Pavel Etingof, Olivier Schiffmann
The purpose of this note is to define and construct highest weight modules for Felder's elliptic quantum groups. This is done by using exchange matrices for intertwining operators between (not necessarily finite-dimensional) modules over quantum affine algebras.
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We generalize the results of [KMST] concerning equivariant quantization by means of Verma modules $M(λ)$ for generic weight $λ$ to the case of general $λ$. We consider the relationship between the Shapovalov form on an irreducible highest weight module of a semisimple complex Lie algebra, fusion elements, and equivariant quantization. We also discuss some limiting properties of fusion elements. [KMST] E. Karolinsky and A. Stolin, Dynamical Yang-Baxter equations, quasi-Poisson homogeneous spaces, and quantization, Lett. Math. Phys., 71 (2005), p.179-197; e-print math.QA/0309203.
Using certain results for the vertex operator algebras associated with affine Lie algebras we obtain recurrence relations for the characters of integrable highest weight irreducible modules for an affine Lie algebra. As an application we show that in the simply-laced level 1 case, these recurrence relations give the known characters, whose principal specializations naturally give rise to some multisum Macdonald identities.
A representation of the Quantum Toroidal Algebra of type sl(N) is constructed on every irreducible integrable highest weight module of the Quantum Affine Algebra of type gl(N). As an intermediate step in the construction, we obtain a quantum analogue of the classical level-rank duality, describing the reciprocal decomposition of the q-Fock space with respect to mutually commutative actions of U_q(gl(N)^) of level L, and U_q(sl(L)^) of level N.
We give a criterion for complete reducibility of tensor product $V\otimes Z$ of two irreducible highest weight modules $V$ and $Z$ over a classical or quantum semi-simple group in terms of a contravariant symmetric bilinear form on $V\otimes Z$. This form is the product of the canonical contravariant forms on $V$ and $Z$. Then $V\otimes Z$ is completely reducible if and only if the form is non-degenerate when restricted to the sum of all highest weight submodules in $V\otimes Z$ or equivalently to the span of singular vectors.
We show that the Weyl-Kac type character formula holds for the integrable highest weight modules over the quantized enveloping algebra of any symmetrizable Kac-Moody Lie algebra, when the parameter $q$ is not a root of unity.
Canonical bases of tensor products of integrable highest weight modules arising from framed constructions
本文通过框架构造方法，证明了量子群中可积最高权模块张量积的典范基可通过更大量子群的典范基获得，并证明其正性性质。
We shall derive Kazhdan-Lusztig type character formula for the irreducible modules with arbitrary non-critical highest weights over affine Lie algebras from the rational case by using the translation functor, the Enright functor and Jantzen's deformation argument.
We completely determine the minimal polynomial of an arbitrary simple highest weight module $L(λ)$ over a complex classical Lie algebra $\mathfrak{g}\subseteq\mathfrak{gl}_N$ relative to its defining module $π=\mathbb{C}^{N}$. These results are applied to ordering on primitive ideals and algebraic properties of Howe duality correspondence.
In [19], Zheng studied the bounded derived categories of constructible $\bar{\mathbb{Q}}_l$-sheaves on some algebraic stacks consisting of the representations of a enlarged quiver and categorified the integrable highest weight modules of the corresponding quantum group by using these categories. In this paper, we shall generalize Zheng's work to highest weight modules of a subalgebra of the double Ringel-Hall algebra associated to a quiver in a functional version.
In this paper, we prove Khovanov-Lauda's cyclotomic categorification conjecture for all symmetrizable Kac-Moody algebras. Let $U_q(g)$ be the quantum group associated with a symmetrizable Cartan datum and let $V(Λ)$ be the irreducible highest weight $U_q(g)$-module with a dominant integral highest weight $Λ$. We prove that the cyclotomic Khovanov-Lauda-Rouquier algebra $R^Λ$ gives a categorification of $V(Λ)$.
Associating modules for the $h$-Yangian and quantum elliptic algebra in type $A$ with $h$-adic quantum vertex algebras
本文研究了将h-Yangian和量子椭圆代数在类型A中的模块与h-adic量子顶点代数关联，并在临界水平构造了新的中心元素。
We decompose the level-1 irreducible highest weight modules of the quantum affine algebra $U_q(\hat{sl}_n)$ with respect to the level-0 $U'_q (\hat{sl}_n)$--action defined in q-alg/9702024. The decomposition is parameterized by the skew Young diagrams of the border strip type.
