Bimodule structure of the mixed tensor product over $U_{q} s\ell(2|1)$ and quantum walled Brauer algebra

We study a mixed tensor product $\mathbf{3}^{\otimes m} \otimes \mathbf{\overline{3}}^{\otimes n}$ of the three-dimensional fundamental representations of the Hopf algebra $U_{q} s\ell(2|1)$, whenever $q$ is not a root of unity. Formulas for the decomposition of tensor products of any simple and projective $U_{q} s\ell(2|1)$-module with the generating modules $\mathbf{3}$ and $\mathbf{\overline{3}}$ are obtained. The centralizer of $U_{q} s\ell(2|1)$ on the chain is calculated. It is shown to be the quotient $\mathscr{X}_{m,n}$ of the quantum walled Brauer algebra. The structure of projective modules over $\mathscr{X}_{m,n}$ is written down explicitly. It is known that the walled Brauer algebras form an infinite tower. We have calculated the corresponding restriction functors on simple and projective modules over $\mathscr{X}_{m,n}$. This result forms a crucial step in decomposition of the mixed tensor product as a bimodule over $\mathscr{X}_{m,n}\boxtimes U_{q} s\ell(2|1)$. We give an explicit bimodule structure for all $m,n$.


INTRODUCTION
Over the course of the last twenty years Logarithmic conformal field theory (LCFT) has established itself as an area of extensive interaction between models of statistical physics such as percolation, the sand pile model, dense polymers as well as other models with nonlocal observables on the one hand, and modern topics in mathematics such as Nichols algebras, quantum groups, braided categories, VOA theory and diagram algebras on the other. One of the most developed approaches [1,2,3] to constructing LCFT is based on the intersection of screening operator kernels. In this approach one chooses a lattice vertex operator algebra (VOA) and fixes a set of fields v i , which correspond to representations of VOA and are called screening currents.
The zero modes of these currents s i " ű v i are called screenings. Under certain integer valuedness conditions on scalar products of the screening currents momenta, the screenings form a finite-dimensional Nichols algebra (see examples in [4,5]). Under these conditions the intersection of the screening kernels is a vacuum module of a rational LCFT: Vac " Ş i Ker s i . In this case, LCFT is a representation space of the rational W -algebra W " Vac.
The algebra W has only a finite number of irreducible representations. The set of simple and projective W-modules is closed under fusion and the characters of the W-irreducible modules generate a finite-dimensional representation of the modular group.
Another source of LCFT is given by various lattice models [6,7,8,9,10,11,12]. CFT appears naturally as a scaling limit of lattice models in the critical point, see e.g. [13]. Then, a mathematically rigorous program on algebraic construction of the scaling limits was initiated in [14]. If one considers nonlocal observables (for example, the cluster probability in percolation theory [15,16,17,18]) in the lattice model, then in the scaling limit an LCFT is in general expected to appear, and in several models [19,20,11,21] its appearance is shown explicitly.
The standard approach to studying the lattice models is the transfer-matrix method [22,23]. In this approach a connection with a spin chain is established by the Hamiltonian limit. Another feature of lattice models with nonlocal observables is that in the Hamiltonian limit there exist nontrivial Jordan blocks in the Hamiltonian [24,25,26,27,28] (see discussion on the Jordan blocks problem in the algebraic Bethe ansatz approach in [29]). From the side of LCFT the existence of nontrivial Jordan blocks in the Hamiltonian is expressed in the fact that the conformal dimension operator L 0 becomes non-diagonalizable and conformal blocks admit logarithmic terms.
In both approaches a quantum group plays a crucial role [30,31,8,12]. In the first case quantum group appears as a double bosonization of the algebra generated by screenings [32]. In the second case the spin-chain can be constructed as tensor product of fundamental representations of the quantum group.
For the simplest case of p1, p q LCFT models [33,34,35,36], the corresponding spinchain T N is a tensor product of two-dimensional representations of the quantum group U q s ℓp2q, [37], and the T N is called the Heisenberg spin-chain.
