}}_n} M\quad\text{and} \quad (\sigma^B_j)^{-1}(M):= R_j' \otimes_{{\mathscr{B}}_n} M.\end{align*} $$
2.3 Nil-categorification of the type B Temperley–Lieb algebra
Recall that the type
$B_n$
Temperley–Lieb algebra
$TL_v(B_n)$
over
${{\mathbb Z}}[v,v^{-1}]$
(in the sense of [Reference GreenGre97, Proposition 1.3]) can be described explicitly as the algebra generated by
$E_1, ..., E_n$
with the following relations:
$$ \begin{align*} E_j^2 &= vE_j + v^{-1}E_j; \\ E_j E_k &= E_k E_j \quad \text{if } |j-k|> 1; \\ E_j E_k E_j &= E_k \quad \text{if } |j-k| = 1 \text{ and } j,k > 1; \\ E_j E_k E_j E_k &= 2E_j E_k \quad \text{if } \{j, k\} = \{1,2\}. \end{align*} $$
The bimodules in the following proposition satisfy a further quotient of the aforementioned relations (see Equation (2-6)).
Proposition 2.8. Define
${\mathcal U }_j := P_j^B \otimes _{\mathbb {K}_j} {}_j P^B (1)$
, where
$\mathbb {K}_1 = {\mathbb R}$
and
$\mathbb {K}_j = {\mathbb C}$
when
$j \geq 2$
. The following are isomorphic as
${{\mathbb Z}}$
-graded
$({\mathscr {B}}_n, {\mathscr {B}}_n)$
-bimodules:
$$ \begin{align} {\mathcal U }_j \otimes_{{\mathscr{B}}_n} {\mathcal U }_j &\cong {\mathcal U }_j(1) \oplus {\mathcal U }_j(-1); \end{align} $$
$$ \begin{align} {\mathcal U }_j \otimes_{{\mathscr{B}}_n} {\mathcal U }_k &\cong 0 \quad \text{if } |j-k|> 1;\end{align} $$
$$ \begin{align} {\mathcal U }_j \otimes_{{\mathscr{B}}_n} {\mathcal U }_k \otimes_{{\mathscr{B}}_n} {\mathcal U }_j &\cong {\mathcal U }_k \quad \text{if } |j-k| = 1 \text{ and } j,k > 1;\end{align} $$
$$ \begin{align} {\mathcal U }_j \otimes_{{\mathscr{B}}_n} {\mathcal U }_k \otimes_{{\mathscr{B}}_n} {\mathcal U }_j \otimes_{{\mathscr{B}}_n} {\mathcal U }_k &\cong ({\mathcal U }_j \otimes_{{\mathscr{B}}_n} {\mathcal U }_k) \oplus ({\mathcal U }_j \otimes_{{\mathscr{B}}_n} {\mathcal U }_k) \quad \text{if } \{j, k\} = \{1,2\}. \end{align} $$
Proof. The isomorphisms in Equations (2-5), (2-6) and (2-7) can be demonstrated using the exact same proof as in the type A case [Reference Khovanov and SeidelKS02, Theorem 2.2]. For the fourth isomorphism in Equation (2-8) with
$j= 1, k=2$
(the other case is similar), we have the following chain of isomorphisms:
$$ \begin{align*} &{\mathcal U }_1 \otimes_{{\mathscr{B}}_n} {\mathcal U }_2 \otimes_{{\mathscr{B}}_n} {\mathcal U }_1 \otimes_{{\mathscr{B}}_n} {\mathcal U }_2 \\[2pt] &\quad\cong P_1^B \otimes_{\mathbb{R}} ( {}_1 P^B \otimes_{{\mathscr{B}}_n} P_2^B) \otimes_{\mathbb{C}} ({}_2 P^B \otimes_{{\mathscr{B}}_n} P_1^B) \otimes_{\mathbb{R}} {}_1 P^B \otimes_{{\mathscr{B}}_n}P_2^B \otimes_{\mathbb{C}} {}_2 P^B (4) \\[2pt] &\quad\cong P_1^B \otimes_{\mathbb{R}} ({}_\mathbb{R} \mathbb{C}_\mathbb{C} \otimes_{\mathbb{C}} {}_\mathbb{C} \mathbb{C}_\mathbb{R}) \otimes_{\mathbb{R}} {}_1 P^B \otimes_{{\mathscr{B}}_n}P_2^B \otimes_{\mathbb{C}} {}_2 P^B (2) \\[2pt] &\quad\cong P_1^B \otimes_{\mathbb{R}} (\mathbb{R} \oplus \mathbb{R}) \otimes_{\mathbb{R}} {}_1 P^B \otimes_{{\mathscr{B}}_n}P_2^B \otimes_{\mathbb{C}} {}_2 P^B (2) \\[2pt] &\quad\cong \left(P_1^B \otimes_{\mathbb{R}} {}_1 P^B \otimes_{{\mathscr{B}}_n}P_2^B \otimes_{\mathbb{C}} {}_2 P^B (2) \right) \oplus \left(P_1^B \otimes_{\mathbb{R}} {}_1 P^B \otimes_{{\mathscr{B}}_n}P_2^B \otimes_{\mathbb{C}} {}_2 P^B (2) \right) \\[2pt] &\quad\cong ({\mathcal U }_1 \otimes_{{\mathscr{B}}_n} {\mathcal U }_2) \oplus ({\mathcal U }_1 \otimes_{{\mathscr{B}}_n} {\mathcal U }_2), \end{align*} $$
where we use Proposition 2.3 repeatedly.
Definition 2.9. Let
${{\mathcal B}}$
denote the monoidal category over
$- \otimes _{{\mathscr {B}}_n} -$
generated by the
${\mathcal U }_j$
defined in Proposition 2.8, with monoidal unit
${\mathscr {B}}_n$
. We use
${\mathcal K } ar(\overline {{{\mathcal B}}})$
to denote the Karoubi envelope of the additive closure
$\overline {{{\mathcal B}}}$
of
${{\mathcal B}}$
.
Corollary 2.10. The monoidal, additive category
${\mathcal K } ar(\overline {{{\mathcal B}}})$
categorifies a quotient of
$TL_v(B_n)$
.
3 Soergel bimodules and the 2-braid group
In this section, we describe the diagrammatic version of the category of Soergel bimodules as given in [Reference Elias and WilliamsonEW16]. We only state the minimal details required to understand the category in the type B case and only with respect to a particular realisation (see next paragraph). For the general theory, we refer the reader to [Reference Elias and WilliamsonEW16] and the references therein. We also recall the definition of the 2-braid group from [Reference Rouquier, de la Peña and BautistaRou06] in the type B case.
Throughout this section,
$(W, S=\{s_1,s_2,...,s_n\})$
denotes the type
$B_n$
Coxeter system, with W the type
$B_n$
Coxeter group. We fix the following balanced, but nonsymmetric realisation of
$(W, S)$
: we define
$\mathfrak {h} := \bigoplus _{s_i \in S} {\mathbb R}\alpha _{s_i}^\vee $
with the set of coroots
$\{\alpha _{s_i}^\vee : s_i \in S\} \subset \mathfrak {h}$
and roots
$\{\alpha _{s_i} : s_i \in S\} \subset \mathfrak {h}^* = {\text {Hom}}_{\mathbb R}(\mathfrak {h}, {\mathbb R})$
, where
$$ \begin{align*} a_{s_i,s_j}:= \langle\alpha_{s_i}^\vee, \alpha_{s_j}\rangle = \begin{cases} 2 &\text{ if } i=j; \\ -1 &\text{ if } i,j \geq 2, |i-j| = 1; \\ -1 &\text{ if } i=1, j=2; \\ -2 &\text{ if } i=2, j=1; \\ 0 &\text{ otherwise}. \end{cases} \end{align*} $$
Attached to this realisation, we set
$R:= \bigoplus _{k \geq 0} S^k(\mathfrak {h}^*)$
as the
${{\mathbb Z}}$
-graded symmetric
${\mathbb R}$
-algebra on
$\mathfrak {h}^*$
with deg
$(\mathfrak {h}^*)$
=2. Note that W acts naturally on
$\mathfrak {h}^*$
and hence acts on R.
