Hypercubic Decomposition of
Verma Supermodules and
Semibricks Realizing the
Khovanov Algebra of Defect One

Classical BGG category $\mathcal{O}$ has been historically important in representation theory. Its super analog has been extensively studied in recent years and is regarded as a highly non-trivial and intriguing object. It is remarkable that, regarding character theory, our understanding has greatly advanced due to successes such as categorification theory. (For example, see [8, 5].) In general, many classical aspects of category $\mathcal{O}$ related to $\mathfrak{sl}(2)$ have been generalized to the super analog by numerous authors [10, 16, 17]. Yet, the theory unique to the super case—particularly the theory surrounding $\mathfrak{gl}(1|1)$, an intriguing counterpart to $\mathfrak{sl}(2)$ in the super world—still appears to have a room for further exploration.

Indeed, the main result of our present work provides a new direction concerning the $\mathfrak{gl}(1|1)$-like aspects of the super category $\mathcal{O}$ through appropriate ”semibricks”. Experts in the super category $\mathcal{O}$ are well aware that the atypical block contains long exact sequences of Verma modules $M^{\mathfrak{b}}(\lambda)$ (induced from Borel subalgebra $\mathfrak{b}$) and this observation evokes a connection to $\mathfrak{gl}(1|1)$. The theorem stated below is nothing but a formalization of this intuition.

A semibrick is, in simple terms, a class of objects in an abelian category that satisfy Schur’s lemma. By a classical result of Ringel [18] (presented here as Theorem 3.13), it is known that taking the “filtration closure” of a semibrick yields an extension-closed abelian full subcategory (i.e. wide subcategory). In case of the module category of finite-dimensional algebras, this concept has been shown to correspond bijectively to key structures and has been actively studied in that context in recent years [1].

This theorem implies that the principal block of $\mathfrak{gl}(1|1)$ is embedded into the super category $\mathcal{O}$ in many ways. A crucial point here is that this embedding forms a wide subcategory, not a Serre subcategory. Indeed, the left-hand side category generally does not contain the actual simple modules $L^{\mathfrak{b}}(\lambda)$, among others.

In general, the problems involving $L^{\mathfrak{b}}(\lambda)$, including the super Kazhdan-Lusztig conjecture [8, 5], are highly nontrivial. On the other hand, if we treat ”bricks” as simple objects instead of actual simple modules and take a coarse perspective on the super category $\mathcal{O}$, this theorem suggests that the situation becomes much more familiar and well-understood. As a corollary, all problems concerning $Ext^{1}$ groups between Verma modules or their variants contained in this category reduce to problems in $\mathfrak{gl}(1|1)$.

It is notable that this theorem only becomes apparent through the process of changing Borel subalgebras. In particular, when constructing a module corresponding to indecomposable projective modules, we considered a module induced from the smaller solvable Lie superalgebra, which is the intersection of two adjacent Borel subalgebras.

This diagram depicts the key modules and homomorphisms in Theorem 1.1.

An exact sequence (this is an almost split sequence in $\operatorname{Filt}H(\alpha,\lambda)$ in the sense of Auslander-Reiten [2])

shows that Verma modules have a natural decomposition. (These modules also appear in references such as [19] and [8].)

More generally, by simultaneously handling $n$ mutually orthogonal odd isotropic simple roots, $M^{\mathfrak{b}}(\lambda)$ can be decomposed into $2^{n}$ $\mathfrak{b}$-highest weight modules of the form $M^{\mathfrak{b}+r_{J}\mathfrak{b}}(\mu)$, based on the intuition of hyperqube $Q_{n}$. As a consequence, similarly, we can construct a semibrick that realizes the principal block for $\mathfrak{gl}(1|1)^{\oplus n}$ (Theorem 3.24).

As we see from Example 3.16 and Example 3.17, we can also construct a natural semibrick in a different direction, distinct from Theorem 1.1. In particular, we can realize many categories of modules over the preprojective algebra of type $A_{2}$. For such a semibrick, the Verma modules play the role of indecomposable projective modules. This construction may be generalized further and is considered an interesting problem in itself, closely related to the contents of [14], especially [14, Remark 4.13].