An interesting generalization of the Heisenberg spin-chain is a spin-chain based on the algebra U q s ℓpM |N q [38]. Such spin-chains describe interaction between spin and other degrees of freedom. For instance, U q s ℓp2|1q-spin-chain is related with (integrable) t-J model which includes the interaction between spin and charge degrees of freedom, [39,40,41].
On the side of LCFT, models related with the quantum group U q s ℓp2|1q are constructed in [32] by the approach based on intersection of kernels of the screening operators. Rational W -algebras W containing as a subalgebraŝ l p2q k at a rational level k naturally occur in these models. At the same time models overŝ l p2q k are not rational. In this caseŝ l p2q k is an analog of the Virasoro algebra in p1, p q-models. More on LCFT withŝ l p2q k see in [42,43,44].
In order to investigate how LCFT with quantum group U q s ℓp2|1q appears in the scaling limit of the spin-chain, it is natural to follow the approach proposed in [37].
In the present paper we study U q s ℓp2|1q mixed tensor product which is the space of states for the spin-chains with U q s ℓp2|1q symmetry. But the case when q is a root of unity q " e i π{p , is more complicated and we leave it for a separate work. Therefore in the present paper we consider only the algebra with a generic value of the parameter q.
It is useful to make an analogy with the Heisenberg U q s ℓp2q-spin-chain with generic q. Its centralizer CpU q s ℓp2qq on the chain is the Temperley-Lieb algebra CpU q s ℓp2qq " T L N with the same value of the parameter q. Thus, the spin-chain space of states can be expressed as a bimodule T N " where V i and M i are some simple U q s ℓp2q-and T L N -modules. When N Ñ 8, the algebra T L N conjecturally converges to the Virasoro algebra. When q is a root of unity the centralizer of the Temperley-Lieb algebra T L N is the Lusztig limit LU q s ℓp2q of U q s ℓp2q. In this case the bimodule decomposition of the spin chain contains non semisimple summands [12].
When q is a root of unity, the algebra LU q s ℓp2q contains the restricted quantum group U q s ℓp2q as a subalgebra, see details in [45]. In [37] it is shown that the centralizer of U q s ℓp2q on the spin-chain T N is the algebra W N , which contains the algebra T L N . In the limit N Ñ 8 the algebra W N gives the triplet algebra W built by the lattice VOA construction.
In case of U q s ℓp2|1q we take its (mutually dual) fundamental representations which are three-dimensional and denote them by 3 3 3 and 3 3 3. We study the mixed tensor product The tensor product T m ,n is the space of states of different integrable spin-chains with U q s ℓp2|1q symmetric hamiltonians, examples of which are considered in [46,7,47,48]. We let X m ,n denote the centralizer of U q s ℓp2|1q on T m ,n , CpU q s ℓp2|1qq " X m ,n . It is shown in [49,50,51,52] that X m ,n is isomorphic to some quotient of the quantum walled Brauer algebra qwB m ,n . In this paper we do not give an explicit description of X m ,n itself, but describe simple and projective modules over X m ,n . We find the decomposition of the chain T m ,n as a bimodule over U q s ℓp2|1q and X m ,n . Even for generic values of q, the bimodule is not semisimple. We give the bimodule in an explicit form in Theorem 5.3.
The quantum walled Brauer algebra qwB m ,n was introduced in [53,54,55]. The two-parametric algebra qwB m ,n was introduced in [56] and the structure of the simple modules was described implicitly. Modules over qwB m ,n and its classical analogue wB m ,n were investigated in [57,58,59,60,61].
For arbitrary values M , N the algebra U q s ℓpM |N q on the appropriate mixed tensor product (which is the tensor product of its fundamental representations) is centralized by some quotient of qwB m ,n , see also [52,62]. If N " 0 the bimodule is semisimple. We study the simplest non-semisimple case N " 1.
The outline of the article is as follows. In Sec. 2 we define the algebra U q s ℓp2|1q and classify its finite-dimensional simple and projective modules. In Sec. 3 we describe the mixed tensor product and introduce the centralizer X m ,n . First, we prove the formulas for the tensor products of modules needed to the mixed tensor product decomposition. Next, we show that the centralizer is a quotient of the algebra qwB m ,n . In Sec. 4 we describe simple and projective modules over X m ,n and the restriction functors on them. In the last Sec. 5 we describe the bimodule structure and give a sketch of a proof for the bimodule decomposition formula.