We give a general way of representing the crystal (base) corresponding to the intgrable highest weight modules of quantum Kac-Moody algebras, which is called polyhedral realizations. This is applied to describe explicitly the crystal bases of integrable highest weight modules for arbitrary rank 2 Kac-Moody algebra cases, the classical A_n-case and the affine A^{(1)}_{n-1}-case.
In this paper, we investigate the structure of highest weight modules over the quantum queer superalgebra $U_q(q(n))$. The key ingredients are the triangular decomposition of $U_q(q(n))$ and the classification of finite dimensional irreducible modules over quantum Clifford superalgebras. The main results we prove are the classical limit theorem and the complete reducibility theorem for $U_q(q(n))$-modules in the category $O_q^{\geq 0}$.
It is proved that the regularity of parafermion vertex operator algebras associated to integrable highest weight modules for affine Kac-Moody algebra A_1^{(1)} implies the C_2-cofiniteness of parafermion vertex operator algebras associated to integrable highest weight modules for any affine Kac-Moody algebra. In particular, the parafermion vertex operator algebra associated to an integrable highest weight module of small level for any affine Kac-Moody algebra is C_2-cofinite and has only finitely many irreducible modules. Also, the parafermion vertex operator algebras with level 1 are determin…
This is the third of a series of articles devoted to the study of relaxed highest weight modules over vertex operator algebras. Relaxed highest weight modules over affine vertex algebras associated to higher rank Lie algebras $A_\ell$ are extensively studied. In particular, the string functions of simple relaxed highest weight modules whose top spaces are simple cuspidal $A_\ell$-modules are shown to be the quotients by a power of the Dedekind eta series of the $q$-characters of simple ordinary modules over affine W-algebras associated with the minimal nilpotent elements of $A_\ell$.
Affine highest weight structures on module categories over quiver Hecke algebras
本文证明任意类型箭图Hecke代数的有限生成分次模范畴具有多种Kleshchev分层，从而得到与Weyl群元素相关的量子幺正子群范畴为仿射最高权范畴，推广了多项早期工作，关键利用行列式模和R-矩阵技术。
The characters of the unitarizable highest weight modules over the N=2 superconformal algebras are presented. This is a slightly extended version of an Encyclopedia entry.
The quantized version of a discrete Knizhnik-Zamolodchikov system is solved by an extension of the generalized Bethe Ansatz. The solutions are constructed to be of highest weight which means they fully reflect the internal quantum group symmetry.
We consider the quantized Knizhnik-Zamolodchikov-Bernard difference equation (qKZB) with step $p$ and values in a tensor product of finite dimensional evaluation modules over the elliptic quantum group $E_{τ,η}(sl_2)$, the equation defined in terms of elliptic dynamical R-matrices. We solve the equation in terms of multidimensional q-hypergeometric integrals and describe its monodromy properties. We identify the space of solutions of the qKZ equation with the space of functions with values in the tensor product of the corresponding modules over the elliptic quantum group $E_{p,η}(sl_2)$.
In this paper we construct Bernstein--Gelfand--Gelfand type resolution of simple highest weight modules over the simple $\mathcal{N}=2$ vertex operator superalgebra of central charge $c_{p,p'}=3\left(1-\frac{2p'}{p}\right)$ by means of the Kazama--Suzuki coset construction. As an application, we compute the twisted Zhu algebras of the simple $\mathcal{N}=2$ vertex operator superalgebra. We also compute the Frenkel--Zhu bimodule structure associated with a certain simple highest weight module of central charge $c_{3,2}=-1$.