3.1 Soergel bimodules as a diagrammatic category
Let
${\mathcal D}_S,$
or simply
${\mathcal D}$
, denote the
${\mathbb R}$
-linear monoidal category (with respect to the previous realisation) defined as follows.
-
(i) Objects: finite sequences
of letters in S, with monoidal structure given by concatenation and monoidal unit given by the empty sequence
$\emptyset $
. -
(ii) Homomorphisms: the Hom space Hom
$_{{\mathcal D}}(\underline {w}, \underline {v})$
is the
${\mathbb R}$
-span of the Soergel graphs decorated by homogeneous
$f \in R$
modulo the relations listed in the following (see Equations (3-1) to (3-24)), where the graphs have bottom boundary
$\underline {w}$
and top boundary
$\underline {v}$
(see [Reference Elias and WilliamsonEW16, Definition 5.1] for the precise definition). In our type B cases, the possible vertices of the Soergel graphs consist of the following types:-
(a) univalent vertices (dots);
-
(b) trivalent vertices connecting three edges of the same colour ;
-
(c)
$2 m_{st}$
-valent vertices connecting edges which alternate in colour between two elements
$s,t$
of S, where in our case,
$m_{st} = 2, 3$
or
$4$
.Figure 2 The possible vertices in a Soergel graph of type B.
Each Soergel graph has a degree, which accumulates
$+1$
for each dot
$\mathrm{(a)}$
,
$-1$
for each trivalent vertex
$\mathrm{(b)}$
,
$0$
for each
$2m_{st}$
-valent vertex
$\mathrm{(c)}$
, and the degree of each decoration
$f\in R$
. The Hom spaces are then graded by the degrees of their Soergel graphs. As one easily checks, the following relations are indeed homogeneous. We remind the reader that Soergel graphs are by definition invariant under isotopies that preserve their top and bottom boundaries, so the bent and rotated versions of the following relations also hold.-
(a) The polynomial relations:
-
• the barbell relation
(3-1) -
• the polynomial forcing relation
(3-2)where
is the Demazure operator defined by
.
-
-
(b) The one-colour relations:
-
• the needle relation
(3-3) -
• the Frobenius relations
(3-4)(3-5)
-
-
(c) The two-colour relations:
-
• the two-colour associativity
(3-6)(3-7) -
• the dot-crossing relations
(3-8)(3-9)(3-10)where
-
-
(d) The three-colour relations or ‘Zamolodzhikov’ relations:
-
• A sub-Coxeter system of type
$A_1 \times I_2(m),$
that is, the Coxeter graph of the parabolic subgroup generated by
is
(3-11)(3-12)(3-13)(3-14)(3-15)
-
-
This concludes the definition of
${\mathcal D}.$
Note that the one-colour relations from Equations (3-4) and (3-5) combined with isotopy invariance encode the datum of the objects
$\color {purple} s$
being Frobenius objects. (Note that they are actually nilpotent, as they also satisfy the relation:
, which is implied by the other relations.) For completeness, we spell out the corresponding one-colour relations that come with the categeory
$\mathcal{D}$
. These relations are expressed in term of cups and caps defined as follows:
The trivalent vertices represent multiplications or comultiplications while the dots represent units or counits. To obtain Equation (3-4), we can rotate Equation (3-16) (respectively Equation (3-17)) using a cup (respectively a cap) and apply Equation (3-22) (respectively Equation (3-21)). However, Equations (3-21)–(3-24) record the cyclicity of all morphisms. Therefore, Equation (3-5) can be replaced by Equations (3-18) and (3-19). Here is the list of them:
This marks the end of the supplementary one-colour relations.
It is a theorem of Elias and Williamson [Reference Elias and WilliamsonEW16, Theorem 6.28] that the Karoubi envelope
${\mathcal K } ar(\overline {{\mathcal D}})$
of the graded additive closure
$\overline {{\mathcal D}}$
of
${\mathcal D}$
is equivalent to the category of Soergel bimodules as additive, monoidal categories. In particular,
${\mathcal K } ar(\overline {{\mathcal D}})$
categorifies the type
$B_n$
Hecke algebra.