2.
THE HOPF ALGEBRA U q s ℓp2|1q 2.1. Definition of U q s ℓp2|1q. Quantum analogues of superalgebras s ℓp2|1q and g ℓp2|1q was studied intensively in [63,64,65,66]. We describe the Hopf algebra U q s ℓp2|1q by a system of generators and relations. In this section and in the entire paper we assume that the parameter q is not a root of unity. We choose the generators adapted to the Hopf subalgebra structure U q s ℓp2|1q Ą U q g ℓp2q Ą U q s ℓp2q (such that the embeddings become tautological); we extensively use these subalgebras while working with U q s ℓp2|1q modules in the sequel. The Hopf subalgebra U q s ℓp2q in U q s ℓp2|1q is generated as an associative algebra by E , K , and F with the relations (2.1) The larger algebra U q g ℓp2q contains an additional generator k satisfying the relations We call the generators E , F , K and k bosonic. There are two additional generators B and C , which extend U q g ℓp2q to U q s ℓp2|1q, and which we call fermionic, or simply fermions. The relations that involve the fermions B and C are where we use q -integers defined as rns " q n´q´n q´q´1 . The Hopf-algebra structure of U q s ℓp2|1q (the coproduct, the antipode, and the counit) is given by S pB q "´k B , S pF q "´K F, S pC q "´C k´1, S pE q "´E K´1, (2.5) εpB q " 0, εpF q " 0, εpC q " 0, εpE q " 0, (2.6) with k and K being group-like.

2.2.
Simple U q s ℓp2|1q modules. We consider a subcategory of U q s ℓp2|1q-modules with k eigenvalues of the form q´n for n P . The subcategory is closed under tensor products. The simple finite-dimensional U q s ℓp2|1q-modules can be labeled as They have dimensions The modules with r " 0 and r " s are atypical, and others are typical. In [63] it was shown that every finite-dimensional irreducible module over the general linear Lie superalgebra g ℓpn|1q can be deformed into an irreducible module over U q g ℓpn|1q.
Notations "typical" and "atypical" for modules in the present work are inherited from the theory of Lie superalgebras (see, for example [67]).

2.2.1.
U q s ℓp2|1q-action on simple modules. We describe (following [68]) the action of U q s ℓp2|1q on its simple modules explicitly, using the basis adapted to the decomposition into U q g ℓp2q-modules. Each U q s ℓp2|1q-module decomposes into a direct sum of simple U q g ℓp2q-modules X α,β s ,r , where α, β "˘, s ě 1, r P . Their dimensions are dim X α,β s ,r " s . Eigenvalues of generators K and k on the highest weight vector in the module X α,β s ,r are αq s´1 and β q´r correspondingly.

Ext 1 1 1 spaces for atypical modules.
For two modules Z 1 and Z 2 , we define Ext 1 pZ 2 , Z 1 q as a linear space with basis identified with nontrivial short exact sequences modulo a certain equivalence relation [69].
The groups Ext 1 vanish for the typical U q s ℓp2|1q modules. For the atypical modules, the Ext 1 pZ 1 , Z 2 q group is at most 1-dimensional. Whenever Ext 1 pZ 1 , Z 2 q is nontrivial, we describe the algebra action in terms of generators: the action of a U q s ℓp2|1qgenerator A on Z 1 i Z 2 is given by A is the direct sum of actions of U q s ℓp2|1q-generators on the simple modules and ξ A " ξ Z 1 ,Z 2 A : Z 1 Ñ Z 2 are linear maps.
We list the ξ A maps in terms of the bases introduced above. The formulas can be somewhat uniformized by adopting the following convention for the 1-dimensional modules Z α,´β 1,0 : we denote this module also by Z α,β 0,0 , with a basis vector |α, 0, β , 0y Ñ 0 " |α, 1,´β , 0y Ð 0 (and, formally, with |α, 0, β , 0y Ð m " 0, m ‰ 0 s ,0 and Z α,β s ,s (where, as before, α, β "˘1 and s ě 1). We describe the projective covers in terms of Loewy graphs. The reconstruction of the U q s ℓp2|1q-action on a projective module from its Loewy graph is described in detail in [68,Sec. 6]. The action ρ A pνq of a generator A on a vector ν has three parts: where ρ p0q A pνq is the action of A in the irreducible subquotient, ξ A is determined in 2.3, and for the map η A we give explicit formulas after each Loewy graph (whenever η A is nonzero). Here c pνq are some coefficients depending on a pair of simple subquotients in the projective module in question. We write them on edges in Loewy graphs (see [68] for a detailed explanation).
It is convenient to distinguish between two series and two exceptional cases of projective covers. The first series is R

