The $\mathfrak {sl}_{3}$-web algebra

Abstract

In this paper we use Kuperberg’s $\mathfrak {sl}_3$-webs and Khovanov’s $\mathfrak {sl}_3$-foams to define a new algebra $K^S$, which we call the $\mathfrak {sl}_3$-web algebra. It is the $\mathfrak {sl}_3$ analogue of Khovanov’s arc algebra. We prove that $K^S$ is a graded symmetric Frobenius algebra. Furthermore, we categorify an instance of $q$-skew Howe duality, which allows us to prove that $K^S$ is Morita equivalent to a certain cyclotomic KLR-algebra of level 3. This allows us to determine the split Grothendieck group $K^{\oplus }_0(\mathcal {W}^S)_{\mathbb {Q}(q)}$, to show that its center is isomorphic to the cohomology ring of a certain Spaltenstein variety, and to prove that $K^S$ is a graded cellular algebra.

Appendix: Filtered and graded algebras and modules

In this appendix, we have collected some basic facts about filtered algebras, the associated graded algebras and the idempotents in both. Our main sources are [60] and [61]. In this appendix, everything is defined over an arbitrary commutative associative unital ring $R$.

Let $A$ be a finite dimensional, associative, unital $R$-algebra together with an increasing filtration of $R$-submodules

$$\begin{aligned} \{0\}\subset A_{-p}\subset A_{-p+1}\subset \cdots \subset A_0\subset \cdots \subset A_{m-1}\subset A_m=A. \end{aligned}$$

Actually, for any $t\in \mathbb {Z}$ we have a subspace $A_t$, where we extend the filtration above by

$$\begin{aligned} A_t= {\left\{ \begin{array}{ll} \{0\},&{} \text {if}\quad t<-p,\\ A,&{}\text {if}\quad p\ge m. \end{array}\right. } \end{aligned}$$

Note that in the language of [60], such a filtration is discrete, separated, exhaustive and complete. If $1\in A_0$ and the multiplication satisfies $A_iA_j\subseteq A_{i+j}$, we say that $A$ is an associative, unital, filtered algebra.

The associated graded algebra is defined by

$$\begin{aligned} E(A)=\bigoplus _{i\in \mathbb {Z}} A_i/A_{i-1}, \end{aligned}$$

and is also associative and unital. Although $A$ and $E(A)$ are isomorphic $R$-modules, they are not isomorphic as algebras.

A finite dimensional, filtered $A$-module is a finite dimensional, unitary $A$-module $M$ with an increasing filtration of $R$-submodules

$$\begin{aligned} \{0\}\subset M_{-q}\subset M_{-q+1}\subset \cdots \subset M_t=M, \end{aligned}$$

such that $A_iM_j\subseteq M_{i+j}$, for all $i,j\in \mathbb {Z}$, after extending the finite filtration to a $\mathbb {Z}$-filtration as above.

We define the $t$-fold suspension $M\{t\}$ of $M$, which has the same underlying $A$-module structure, but a new filtration defined by

$$\begin{aligned} M\{t\}_r=M_{r+t}. \end{aligned}$$

Given a filtered $A$-module $M$, the associated graded module is defined by

$$\begin{aligned} E(M)=\bigoplus _{i\in \mathbb {Z}} M_i/M_{i-1}. \end{aligned}$$

An $A$-module map $f:M\rightarrow N$ is said to preserve the filtrations if $f(M_i)\subseteq N_i$, for all $i\in \mathbb {Z}$. Any such map $f:M\rightarrow N$ induces a grading preserving $E(A)$-module map $E(f):E(M)\rightarrow E(N)$ in the obvious way.

This way, we get a functor

$$\begin{aligned} E:A\text {-}\mathrm{\mathbf{Mod}}_{\mathrm{fl}} \rightarrow E(A)\text {-}\mathrm{\mathbf{Mod}}_{\mathrm{gr}}, \end{aligned}$$

where $A\text {-}\mathbf{Mod}_{\mathrm{fl}}$ is the category of finite dimensional, filtered $A$-modules and filtration preserving $A$-module maps and $E(A)\text {-}\mathbf{Mod}_{\mathrm{gr}}$ is the category of finite dimensional, graded $E(A)$-modules and grading preserving $E(A)$-module maps.

Recall that $A\text {-}\mathbf{Mod}_{\mathrm{fl}}$ is not an abelian category, e.g. the identity map $M\rightarrow M\{1\}$ is a filtration preserving bijective $A$-module map, but does not have an inverse in $A\text {-}\mathbf{Mod}_{\mathrm{fl}}$.

In order to avoid such complications, one can consider a subcategory with fewer morphisms. An $A$-module map $f:M\rightarrow N$ is called strict if

$$\begin{aligned} f(M_i)=f(M)\cap N_i \end{aligned}$$

holds, for all $i\in \mathbb {Z}$. Let $A\text {-}\mathbf{Mod}_{\mathrm{st}}$ be the subcategory of filtered $A$-modules and strict $A$-module homomorphisms.

Lemma 5.34

The restriction of $E$ to $A\text {-}\mathbf{Mod}_{\mathrm{st}}$ is exact.