3.2 The 2-braid group
The elementary Rouquier complexes corresponding to a simple reflection
$s \in S$
are defined as follows (see [Reference Rouquier, de la Peña and BautistaRou06]):
with both
$B_s$
in cohomological degree
$0.$
The
$2$
-braid group 2-
${{\mathcal B}} r$
of type
$B_n$
is the full monoidal subcategory of
${\text {Kom}}^b({\mathcal K } ar(\overline {{\mathcal D}}))$
generated by
$F_s$
and
$E_s$
for all
$s \in S$
. Observe that the set of isomorphism classes of objects in
$2$
-
${{\mathcal B}} r,$
denoted by Pic
$(2$
-
${{\mathcal B}} r)$
, forms a group called the Picard group of the monoidal category
$2$
-
${{\mathcal B}} r$
under tensor product composition. Rouquier showed that
$F_s$
and
$E_s$
are inverses of each other, and moreover they satisfy the required braid relations [Reference Rouquier, de la Peña and BautistaRou06, Proposition 9.2, Lemma 9.3]. In particular, the map sending
$\sigma _j$
to the isomorphism class
$F_{s_j} \in $
Pic
$(2$
-
${{\mathcal B}} r)$
for all
$j \geq 1$
is well defined [Reference Rouquier, de la Peña and BautistaRou06, Proposition 9.4]. Rouquier conjectures that this assignment is moreover faithful in general, and we prove this in the type B case.
4 Quotient category and proof of faithfulness
The aim of this section is to show that
${\mathcal K } ar(\overline {{{\mathcal B}}})$
is a quotient category of
${\mathcal K } ar(\overline {{\mathcal D}})$
. This is given by an essentially surjective monoidal functor
$G_0: {\mathcal D} \rightarrow {{\mathcal B}}$
, which leads to an essentially surjective additive, monoidal functor on the Karoubi envelope of their respective additive closures. For clarity, we use
or
(solid line) for a simple reflection j,
(dashed line) for an adjacent simple reflection
$j \pm 1,$
and
(dotted line) for a distant simple reflection k with
$|j-k|>1.$
We define
$G_0$
on generating objects by sending j in S to
${\mathcal U }_j$
and on generating morphisms as in the proof of Theorem 4.3, where the five types of
$({\mathscr {B}}_n, {\mathscr {B}}_n)$
-bimodule homomorphisms needed are defined as follows:
-
•
$(-1)^{j+1} \alpha _j: {\mathcal U }_j \rightarrow {\mathcal U }_j(-1) \oplus {\mathcal U }_j(1) \xrightarrow {\cong } {\mathcal U }_j \otimes _{{\mathscr {B}}_n} {\mathcal U }_j $
is a morphism of degree (
$-1$
) defined by
$$ \begin{align*} e_j \otimes e_j \mapsto (0, e_j \otimes e_j) \mapsto e_j \otimes e_j \otimes_{{\mathscr{B}}_n} e_j \otimes e_j; \end{align*} $$
-
•
$\delta _j: {\mathcal U }_j \otimes _{{\mathscr {B}}_n} {\mathcal U }_j \xrightarrow {\cong } {\mathcal U }_j(-1) \oplus {\mathcal U }_j(1) \rightarrow {\mathcal U }_j$
is a morphism of degree (
$-1$
) defined by
$$\begin{align*} e_j \otimes X_j \otimes_{{\mathscr{B}}_n} e_j \otimes e_j \ & \mapsto (e_j \otimes e_j,0 ) \mapsto e_j \otimes e_j; \\e_j \otimes e_j \otimes_{{\mathscr{B}}_n} e_j \otimes e_j \ \ & \mapsto (0, e_j \otimes e_j) \mapsto 0; \end{align*}$$
-
•
$\epsilon _j : {\mathscr {B}}_n \rightarrow {\mathscr {B}}_n$
is a morphism of degree (2) defined by
$$ \begin{align*} 1 \mapsto \begin{cases} (-1)^{j+1} (2X_j + 2X_{j+1}) &\text{for } j=1;\\ (-1)^{j+1} (2X_j + X_{j-1} + X_{j+1}) &\text{for } 1<j < n; \\ (-1)^{j+1} ( 2X_j + X_{j-1}) &\text{for } j = n; \end{cases} \end{align*} $$
-
•
$(-1)^{j+1}\beta _j : {\mathcal U }_j \rightarrow {\mathscr {B}}_n$
is a morphism of degree (1) defined by
$$ \begin{align*} e_j \otimes e_j \mapsto e_j; \end{align*} $$
-
•
$\gamma _j : {\mathscr {B}}_n \rightarrow {\mathcal U }_j$
is a morphism of degree (1) defined by
$$ \begin{align*} 1 \mapsto \begin{cases} X_j \otimes e_j + e_j \otimes X_j + (j+1|j) \otimes (j|j+1) \\ \hspace{8mm} + (-ie_{j+1})(j+1|j) \otimes (j|j+1)(ie_{j+1}) &\text{for } j=1;\\ X_j \otimes e_j + e_j \otimes X_j + (j-1|j) \otimes (j|j-1) + (j+1|j) \otimes (j|j+1) &\text{for } 1<j < n; \\ X_j \otimes e_j + e_j \otimes X_j + (j-1|j) \otimes (j|j-1) &\text{for } j = n. \end{cases} \end{align*} $$
Note that the last two maps
$\beta _j$
and
$\gamma _j$
are exactly the bimodule maps in Definition 2.6.