THE MIXED TENSOR PRODUCT
We study the mixed tensor product ("spin-chain") (1.1), where 3 3 3 " Z 1,´1 1,1 and 3 3 3 " Z 1,1 2,0 are the two three-dimensional simple U q s ℓp2|1q-modules. We are interested in decomposing T m ,n as a bimodule over U q s ℓp2|1q and its centralizer X m ,n . As a necessary first step, we decompose tensor products of relevant U q s ℓp2|1q-modules with the fundamental modules Z where we write α 12 " α 1 α 2 and β 12 " β 1 β 2 .

3.1.1.
It follows, in particular, that the set of simple modules and their projective covers is closed under tensor product decompositions.
We consider the U q s ℓp2|1q-modules in the left-hand side of the tensor product as U q g ℓp2q-modules (as explained in 2.2.1) and calculate their tensor product using the results in [45]. For the tensor product Z α 1 ,β 1 Decomposition (3.1) contains six U q g ℓp2q-modules. Taking into account that a typical module contains four U q g ℓp2q-summands and an atypical one contains two, the module in (3.1) can be the direct sum of either three atypical U q s ℓp2|1q-modules or one typical and one atypical module. Explicitly writing the decompositions of possible U q s ℓp2|1q-modules shows that there exists only one U q s ℓp2|1q-module that has the decomposition (3.1). The second and the fifth summands can be combined into Z α 12 ,´β 12 s`1,s`1 and the other four summands give Z α 12 ,β 12 s ,s`1 . Thus, we have We next consider the product Z involves only projective modules, which, as we recall from 2.4.2, consist of all typical simple modules and the R α,β s ,r . There are several U q s ℓp2|1q-modules that have the U q g ℓp2q-decomposition (3.2), but only one of them is projective. 1 Thus, we have are worked out similarly. We consider U q g ℓp2q-decompositions of both tensorands and calculate tensor products of U q g ℓp2qmodules. This gives a long direct sum of simple and projective U q g ℓp2q-modules that each time are combined uniquely into a sum of projective U q s ℓp2|1q-modules.

Remark.
Decomposition of all tensor products of finite dimensional s ℓp2|1qrepresentations into their indecomposable building blocks was found in [70].

3.1.3.
We calculate decomposition of T m ,n iteratively using Theorem 3.1. The multiplicities of U q s ℓp2|1q-indecomposable modules are dimensions of X m ,n -modules, which we discuss below.

3.2.
The centralizer of U q s ℓp2|1q on the mixed tensor product. We fix bases in the 3 3 3 and 3 3 3 modules in accordance with 2.2.1 and introduce a shorthand notation for them: In the tensor products of two U q s ℓp2|1q modules, we then have the operators that commute with U q s ℓp2|1q and are explicitly given by 1 For example, the direct sum of simple U q s ℓp2|1q-modules 2Z is compatible with the U q g ℓp2q-decomposition (3.2), but is not a projective U q s ℓp2|1q-module.
g :¨f and h :¨v On T m ,n , we define the operators These are the generators of a quantum walled Brauer algebra, which we discuss in the next subsection.

3.3.1.
The algebra qwB m ,n is the associative unital algebra generated by g i , E, h j , where 1 ď i ď m´1 and 1 ď j ď n´1, with relations (see [53,54,55]) These relations involve complex parameters γ, δ, and θ , and we sometimes use the notation qwB m ,n pγ, δ, θ q for the algebra, although one parameter can be eliminated from the relations by renormalizing the generators. We write the relations in the present form for more convenient comparison with different choices in literature.