The small elliptic quantum group $e_{τ,γ}(sl_N)$, introduced in the paper, is an elliptic dynamical analogue of the universal enveloping algebra $U(sl_n)$. We define highest weight modules, Verma modules and contragradient modules over $e_{τ,γ}(sl_N)$, the dynamical Shapovalov form for $e_{τ,γ}(sl_N)$ and the contravariant form for highest weight $e_{τ,γ}(sl_N)$-modules. We show that any finite-dimensional $sl_N$-module and any Verma module over $sl_N$ can be lifted to the corresponding $e_{τ,γ}(sl_N)$-module on the same vector space. For the elliptic quantum group $E_{τ,γ}(sl_N)$ we construct…
We define a filtration indexed by the integers on the tensor product of an integrable highest weight module and a loop module for a quantum affine algebra. We prove that the filtration is either trivial or strictly decreasing and give sufficient conditions for this to happen. In the first case we prove that the module is irreducible and in the second case we prove that the intersection of all the modules is zero, thus allowing us to define the completed tensor product. In certain special cases, we identify the subsequent quotients of filtration. These are certain highest weight integrable modu…
We classify positive energy representations with finite degeneracies of the Lie algebra $W_{1+\infty}\/$ and construct them in terms of representation theory of the Lie algebra $\hatgl ( \infty R_m )\/$ of infinite matrices with finite number of non-zero diagonals over the algebra $R_m = \C [ t ] / ( t^{m + 1} )\/$. The unitary ones are classified as well. Similar results are obtained for the sin-algebras.
The Bershadsky-Polyakov algebras are the minimal quantum hamiltonian reductions of the affine vertex algebras associated to $\mathfrak{sl}_3$ and their simple quotients have a long history of applications in conformal field theory and string theory. Their representation theories are therefore quite interesting. Here, we classify the simple relaxed highest-weight modules, with finite-dimensional weight spaces, for all admissible but nonintegral levels, significantly generalising the known highest-weight classifications [arxiv:1005.0185, arxiv:1910.13781]. In particular, we prove that the simple…
Combinatorial bases of standard modules of twisted affine Lie algebras in types $A_{2l-1}^{(2)}$ and $D_{l+1}^{(2)}$: rectangular highest weights
本文利用顶点代数技术为两类扭仿射李代数的矩形最高权标准模构造组合基，并计算特征标公式，进而得到两个新的组合恒等式族。
In the present paper, using the technique of localization, we determine the center of the quantum Schrödinger algebra $§_q$ and classify simple modules with finite-dimensional weight spaces over $§_q$, when $q$ is not a root of unity. It turns out that there are four classes of such modules: dense $U_q(\mathfrak{sl}_2)$-modules, highest weight modules, lowest weight modules, and twisted modules of highest weight modules.
In arXiv:1001.2562 a certain non-commutative algebra $A$ was defined starting from a semi-simple algebraic group, so that the derived category of $A$-modules is equivalent to the derived category of coherent sheaves on the Springer (or Grothendieck-Springer) resolution. Let $\hat{\g}$ be the affine Lie algebra corresponding to the Langlands dual Lie algebra. Using results of Frenkel and Gaitsgory arXiv:0712.0788 we show that the category of $\hat{\g}$ modules at the critical level which are Iwahori integrable and have a fixed central character, is equivalent to the category of modules over a q…
Lusztig sheaves and tensor products of integrable highest weight modules
本文通过引入 N-框架箭图并局部化 Lusztig 层，范畴化实现了量子包络代数可积最高权模的张量积，证明其简单反常层基与 Lusztig 和 Bao-Wang 的典范基一致，并给出 Yang-Baxter 方程的范畴化解释。
The present paper establishes a connection between the Lie algebra W_{1+infty} and the bispectral problem. We show that the manifolds of bispectral operators obtained by Darboux transformations on powers of Bessel operators are in one to one correspondence with the manifolds of tau-functions lying in the W_{1+infty}-modules M_beta introduced in our previous paper hep-th/9510211. An immediate corollary is that they are preserved by hierarchies of symmetries generated by subalgebras of W_{1+infty}. This paper is the last of a series of papers (hep-th/9510211, q-alg/9602010, q-alg/9602011) on the…
We define analogues of Verma modules for finite W-algebras. By the usual ideas of highest weight theory, this is a first step towards the classification of finite dimensional irreducible modules. Motivated by known results in type A, we then formulate some precise conjectures in the case of nilpotent orbits of standard Levi type.
Relaxed highest-weight modules play a central role in the study of many important vertex operator (super)algebras and their associated (logarithmic) conformal field theories, including the admissible-level affine models. Indeed, their structure and their (super)characters together form the crucial input data for the standard module formalism that describes the modular transformations and Grothendieck fusion rules of such theories. In this article, character formulae are proved for relaxed highest-weight modules over the simple admissible-level affine vertex operator superalgebras associated to…
We introduce a category O of modules over the elliptic quantum group of sl_N with well-behaved q-character theory. We construct asymptotic modules as analytic continuation of a family of finite-dimensional modules, the Kirillov--Reshetikhin modules. In the Grothendieck ring of this category we prove two types of identities: generalized Baxter relations in the spirit of Frenkel--Hernandez between finite-dimensional modules and asymptotic modules; three-term Baxter TQ relations of infinite-dimensional modules.
Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra with highest root $θ$ and let $\mathfrak{g}[t]$ be the corresponding current algebra. In this paper, we consider the $\mathfrak{g}[t]$-stable Demazure modules associated to integrable highest weight representations of the affine Lie algebra $\widehat{\mathfrak{g}}$. We prove that the fusion product of Demazure modules of a given level with a single Demazure module of a different level and with highest weight a multiple of $θ$ is a generalized Demazure module, and also give defining relations. This also shows that the fusion …
In the first part of the paper we give the denominator identity for all simple finite-dimensional Lie super algebras $\frak g\/$ with a non-degenerate invariant bilinear form. We give also a character and (super) dimension formulas for all finite-dimensional irreducible $\frak g\/$-modules of atypicality $\leq 1\/$ . In the second part of the paper we give the denominator identity for the affine superalgebras $\hat{\frak g}\/$ associated to $\frak g\/$. Specializations of this identity give almost all old and many new formulas for the number of representations of an integer as sums of squares …
In this paper, we consider the classification of irreducible ${\bf Z}$- and ${\bf Z}^2$-graded modules with finite dimensional homogeneous subspaces over the Virasoro-like algebra. We first prove that such a module is a uniformly bounded module or a generalized highest weight module. Then we determine all generalized highest weight irreducible modules. As a consequence, we also determine all the modules with nonzero center. Finally, we prove that there does not exist any nontrivial ${\bf Z}$-graded modules of intermediate series.
Highest weight modules of the double affine Lie algebra $\widehat{\widehat{\mathfrak{sl}}}_{2}$ are studied under a new triangular decomposition. Singular vectors of Verma modules are determined using a similar condition with horizontal affine Lie subalgebras, and highest weight modules are described under the condition that $c_1>0$ and $c_2=0$.
We build representations of the elliptic braid group from the data of a quantum D-module M over a ribbon Hopf algebra U. The construction is modelled on, and generalizes, similar constructions by Lyubashenko and Majid, and also certain geometric constructions of Calaque, Enriquez, and Etingof concerning trigonometric Cherednik algebras. In this context, the former construction is the special case where M is the basic representation, while the latter construction can be recovered as a quasi-classical limit of U=U_t(sl_N), as t limits 1. In the latter case, we produce representations of the doub…
This is the second of a series of articles devoted to the study of relaxed highest-weight modules over affine vertex algebras and W-algebras. The first studied the simple "rank-$1$" affine vertex superalgebras $L_k(\mathfrak{sl}_2)$ and $L_k(\mathfrak{osp}(1\vert2))$, with the main results including the first complete proofs of certain conjectured character formulae (as well as some entirely new ones). Here, we turn to the question of classifying relaxed highest-weight modules for simple affine vertex algebras of arbitrary rank. The key point is that this can be reduced to the classification o…
We give the first positive formulas for the weights of every simple highest weight module $L(λ)$ over an arbitrary Kac-Moody algebra. Under a mild condition on the highest weight, we also express the weights of $L(λ)$ as an alternating sum similar to the Weyl-Kac character formula. To obtain these results, we show the following data attached to a highest weight module are equivalent: (i) its integrability, (ii) the convex hull of its weights, (iii) the Weyl group symmetry of its character, and (iv) when a localization theorem is available, its behavior on certain codimension one Schubert cells…
In [1] we have constructed a [n+1/2]+1 parameters family of irreducible representations of the Braid group B_3 in arbitrary dimension using a $q-$deformation of the Pascal triangle. This construction extends in particular results by S.P. Humphries (2000), who constructed representations of the braid group B_3 in arbitrary dimension using the classical Pascal triangle. E. Ferrand (2000) obtained an equivalent representation of B_3 by considering two special operators in the space ${\mathbb C}^n[X].$ Slightly more general representations were given by I. Tuba and H. Wenzl (2001). They involve [n…