The following lemma is crucial.
Lemma 4.1 [Reference Heng and NgeHN23, Lemma 3.11]
Denote
$\mathbb {K}_j := {\mathbb R}$
when
$j = 1$
and
$\mathbb {K}_j := {\mathbb C}$
when
$j \geq 2$
. We have the adjoint pairs
$(P_j^B \otimes _{\mathbb {K}_j} - , \ {}_{j}{P}^B \otimes _{{\mathscr {B}}_n} -)$
and
$( {}_{j}{P}^B (2) \otimes _{{\mathscr {B}}_n} - , \ P_j^B \otimes _{\mathbb {K}_j} - )$
.
Proposition 4.2. Denote
$\mathbb {K}_j := {\mathbb R}$
when
$j = 1$
and
$\mathbb {K}_j := {\mathbb C}$
when
$j \geq 2$
. We have the following identification of grading preserving
$({\mathscr {B}}_n, {\mathscr {B}}_n)$
-bimodule morphism spaces:
-
(1)
${\mathcal U }_j \rightarrow {\mathscr {B}}_n$
of degree
$(1)$
is isomorphic to
$\mathbb {K}_j \beta _j;$
-
(2)
${\mathscr {B}}_n \rightarrow {\mathcal U }_j$
of degree
$(1)$
is isomorphic to
$\mathbb {K}_j \gamma _j;$
-
(3)
${\mathcal U }_j \rightarrow {\mathcal U }_j \otimes _{{\mathscr {B}}_n} {\mathcal U }_j $
of degree
$(-1)$
is isomorphic to
$\mathbb {K}_j \alpha _j;$
-
(4)
${\mathcal U }_j \otimes _{{\mathscr {B}}_n} {\mathcal U }_j \rightarrow {\mathcal U }_j $
of degree
$(-1)$
is isomorphic to
$\mathbb {K}_j \delta _j;$
-
(5)
${\mathscr {B}}_n \rightarrow {\mathscr {B}}_n$
of degree
$(2)$
is isomorphic to
$\oplus _{1 \leq j \leq n} \mathbb {K}_jX_j,$
where
$X_j$
is interpreted with left multiplication (equivalently right multiplication) by
$X_j$
.
Proof. Item (5) follows directly from the fact that the map is completely determined by the image of
$1 \in {\mathscr {B}}_n$
together with the degree restriction.