Remark.
The algebra qwB m ,n has a presentation by tangle diagrams, see [61].
It can be considered as a classical limit of quantum walled Brauer algebra qwB m ,n .
To get this limit from the algebra with relations 3.3.1 we can do the following. By renormalization of the generators, parameter γ can be set to γ "´1. We introduce a complex parameter r : θ "´δ r so that the relation reads EE "´δ r´1 δ´1 E. Then we consider the limit δ Ñ 1. The dependent on parameters algebra relations become Such an algebra is called the (classical) walled Brauer algebra with (only one) parameter r . We use the notation wB m ,n p´r q for it.  θ " δ γ remains invariant. This relation means that we consider a degenerate case in which the algebra becomes non-semisimple as we discuss below.
3.3.6. Corollary. The endomorphism algebra of U q s ℓp2|1q-module T m ,n is isomorphic to the quotient of the algebra qwB m ,n with special parameters (3.3).
One can consider an algebra U q s ℓpM |N q for arbitrary positive integers M and N . Let V and V ‹ be fundamental representation of U q s ℓpM |N q and its dual. We let X M ,N m ,n denote the algebra of endomorphisms of U q s ℓpM |N q on mixed tensor product V ‹bm b V bn . As was shown in [71] (see also [50,51,49,52]) there is a surjective homomorphism (3.5) Ψ M ,N m ,n : qwB m ,n pγ "´1, δ " q´2, θ "´q´2 pM´N q q Ñ X M ,N m ,n .
Here the parameter q is the same as in the algebra U q s ℓpM |N q. In the classical limit we conclude that the algebra of endomorphisms of s ℓpM |N q on mixed tensor product of its fundamental representations is a quotient of the algebra wB m ,n pr q with r " N´M . This is consistent with the results of [49,52] because classical algebras wB m ,n pr q and wB m ,n p´r q are isomorhic to each other. Indeed, the isomorphism is given by the formulas g 1 i "´g i , h 1 j "´h j and E 1 "´E. We note that for N " 0 the algebra X M ,0 m ,n is semisimple and ker Ψ M ,0 m ,n contains the whole radical of qwB m ,n , see [54].
At the end of this section we formulate two statements important for the sequel. The walled Brauer algebra has quasihereditary structure, see [58]. According to our first conjecture we suppose qwB m ,n with generic values of the parameter δ γ to be also quasihereditary.
In the following sections we consider only the case M " 2, N " 1 and use the notation X m ,n for X 2,1 m ,n . The second important statement is (see also [72]  The conjecture about quasihereditary structure in the general case X M ,N m,n can apparently be formulated but is beyond the scope of this paper.
A bipartition is a pair of partitions λ " pλ L , λ R q. Let Λ be the set of all bipartitions. For each integer 0 ď f ď minpm, nq, we set where |λ| is the sum of elements of a partition, and The set Λ m ,n is in bijective correspondence with the set of qwB m ,n Specht modules [56]. We let S pλq denote the qwB m ,n -Specht module corresponding to the bipartition λ.
The following claim is given in [52]

Theorem.
For generic values of the qwB m ,n parameters, each Specht module is simple, and the sets of Specht and simple modules coincide.

4.2.
Modules over qwB m ,n with special parameters. We now consider the cathegory of qwB m ,n modules with the parameters related as in (3.4). The algebra is then nonsemisimple, and some of the Specht modules S pλq become reducible.
Let D pλq and K pλq be the simple head and the projective cover for S pλq. Below we also use the notation D The decomposition multiplicities d λ,µ " " S pµq : D pλq ‰ for the S pλq-modules in terms of their simple subquotients are determined in [58]. Because of the quasihereditary structure of qwB m ,n each projective module K pλq has a filtration by Specht modules. Letd λµ " " K pλq : S pµq ‰ be the multiplicity of a given Specht module S pλq in the filtration; then, by the Brauer-Humphreys reciprocity (see [58] and references therein) We use this statement to construct projective modules for X m ,n in the next subsection.

Modules in the decomposition of the mixed tensor product.
As a X m ,n bU q s ℓp2|1qbimodule, the mixed tensor product T m ,n decomposes into a direct sum of indecomposable bimodules.

Definition.
For non-negative integers p , q , a partition µ is called a pp, q q-hook partition if it doesn't contain a box in the pp`1, q`1q-position, i.e. µ p`1 ă q`1.