Classification of irreducible highest weight modules for the parafermion vertex algebras $N_k(\mathfrak{sl}_2)$ at arbitrary level
该文分类了任意level下parafermion顶点代数$N_k(\boldsymbol{\frak{sl}}_2)$的不可约最高权模，给出Zhu代数的表示，在非整数可允许level及临界level完成分类，补充了正整数level的已有结果。
In this paper, a notion of cyclotomic (or level $k$) walled Brauer algebras $\mathscr B_{k, r, t}$ is introduced for arbitrary positive integer $k$. It is proven that $\mathscr B_{k, r, t}$ is free over a commutative ring with rank $k^{r+t}(r+t)!$ if and only if it is admissible. Using super Schur-Weyl duality between general linear Lie superalgebras $\mathfrak{gl}_{m|n}$ and $\mathscr B_{2, r, t}$, we give a classification of highest weight vectors of $\mathfrak{gl}_{m|n}$-modules $M_{pq}^{rt}$, the tensor products of Kac-modules with mixed tensor products of the natural module and its dual. …
On multiplicity-free weight modules over quantum affine algebras
本文旨在构造并研究量子仿射代数的无重数权模，通过引入关于可对称化广义 Cartan 矩阵的可移性条件，探讨了其在代数结构与无穷维模实现中的应用，并计算了相关最高 $\ell$-权。
Using the bases of principal subspaces for twisted affine Lie algebras except $A_{2l}^{(2)}$ by Butorac and Sadowski, we construct bases of the highest weight modules of highest weight $kΛ_0$ and parafermionic spases for the same affine Lie algebras. As a result, we obtain their character formulas conjectured in arXiv:math/0102113.
Let $V(Λ_i)$ (resp., $V(-Λ_j)$) be a fundamental integrable highest (resp., lowest) weight module of $U_q(\hat{sl}_{2})$. The tensor product $V(Λ_i)\otimes V(-Λ_j)$ is filtered by submodules $F_n=U_q(\hat{sl}_{2})(v_i\otimes \bar{v}_{n-i})$, $n\ge 0, n\equiv i-j\bmod 2$, where $v_i\in V(Λ_i)$ is the highest vector and $\bar{v}_{n-i}\in V(-Λ_j)$ is an extremal vector. We show that $F_n/F_{n+2}$ is isomorphic to the level 0 extremal weight module $V(n(Λ_1-Λ_0))$. Using this we give a functional realization of the completion of $V(Λ_i)\otimes V(-Λ_j)$ by the filtration $(F_n)_{n\geq0}$. The subsp…
For a field $F$ of characteristic zero and an additive subgroup $G$ of $F$, a Lie algebra $B(G)$ of lock type is defined with basis $\{L_{a,i},c|a \in G, i>-2\}$ and relations $[L_{a,i},L_{b,j}]=((i+1)b-(j+1)a)L_{a+b,i+j}+a\d_{a,-b}\d_{i+j,-2}c, [c,L_{a,i}]=0.$ Given a total order $\succ$ on $G$ compatible with its group structure, and any $Λ\in B(G)_0^*$, a Verma $B(G)$-module $M(Λ,\succ)$ is defined, and the irreducibility of $M(Λ,\succ)$ is completely determined. Furthermore, it is proved that an irreducible highest weight $B(Z)$-module is quasifinite if and only if it is a proper quotient …
We give an integral representation for solutions of the elliptic quantum Knizhnik-Zamolodchikov-Bernard difference equations, in the case of sl(2). The result is based on a geometric construction of highest weight representations of the elliptic quantum group associated to sl(2). We also obtain Bethe ansatz eigenfunctions for the corresponding integrable systems of difference operators, and prove their completeness in some cases.
We announce a systematic way for constructing bispectral algebras of commuting differential operators of any rank N. It enables us to obtain all previously known classes and examples of bispectral operators. Moreover, we give a representation-theoretic explanation of the results including those of Duistermaat and Grünbaum. The manifold of bispectral operators of any order is preserved by an hierarchy of symmetries. We point out that our methods provide a completely algorithmic procedure for obtaining bispectral algebras. We conjecture that the class built in the present paper exhausts all bisp…
The aim of the paper is to study modules for the twisted Heisenberg-Virasoro algebra $\mathcal H$ at level zero as modules for the $W(2,2)$-algebra by using construction from [J. Pure Appl. Algebra 219 (2015), 4322-4342, arXiv:1405.1707]. We prove that the irreducible highest weight ${\mathcal H}$-module is irreducible as $W(2,2)$-module if and only if it has a typical highest weight. Finally, we construct a screening operator acting on the Heisenberg-Virasoro vertex algebra whose kernel is exactly $W(2,2)$ vertex algebra.