The morphisms
$\beta _j, \gamma _j, \alpha _j$
and
$\delta _j$
in items (1)–(4) are indeed nontrivial
$({\mathscr {B}}_n,{\mathscr {B}}_n)$
-bimodule morphisms of the respective spaces; it is therefore sufficient to show that Hom-spaces are all one-dimensional. This follows from an easy computation using the adjoint pair
$(P_j^B \otimes _{\mathbb {K}_j} - , \ {}_{j}{P}^B \otimes _{{\mathscr {B}}_n} -)$
in Lemma 4.1. For example, item (4) follows from the following identification of Hom-spaces:
$$ \begin{align*} \text{Hom}_{{\mathscr{B}}_n\text{-bimod}}({\mathcal U }_j \otimes_{{\mathscr{B}}_n} {\mathcal U }_j, {\mathcal U }_j(-1)) &= \text{Hom}_{{\mathscr{B}}_n\text{-bimod}}(P_j \otimes_{\mathbb{K}_j} {}_jP \otimes_{{\mathscr{B}}_n} P_j \otimes_{\mathbb{K}_j} {}_jP(2), P_j \otimes_{\mathbb{K}_j} {}_jP) \\ &\cong \text{Hom}_{\text{\,mod-}{\mathscr{B}}_n}( {}_jP \otimes_{{\mathscr{B}}_n} P_j \otimes_{\mathbb{K}_j} {}_jP(2), {}_jP \otimes_{{\mathscr{B}}_n} P_j \otimes_{\mathbb{K}_j} {}_jP) \\ &\cong \text{Hom}_{\text{\,mod-}{\mathscr{B}}_n}\big( {}_jP(2) \oplus {}_jP, {}_jP \oplus {}_jP(-2) \big). \end{align*} $$
Since
$\text {Hom}_{\text {mod-}{\mathscr {B}}_n}( {}_jP, {}_jP (k)) \cong \mathbb {K}_j$
if and only if
$k = 0$
or
$2$
,
$$ \begin{align*} \text{Hom}_{{\mathscr{B}}_n\text{-bimod}}({\mathcal U }_j \otimes_{{\mathscr{B}}_n} {\mathcal U }_j, {\mathcal U }_j(-1)) \cong \mathbb{K}_j \end{align*} $$
as required. We leave the other cases to the reader.
Theorem 4.3. There is an essentially surjective monoidal functor
$G_0 : {\mathcal D} \rightarrow {{\mathcal B}}$
that sends:
-
(i) the empty sequence
$\emptyset $
in S to
${\mathscr {B}}_n,$
the monoidal identity in
${{\mathcal B}};$
-
(ii)
$s_j$
in S to
${\mathcal U }_j;$
and -
(iii) a sequence
$s_{i_1} s_{i_2} \cdots s_{i_k}$
in S of length
$k \geq 2$
to
$G_0( i_1 i_2 \cdots i_{k-1}) \otimes _{{\mathscr {B}}_n} G_0(i_k).$
Proof. The assignment of
$G_0 : {\mathcal D} \rightarrow {{\mathcal B}}$
on objects is done as in the statement of the theorem, whereas on generating morphisms, it is given in Figure 3.
Figure 3 The assignment of
$G_0: {\mathcal D} \rightarrow {{\mathcal B}}$
on generating morphisms. Note that
$a_j$
,
$b_j$
,
$c_j$
,
$d_j$
,
$f^j_k \ \in {\mathbb R}$
for
$j = 1$
and
$\in {\mathbb C}$
for
$j> 1.$
We want to find a set of scalars such that the restrictions imposed by the relations in
${\mathcal D}$
are satisfied. To illustrate, let us consider the barbell relation in Equation (3-1) for
$s_1$
:
By definition,
$$ \begin{align*} 1 & \mapsto c_1 \left( X_1 \otimes e_1 + e_1 \otimes X_1 + (2|1) \otimes (1|2) + (-ie_2)(2|1) \otimes (1|2)(ie_2) \right)\\ & \mapsto c_1b_1 \left( X_1 + X_1 + X_2 + X_2 \right) \\ & = c_1b_1 \left( 2X_1 + 2X_2 \right). \end{align*} $$
Equating with the right-hand side, we get
$c_1b_1 ( 2X_1 + 2X_2 ) = \sum ^n_{k=1} f^1_k X_k$
which, in turn, implies
$2c_1 b_1 = f^1_1 $
and
$2c_1 b_1 = f^1_2.$
By the same token, let us look at the type B 8-valences relation in Equation (3-10): suppose solid lines (
, left-most strands) encodes