Definition.
(see [73]) For non-negative integers p , q a bipartition λ " pλ L , λ R q is called a pp, q q-cross bipartition if there exist non-negative integers p 1 , p 2 , q 1 , q 2 such that λ L is a pp 1 , q 1 q-hook partition, λ R is a pp 2 , q 2 q-hook partition and p 1`p2 ď p , Let Cr m ,n be the subset of all p2, 1q-cross bipartitions in Λ m ,n . Assuming the Conjecture 1 (3.4) and applying the statements from [52], [73] for M " 2, N " 1, we have

Proposition.
If λ P Cr m ,n then ker Ψ 2,1 m ,n acts as zero on D pλq. The modules D pλq, λ P Cr m ,n give a complete set of simple X m ,n -modules.

Proposition.
Each X m ,n -simple module D pλq, λ P Cr m ,n occurs as a subquotient in the bimodule decomposition of T m ,n .
In the following we use notation a " |m´n|. For bipartitions from Cr m ,n we intro- We call these bipartitions atypical. If λ P At m ,n we call corresponding modules S pλq and D pλq atypical also.
We define the operationĜ from the set of qwB m ,n -modules to the set of qwB n ,mmodules. The operationĜ acts on the simple qwB m ,n -module by the formula i.e. it changes left and right partitions in a bipartition. We note thatĜ A a s "Â a s , and similarly for B a s , C a s . When applied to projective modules, the operationĜ acts on each simple subquotient by the formula (4.5) and does not change the structure of the Loewy graph. It is obvious that The action of the algebra X m ,n on an arbitrary qwB m ,n -module is not defined in general. In particular, it is not defined on some qwB m ,n -Specht modules, that contain D pλ 1 q, λ 1 R Cr m ,n as a subquotient. For λ P Cr m ,n we define a Specht module over X m ,n (abusing notation we use the same symbol S pλq for it) as a factor of corresponding qwB m ,n -Specht module S pλq over all suquotients D pλ 1 q with λ 1 R Cr m ,n .
Similarly we let K pλq denote the projective cover for X m ,n -module S pλq. This projective cover is a subquotient of qwB m ,n projective module K pλq.
Assuming the Conjecture 2 (3.5), we have the equality of multiplicitiesd λ,µ " d λ,µ for X m ,n in analogy with (4.3). Using [58] and Proposition 4.3.3, we have the following Theorem. We write down the structure of the Loewy graphs for X m ,n -projective modules (analogously to the formulas 2.11-2.13 for U q s ℓp2|1q-projective modules). They are oriented graphs where arrows mean the action of the algebra X m ,n . States from the subquotient at the beginning of an arrow are mapped to the states in the subquotient at the end of an arrow and (possibly) in the subquotients further the arrows. Investigation of Ext 1 1 1 spaces for the algebra X m ,n and the detailed action of all X m ,n -generators on projective modules are beyond the scope of this paper.

Theorem.
For λ P Cr m ,n , λ R At m ,n , the projective module over X m ,n coincides with the simple module: K pλq " D pλq. For λ P At m ,n , we have the following structure of projective modules over X m ,n for m ą n: D pA a n q K pA a n q " D pA a n´1 q , a ě 1, n ě 1, D pA a n q K pA a 0 q " D pA a 0 q , a ě 1, n ě 1, D pB a n q K pB a n q " D pB a n´1 q , a ě n, n ě 2, D pC n´2 n´2 q for m " n: Structure of projective modules K pĈ 0 s q, 0 ď s ď n´2 for m " n and all projective modules for m ă n can be obtained from this using the formula (4.6).