I would like to express my heartfelt gratitude to my supervisor, Syu Kato, for his patient and extensive guidance throughout the preparation of master’s thesis, as well as for his helpful suggestions and constructive feedback. The author is also grateful to Istvan Heckenberger for pointing out some errors. The author is also grateful to Shun-Jen Cheng for valuable discussions, which helped in identifying several mistakes. The author would like to thank the Kumano Dormitory community at Kyoto University for their generous financial and living assistance.

Below, we summarize fundamental results derived from the PBW theorem and Frobenius reciprocity.

Hereafter, when we write $\operatorname{Ext}^{1}$, we mean $\operatorname{Ext}^{1}_{\mathcal{O}}$ unless otherwise specified. The following Lemma is a generalization of [15, Propsition 3.1] and serves as a fundamental result in this subsection.

In this subsection, we focus on $\lambda$-adjusted Borel subalgebras, which are important for our study.

For (6), it is well known, as also stated in [8, Lemma 6.9].

For (7), since the characters are equal by (6), it is clear that the dimension of the Hom space is 1.

The image of this homomorphism is a module generated by $E_{J}^{\mathfrak{b}}v_{\lambda-\Sigma^{\mathfrak{b}}_{J}}^{r_{J}\mathfrak{b}}$, so the claim follows from Theorem 2.1, Proposition 3.4 and Proposition 3.5.

For (8), since we have $\#\bigl{\{}D\subset\Delta^{\mathfrak{b}+}\mid\sum_{\beta\in D}\beta=\Sigma_{J}^{\mathfrak{b}}\bigr{\}}=1,$ it follows that $\dim M^{\mathfrak{b}}(\lambda+\Sigma_{J}^{\mathfrak{b}})_{\lambda}=1.$ Thus, the Hom space is one-dimensional.

The image of the nonzero homomorphism is a submodule of $M^{\mathfrak{b}}(\lambda+\Sigma^{\mathfrak{b}}_{J})$ generated by $F_{J}^{\mathfrak{b}}v_{\lambda+\Sigma^{\mathfrak{b}}_{J}}^{\mathfrak{b}}$, so the claim follows from Theorem 2.1, Proposition 3.4 and Proposition 3.5. $\square$

By orthogonality, it is easy to see that the submodule generated by elements of the form

that are not equal to $v_{\lambda}^{\mathfrak{b}\cap r_{J}\mathfrak{b}}$ does not contain $v_{\lambda}^{\mathfrak{b}\cap r_{J}\mathfrak{b}}$.

By Theorem 2.1, Proposition 3.4 and Proposition 3.5, we have the following isomorphism:

Thus, $M^{\mathfrak{b}\cap r_{J}\mathfrak{b}}(\lambda)$ has a simple top $L^{\mathfrak{b}}(\lambda)$ and is therefore indecomposable. $\square$

Here, following [12], we review the basics of semibricks.

From here on, let $\mathcal{A}$ denote an abelian category.

• •

  We denote by $\operatorname{sim}\mathcal{A}$ the collection of isomorphism classes of simple objects in $\mathcal{A}$.

• •

  For a collection $\mathcal{C}$ of objects in $\mathcal{A}$, we denote by $\operatorname{Filt}\mathcal{C}$ the subcategory of $\mathcal{A}$ consisting of objects $X\in\mathcal{A}$ such that there exists a chain $0=X_{0}\subset X_{1}\subset\cdots\subset X_{l}=X$ of submodules with $X_{i}/X_{i-1}\in\mathcal{C}$ for each $i$. We call this the filtration closure of $\mathcal{C}$ in $\mathcal{A}$.

We say that an abelian category $\mathcal{A}$ is length if $\operatorname{sim}\mathcal{A}$ is a set and $\mathcal{A}=\operatorname{Filt}(\operatorname{sim}\mathcal{A})$ holds.

For example, categories such as $\mathfrak{g}\text{-smod}$, $s\mathcal{O}$ and $\mathcal{O}$ are length categories, but $\mathfrak{g}\text{-sMod}$ is not a length category.

The following is a classical result by Ringel.

For example, a Serre subcategory is the only wide subcategory that can be obtained by applying $\operatorname{Filt}$ to a subclass of $\operatorname{sim}\mathcal{A}$.