and dashed lines (
) encodes
,
which, by definition of the functor, is equivalent to checking the equation
Observe that there are five terms containing Soergel graphs on the right-hand side. One thing to note here is that
${\mathcal U }_1 \otimes {\mathcal U }_2 \otimes {\mathcal U }_1 \otimes {\mathcal U }_2$
is spanned by
$e_1 \otimes _{\mathbb R} (1|2) \otimes _{{\mathscr {B}}_n} e_2 \otimes _{\mathbb C} e_2 \otimes _{{\mathscr {B}}_n} (2|1) \otimes _{\mathbb R} (1|2) \otimes _{{\mathscr {B}}_n} e_2 \otimes _{\mathbb C} e_2 $
and
$e_1 \otimes _{\mathbb R} (1|2) \otimes _{{\mathscr {B}}_n} e_2 \otimes _{\mathbb C} e_2 \otimes _{{\mathscr {B}}_n} (-ie_2)(2|1) \otimes _{\mathbb R} (1|2) \otimes _{{\mathscr {B}}_n} e_2 \otimes _{\mathbb C} e_2.$
Without the coefficients, looking at the second term in the aforementioned equation and applying it appropriately,
Similarly,
$$ \begin{align*} & e_1 \otimes (1|2) \otimes e_2 \otimes e_2 \otimes (-ie_2)(2|1) \otimes (1|2) \otimes e_2 \otimes e_2 \\ & \ \ \mapsto c_1 a_2 d_2 b_1^2 ( (1|2) \otimes e_2 \otimes (2|1) \otimes (1|2)(-ie_2) \otimes e_2 \otimes e_2 \\ & \ \ \ \ \ + (1|2) \otimes e_2 \otimes (-ie_2)(2|1) \otimes (1|2) \otimes e_2 \otimes e_2). \end{align*} $$
Once the calculations for all the five Soergel graphs have been done, comparing coefficients coming from four basis elements in the codomain yields four defining equations.
Before giving all the relations, we eliminate the unnecessary or redundant relations. Equation (3-3), says
$b_j d_j a_j \beta _j \delta _j \alpha _j = 0$
, which is true as
$\delta _j \alpha _j = 0.$
However, Equation (3-4) (equivalently Equations (3-16) and (3-17)) does not impose any restrictions on the coefficients, whereas Equation (3-5) can be replaced by Equations (3-18) and (3-19). In addition, the relations in Equations (3-6), (3-7), (3-11) (3-12), (3-13), (3-14) and (3-15) are all trivially satisfied as the
$2m_{st}$
-valent vertices are killed for every
$s,t \in S.$
Finally, Equation (3-8) has both sides equal to zero as
${\mathcal U }_j \otimes _{{\mathscr {B}}_n} {\mathcal U }_k = 0$
in
${{\mathcal B}}$
for
$|j-k|> 1 $
.
We now summarise all of the other required relations (these are mostly the same as in the type A case, with the exception of the two-colour relations):
Equation (3-1)
$\implies $
$f^1_1 = 2b_1 c_1, f^1_2 =2 b_1 c_1, f^j_j = 2 b_j c_j, f^j_{j \pm 1} = b_j c_j, f^j_k=0$
for
$j, k \geq 2$
and
$|j-k|> 1;$
Equation (3-2)
$\implies $
$f^1_1 = 2b_1 c_1, f^1_2 = - 2 b_2 c_2, f^j_j = 2 b_j c_j, f^{j \pm 1}_j = -b_j c_j, f^j_k=0$
for
$j,k \geq 2$
and
$|j-k|> 1;$
Equation (3-9) (Type A 6-valences relation)
$\implies $
$d_{j \pm 1} b_j c_j = -b_{j \pm 1}$
for suitable
$j;$
Equation (3-10) (Type B 8-valences relation)
$\implies $
Since
$a_{s_2,s_1} a_{s_1,s_2}-1 = (-2)(-1) -1 =1,$
we simplify the denominator first. For
(
or solid line) left-aligned,
$$ \begin{align*}b_1 - {a_{s_1,s_2}}b_1^2 d_2 a_2 c_1 - {a_{s_2,s_1}}b_2 d_1 c_2 + b_2 d_1 b_1 a_2 c_1 + b_1 d_2 c_2 = 0,\end{align*} $$