4.4.1.
There are two natural embeddings between quantum walled Brauer algebras (see [57]) qwB m´1,n Ñ qwB m ,n , qwB m ,n´1 Ñ qwB m ,n . The first embedding acts by identification of the corresponding generators E, g 1 , g 2 , . . .g m´2 , h 1 , h 2 , . . .h n´1 . The second embedding acts by identification of the generators E, g 1 , g 2 , . . .g m´1 , h 1 , h 2 , . . .h n´2 . These two maps induce two restriction functors res m ,n m´1,n and res m ,n m ,n´1 from the category of qwB m ,n -modules to the categories of qwB m´1,n and qwB m ,n´1modules respectively.
Let addpµq be the set of boxes for a partition µ, which can be added singly to µ such that the result µ`l is a partition. Let rempµq be a set of boxes which can be removed from µ such that µ{l is a partition.
In what follows the sign Ţ denotes the non-direct sum of modules. Following [57], where the classical case q " 1 is considered, we have for modules over qwB m ,n

Proposition.
For λ P Λ m ,n pf q with n ě 1 we have This statement is valid for the algebra qwB m ,n with either generic or special parameters. For qwB m ,n with generic parameters all Ţ become direct sums.
Proof. We discuss the case K pA a s q for 2 ď s ď n´1, a ě 1. Other cases are similar. The projective module K pA a s q has a filtration by two atypical Specht modules, so one can write it as a non direct sum K pA a s q " S pA a s q ě S pA a s´1 q.
Applying the Proposition 4.4.3 one obtains the sum of simple and atypical Specht modules: res m ,n m ,n´1 K pA a s q " res m ,n m ,n´1`S pA a s q ě S pA a s´1 q˘" " S pA a`1 s q ě S rpa , 1 s`1 q, ps qs ě S rpa , 1 s q, ps´1qs ě S pA a`1 s´1 q ě S rpa , 1 s q, ps´1qs ě S rpa , 1 s´1 q, ps´2qs.
In this sum only two modules are atypical, other modules are simple res m ,n m ,n´1 K pA a s q " S pA a`1 s q ě S pA a`1 s´1 q ě D rpa , 1 s`1 q, ps qs ě 2D rpa , 1 s q, ps´1qs ě D rpa , 1 s´1 q, ps´2qs.
These two atypical Specht modules are glued uniquely into a projective module, thus res m ,n m ,n´1 K pA a s q " K pA a`1 s q ' D rpa , 1 s`1 q, ps qs ' 2D rpa , 1 s q, ps´1qs'D rpa , 1 s´1 q, ps´2qs.
To formulate the next theorem we introduce notation a 1 " |m´n`1|.
For λ R At m ,n the generic rule is: Proof. If λ R At m ,n then D pλq " S pλq, and the proof follows from 4.4.3 similarly to the proof of Theorem 4.4.5.
Now we consider λ P At m ,n . We discuss only D pA a s q for a ě 1, 1 ď s ď n´1, other cases are similar. We prove that by induction on s . First, we prove the induction base for s " n´1, then we check the induction step from s to s´1. The X m ,n -module S pA a n q is simple: S pA a n q " D pA a n q, so we have from 4.4.3 (4.8) res m ,n m ,n´1 D pA a n q " res m ,n m ,n´1 S pA a n q " S " pa , 1 n q, pn´1q ‰ " D " pa , 1 n q, pn´1q ‰ . According to 4.4.3 we have for s ă n We write X m ,n -Specht modules as a non-direct sum S pA a s q " D pA a s q Ţ D pA a s`1 q for s ă n. The X m ,n´1 -module S pA a`1 n´1 q " D pA a`1 n´1 q, so from (4.9) for s " n´1 we get  We can also make generalization to the qwB m ,n modules.

Conjecture.
Consider the algebra qwB m ,n with special parameter θ "´p´δ γ q M´N . Let λ P Λ m ,n be an pM , N q-cross bipartition, then res m ,n m ,n´1 D pλq contains only subquotients D pλ 1 q for which λ 1 P Λ m ,n´1 is an pM , N q-cross bipartition.
In other words, the restriction functor for qwB m ,n with special parameters preserves the class of all pM , N q-cross bipartitions. In particular we have the next important consequence for M " 2, N " 1.

4.4.9.
Conjecture. For λ P Cr m ,n the restrictions res m ,n m ,n´1 D pλq for simple modules over qwB m ,n with θ " δ γ are explicitly given by the formulas from theorem 4.4.6 without any changes. and the exeptional case is Then the dimensions are:

5.2.
The bimodule is a direct sum of subbimodules For m " 5 and n " 3, the table of bipartitions λ 5,3 pt , r q reads

5.2.2.
In the next Theorem, we give explicit formulas for the decomposition of T m ,n for m ě n; the case m ă n can be easily recovered from m ą n using operationĜ interchanging m with n T n ,m "Ĝ T m ,n .
The operation is involutive,Ĝ 2 " 1, and additive,Ĝ pX ' Y q "Ĝ pX q 'Ĝ pY q. It acts on the indecomposable summands in the semisimple part T s m ,n by the formula where the actionĜ´D " λ L , λ R ‰¯i s defined in (4.5) and When applied to the atypical part T at m ,n , the operationĜ acts on each simple subquotient by the formula (5.5) and does not change the structure of the Loewy graph.