The following examples arise naturally, although they are in a different direction from the main theorem.

Suppose $\lambda\in\mathfrak{h}^{*}$, and let $\alpha_{1}$ and $\alpha_{3}$ be $\mathfrak{b}$-simple odd isotropic roots such that

Denote by $M_{1}$ and $M_{3}$ the images of the nonzero homomorphisms from $M^{r_{1}\mathfrak{b}}(\lambda-\alpha_{1})$ and $M^{r_{3}\mathfrak{b}}(\lambda-\alpha_{3})$ to $M^{\mathfrak{b}}(\lambda)$, respectively, which are unique up to scalar multiples. Define $M_{2}$ as the quotient $M_{2}=M^{\mathfrak{b}}(\lambda)/(M_{1}+M_{3})$.

If $(\alpha_{1},\alpha_{3})\neq 0$, then we have

Suppose there exist $y,z\in U(\mathfrak{n}^{-})$ such that

Using the notation from the PBW theorem Theorem 2.1, express $y$ in the basis where $x_{n}=f_{1}$ and $x_{n-1}=f_{3}$.

Define $y_{1}$ as the sum of PBW monomials in $y$ where $f_{3}$ does not appear. Since terms containing $f_{1}$ in $y$ can be ignored, we write

for some $y_{2}$, where $y_{2}$ is the sum of PBW monomials in which neither $f_{1}$ nor $f_{3}$ appears.

Then, we obtain

Here, using the notation from the PBW theorem Theorem 2.1, the right-hand side is expressed in the PBW basis where $x_{n}=f_{3}$ and $x_{n-1}=f_{1}$.

Since we can assume that $zf_{3}v_{\lambda}^{\mathfrak{b}}$ is already represented in this PBW basis, it cannot be equal to $yf_{1}v_{\lambda}^{\mathfrak{b}}$ unless $y_{1}=y_{2}=0$.

Therefore, $M_{1}$, which is generated by $f_{1}v_{\lambda}^{\mathfrak{b}}$, and $M_{3}$, which is generated by $f_{3}v_{\lambda}^{\mathfrak{b}}$, must intersect trivially.

If $(\alpha_{1},\alpha_{3})=0$, then we have

Thus, $M_{1}\cap M_{3}$ is the submodule generated by $f_{1}f_{3}v_{\lambda}^{\mathfrak{b}}$, which is $M^{\mathfrak{b}+r_{1}r_{3}\mathfrak{b}}(\lambda-\alpha_{1}-\alpha_{3})$. $\square$

In this subsection, let $\mathfrak{g}=\mathfrak{gl}(1|1)^{\oplus n}$. Then,

By embedding each $i$-th copy of $\mathfrak{gl}(1|1)$ into $\mathfrak{gl}(n|n)$ via the assignments:

and by identifying their Cartan subalgebras. Let $E_{ii}\in\mathfrak{gl}(n|n)$ be associated with the dual basis elements $\varepsilon_{i}$ for $1\leq i\leq 2n$, and define $\delta_{i}=\varepsilon_{n+i}$ for $1\leq i\leq n$. We have

Let $\mathfrak{b}_{\mathrm{st}}$ be the standard Borel subalgebra such that

The graph whose vertices are the Borel subalgebras of $\mathfrak{gl}(1|1)^{\oplus n}$ and whose edges correspond to odd reflections between them can be naturally identified with the hypercube graph $Q_{n}$.

The Verma module is $2^{n}$-dimensional, therefore $s\mathcal{O}=\mathfrak{g}\text{-smod}$.

Let the block containing the trivial module (i.e., the principal block) be denoted by $\mathcal{O}_{0}(\mathfrak{gl}(1|1)^{\oplus n})$.

We denote $\lambda=m_{1}(\varepsilon_{1}-\delta_{1})+\dots+m_{n}(\varepsilon_{n}-\delta_{n})$ more compactly as the tuple $(m_{1},\dots,m_{n})$.

Then,

$I=\{1,2,\dots,n\}$ is clearly a $(\mathfrak{b}_{\text{st}},0)$-hypercubic collection.