$$ \begin{align*}b_1 - a_{s_1,s_2} b_1^2 d_2 a_2 c_1 &= 0,\quad -a_{s_1,s_2} b_1^2 d_2 a_2 c_1 + b_2 d_1 b_1 a_2 c_1 = 0,\\& \quad -a_{s_1,s_2} b_1^2 d_2 a_2 c_1 + b_1 d_2 c_2 = 0,\end{align*} $$
while for
(
or dashed line) left-aligned,
$$ \begin{align*}b_2 - a_{s_2,s_1} b_2^2 d_1 a_1 c_2 - a_{s_1,s_2} b_1 d_2 c_1 + b_1 d_2 a_1 b_2 c_2 + b_2 d_1 c_1 = 0,\end{align*} $$
$$ \begin{align*}b_2 - a_{s_1,s_2} b_1 d_2 c_1 = 0, \ \ - a_{s_1,s_2} b_1 d_2 c_1 + b_1 d_2 a_1 b_2 c_2 = 0, \ \ -a_{s_1,s_2} b_1 d_2 c_1 + b_2 d_1 c_1 = 0; \end{align*} $$
Equation (3-18)
$\implies $
$a_jb_j = 1$
for all
$1 \leq j \leq n;$
Equation (3-19)
$\implies $
$c_jd_j = 1$
for all
$1 \leq j \leq n;$
Equation (3-20)
$\implies $
$a_jb_jc_jd_j = 1$
for all
$1 \leq j \leq n;$
Equation (3-21)
$\implies $
$a_jb_jd_j = d_j$
for all
$1 \leq j \leq n;$
Equation (3-22)
$\implies $
$a_jc_jd_j = a_j$
for all
$1 \leq j \leq n;$
Equation (3-23)
$\implies $
$b_jc_jd_j = b_j$
for all
$1 \leq j \leq n;$
Equation (3-24)
$\implies $
$a_jb_jc_j = c_j$
for all
$1 \leq j \leq n.$
The solution
$a_j = b_j = (-1)^{j+1}, c_j = d_j = 1, f^1_2 = 2, f^j_j = (-1)^{j+1} 2, f^j_{j \pm 1} = (-1)^{j+1}$
and
$f^j_k = 0$
for
$|j-k|>1$
gives our desired functor.
Corollary 4.4. The functor
$G_0: {\mathcal D} \to {{\mathcal B}}$
in Theorem 4.3 induces an exact monoidal functor
$\overline {G}: {\text {Kom}}^b(Kar(\overline {{\mathcal D}})) \rightarrow {\text {Kom}}^b(Kar(\overline {{\mathscr {B}}}))$
. This functor
$\overline {G}$
sends the generators of the 2-braid groups
$F_{s_j}$
to
$R_j[-1](1)$
, matching Rouquier’s complexes with our twist complexes (up to internal grading shift and cohomological shift).
Corollary 4.5 (Faithfulness of type B 2-braid group)
The group homomorphism
${\mathcal A}(B_n) \rightarrow $
Pic
$(2$
-
${{\mathcal B}} r)$
sending
$\sigma _j \mapsto F_{s_j}$
is faithful.
Proof. By Theorem 2.7, the assignment
$\sigma _j \mapsto R_j[-1](1)$
is faithful as the (gradings-shifted) action it induces on
${\text {Kom}}^b({\mathscr {B}}_n$
-
$p_r g_r \,\mathrm {mod})$
is faithful. Corollary 4.4 shows that this assignment factors through Pic
$(2$
-
${{\mathcal B}} r)$
, which implies that the group homomorphism
${\mathcal A}(B_n) \rightarrow $
Pic
$(2$
-
${{\mathcal B}} r)$
is also faithful.
Remark 4.6. Note that a similar proof strategy on the ‘decategorified level’ will not work, as the Burau representation may not be faithful even in type
$A_n$
; for
$n=5$
, see [Reference BigelowBig99]. To the best of our knowledge, the faithfulness of the Hecke algebra representation of Artin braid groups remains open for large ranks.
Acknowledgements
We would like to thank our supervisor Tony Licata for his guidance throughout. This paper results from a discussion with Thorge Jensen during the conference ‘New Connections in Representation Theory 2020, Mooloolaba’, for which we are deeply grateful. We thank the organisers for making this possible and we thank Thorge for his patience in answering our questions. We would also like to thank the referee for their suggestions and for carefully reading our paper.