We also need to recall a simple result about bases. A basis $\{x_1,\ldots ,x_n\}$ of a filtered algebra $A$ is called homogeneous if, for each $1\le j\le n$, there exists an $i\in \mathbb {Z}$ such that $x_j\in A_i\backslash A_{i-1}$. In that case, $\{\overline{x}_1,\ldots ,\overline{x}_n\}$ defines a homogeneous basis of $E(A)$, where $\overline{x_j}\in A_i/A_{i-1}$. In order to avoid cluttering our notation, we always write $\overline{x}_j$ and then specify in which subquotient we take the equivalence class by saying that it belongs to $A_i/A_{i-1}$.

Given a homogeneous basis $\{y_1,\ldots ,y_n\}$ of the associated graded $E(A)$, we say that a homogeneous basis $\{x_1,\ldots ,x_n\}$ of $A$ lifts $\{y_1,\ldots ,y_n\}$ if $\overline{x}_j=y_j\in A_i/A_{i-1}$ holds, for each $1\le j\le n$ and the corresponding $i\in \mathbb {Z}$. The result in the following lemma is well-known. However, we could not find a reference in the literature, so we provide a short proof here.

Lemma 5.35

Let $A$ be a finite dimensional, filtered algebra and $\{y_1,\ldots ,y_n\}$ be a homogeneous basis of $E(A)$. Then there is a homogeneous basis $\{x_1,\ldots ,x_n\}$ of $A$ which lifts $\{y_1,\ldots ,y_n\}$.

Proof

We prove the lemma by induction with respect to the filtration degree $q$. Suppose $A_q=0$, for all $q<-p$, and $A_q=A$, for all $q\ge m$. Then $E(A_{-p})=A_{-p}$. Since $\{y_1,\ldots ,y_n\}$ is a homogeneous basis of $E(A)$, a subset of this basis forms a basis of $A_{-p}$.

For each $-p+1\le q\le m$, choose elements in $A_q$ which lift the homogeneous subbasis of $E(A_q)$. We claim that the union of the sets of these elements, for all $-p\le q\le m$, form a homogeneous basis of $A$ which lifts $\{y_1,\ldots ,y_n\}$. Call it $\{x_1,\ldots ,x_n\}$. By definition, the $x_j$ lift the $y_j$, for all $1\le j\le n$. It remains to show that the $x_j$ are all linearly independent. This is true for $q=-p$, as shown above.

Let $q>-p$ and suppose that the claim holds for $\{x_1,\ldots ,x_{m_{q-1}}\}$, the subset of $\{x_1,\ldots ,x_n\}$ which belongs to $A_{q-1}$. Let

$$\begin{aligned} \{x_1,\ldots , x_{m_q}\}=\{x_1,\ldots ,x_{m_{q-1}}\}\cup \{x_{m_{q-1}+1},\ldots ,x_{m_q}\} \end{aligned}$$

be the subset belonging to $A_q$. Suppose that

$$\begin{aligned} \sum _{j=1}^{m_q}\lambda _jx_j=0, \end{aligned}$$

(6.1)

with $\lambda _j\in R$. Then we have

$$\begin{aligned} \sum _{j=1}^{m_q}\lambda _j\overline{x}_j=\sum _{j=m_{q-1}+1}^{m_q}\lambda _j\overline{x}_j=\sum _{j=m_{q-1}+1}^{m_q}\lambda _jy_j=0 \in A_q/A_{q-1}. \end{aligned}$$

By the linear independence of the $y_j$, this implies that $\lambda _j=0$, for all $m_{q-1}+1\le j\le m_q$. Thus, the linear combination in (5.28) becomes

$$\begin{aligned} \sum _{j=1}^{m_{q-1}}\lambda _jx_j=0. \end{aligned}$$

By induction, this implies that $\lambda _j=0$, for all $1\le j\le m_{q-1}$.

This shows that $\lambda _j=0$, for all $1\le j\le n$, so the $x_j$ are linearly independent.$\square$

For a proof of the following proposition, see for example Proposition 1 in the appendix of [61].

Proposition 5.36

Let $M$ and $N$ be filtered $A$-modules and $f:M\rightarrow N$ a filtration preserving $A$-linear map. If $E(f):E(M)\rightarrow E(N)$ is an isomorphism, then $f$ is an isomorphism (and therefore strict too).

The most important fact about filtered, projective modules and their associated graded, projective modules, that we need in this paper, is Theorem 6 in [60]. Note that these projective modules are the projective objects in the category $A\text {-}\mathrm{\mathbf{Mod}}_{\mathrm{st}}$.

Theorem 5.37

(Sjödin) Let $P$ be a finite dimensional, graded, projective $E(A)$-module. Then there exists a finite dimensional, filtered, projective $A$-module $P'$, such that $E(P')=P$. Moreover, if $M$ is a finite dimensional, filtered, $A$-module, then any degree preserving $E(A)$-module map $P\rightarrow E(M)\{t\}$, for some grading shift $t\in \mathbb {Z}$, lifts to a filtration preserving $A$-module map $P'\rightarrow M\{t\}$.

We also recall the following corollary of Sjödin (Corollary in [60] after Lemma 20).

Corollary 5.38

Let $M$ be a finite dimensional, filtered, $A$-module, then any finite or countable set of orthogonal idempotents in

$$\begin{aligned} \mathrm{im}(\phi )\subset \mathrm{Hom}_{E(A)}(E(M),E(M)) \end{aligned}$$

can be lifted to $\mathrm{Hom}_{A}(M,M)$, where $\phi$ is the natural transformation

$$\begin{aligned} \phi :E(\mathrm{Hom}_{A}(M,M))\rightarrow \mathrm{Hom}_{E(A)}(E(M),E(M)). \end{aligned}$$