Lusztig sheaves and integrable highest weight modules in the symmetrizable case
本文基于前人工作，研究对称可约情况下的Lusztig sheaves与可积分最高权模的构造及格罗滕迪克群的实现，揭示其与晶格结构的关系。
Two classes of irreducible highest weight modules of the general linear Lie superalgebra $gl(1/\infty)$ are constructed. Within each module a basis is introduced and the transformation relations of the basis under the action of the algebra generators are written down.
We present a criterion for establishing Morita equivalence of monoidal categories, and apply it to the categorical representation theory of reductive groups $G$. We show that the "de Rham group algebra" $\mathcal D(G)$ (the monoidal category of $\mathcal D$-modules on $G$) is Morita equivalent to the universal Hecke category $\mathcal D(N \backslash G/N)$ and to its monodromic variant $\widetilde{\mathcal D}(B \backslash G / B)$. In other words, de Rham $G$-categories, i.e., module categories for $\mathcal D(G)$, satisfy a "highest weight theorem" - they all appear in the decomposition of the …
On the computation of the canonical basis for irreducible highest weight $U_q (\mathfrak{gl}_{\infty})$-modules
本文提出了一种新的方法，用于计算不可约最高权 $U_q (\mathfrak{gl}_{\infty})$-模的 canonical basis，并将其应用于 Calogero-Moser 细胞特征标和 Ariki-Koike 代数的分解矩阵。
We show that subsingular vectors exist in Verma modules over W(2,2), and present a subquotient structure of these modules. We prove conditions for irreducibility of a tensor product of intermediate series module with the highest weight module. Relations to intertwining operators over vertex operator algebra associated to W(2,2) is discussed. Also, we study a tensor product of intermediate series and highest weight module over the twisted Heisenberg-Virasoro algebra, and present series of irreducible modules with infinite-dimensional weight spaces.
Lusztig sheaves and integrable highest weight modules
本文通过带框箭图 Lusztig 层的局部化及相应函子，在 Grothendieck 群中实现量子群的不可约可积最高权模，并以非零简单反常层构造典范基，证明其与 Nakajima 箭图簇实现间的转移矩阵为对角元全为 $\pm 1$ 的上三角矩阵。
In [8], the affine vertex algebra $L_k(\mathfrak{sl}_2)$ is realized as a subalgebra of the vertex algebra $Vir_c \otimes Π(0)$, where $Vir_c$ is a simple Virasoro vertex algebra and $Π(0)$ is a half-lattice vertex algebra. Moreover, all $L_k(\mathfrak{sl}_2)$--modules (including, modules in the category $KL_k$, relaxed highest weight modules and logarithmic modules) are realized as $Vir_c \otimes Π(0)$--modules. A natural question is the generalization of this construction in higher rank. In the current paper, we study the case $\mathfrak{g}= \mathfrak{sl}_3$ and present realization of the VO…
We investigate the relation between the generating functions of the highest weight vectors of the Monster module and the McKay-Thompson series.
We study some non-highest weight modules over an affine Kac-Moody algebra at non-critical level. Roughly speaking, these modules are non-commutative localizations of some non-highest weight "vacuum" modules. Using free field realization, we embed some rings of differential operators in endomorphism rings of our modules. These rings of differential operators act on a localization of the space of coinvariants of any module over the Kac-Moody algebra with respect to a certain level subalgebra. In a particular case this action is identified with the Casimir connection.
We show that the equivariant hypertoric convolution algebras introduced by Braden-Licata-Proudfoot-Webster are affine quasi hereditary in the sense of Kleshchev and compute the Ext groups between standard modules. Together with the main result of arXiv:2009.03981, this implies a number of new homological results about the bordered Floer algebras of Ozsvath-Szabo, including the existence of standard modules over these algebras. We prove that the Ext groups between standard modules are isomorphic to the homology of a variant of the Lipshitz-Ozsvath-Thurston bordered strands dg algebras.
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