Theorem.
The X m ,n bU q s ℓp2|1q-bimodule decomposition of T m ,n , m ě n, has the form T m ,n " T s m ,n ' T at m ,n with the semisimple part m ą n: and the atypical part T at m ,n is given by figures 1-5 in Appendix A.

Verification.
To check the decomposition formula for the bimodule we make two powerful verifications using formulas for tensor product decompositions for U q s ℓp2|1q modules and restrictions for X m ,n modules. We check that T m ,n b 3 3 3 coincides with res m ,n`1 m ,n T m ,n`1 as U q s ℓp2|1q-module in the first verification and as X m ,n -module in the second one. In order to do this we introduce two Grothendieck (forgetful) functors and .
We define the Grothendieck functor on the category of U q s ℓp2|1q-modules which maps an indecomposable module into a direct sum of its simple subquotients. The functor on any U q s ℓp2|1q-module is known from 2.4. For example r , @p, t , r. We define the other Grothendieck functor on the category of X m ,n modules which maps an indecomposable module into a direct sum of its simple subquotients. The functor on any X m ,n -module is known from 4.3.5. For example We introduce the notation T m ,n " T m ,n . The following relation must hold: T m ,n b 3 3 3 " res m ,n`1 m ,n T m ,n`1 .
Because T m ,n has the form T m ,n " À D bR À D bZ, we can calculate T m ,n b3 3 3 using formulas from 3.1. Because T m ,n`1 contains as subquotients only modules D pλq for λ P Cr m ,n`1 , we can calculate res m ,n`1 m ,n T m ,n`1 using formulas from 4.4.6, and then apply the functor . We have checked the validity of relation 5.7 for all m, n whenever m`n ď 25.

5.4.2.
As X m ,n module. The action of on the atypical part T at m ,n has the form m ą n: m " n " 1: We introduce the notation T m ,n " T m ,n . The following relation must hold: pT m ,n b 3 3 3q " res m ,n`1 m ,n T m ,n`1 .
Because T m ,n has the form T m ,n " À K b Z À D b Z, we can calculate T m ,n b3 3 3 using formulas from 3.1. Because T m ,n`1 contains as subquotients only modules K pλq and D pλq for λ P Cr m ,n`1 , we can calculate res m ,n`1 m ,n T m ,n`1 using formulas from 4.4.6 and 4.4.5 and then apply the functor . We have checked the validity of relation 5.8 for all m, n whenever m`n ď 25.

CONCLUSION
In the present work we have studied the U q s ℓp2|1q mixed tensor product and found its decomposition as a bimodule over X m ,n b U q s ℓp2|1q. These results are the basis for a further study of the U q s ℓp2|1q-spin-chain and appropriate LCFT. The next step is studying the mixed tensor product with parameter q at the root of unity. We expect the appearance of the Lusztig limit of algebra U q s ℓp2|1q in that case. We anticipate that X m ,n will remain the centralizer of LU q s ℓp2|1q on the mixed tensor product and some triplet extension of X m ,n will be the centralizer of U q s ℓp2|1q.
Natural ways for further developments of the results presented in this paper: (1) Describe explicitly the algebra X m ,n and identify it with some quotient of qwB m ,n .
Similar problem is posed the algebras X M ,N m ,n of U q s ℓpM |N q-endomorphisms. (4) Classify Ext 1 spaces for modules over the algebra X m ,n and describe explicitly the action of X m ,n -generators on the basis of projective modules K pλq. The solution to this problem will allow one to describe explicitly the X m ,n -action in the bimodule T m ,n .

APPENDIX A. ATYPICAL PART OF THE BIMODULE
In this section we represent the structure of Loewy graph for the indecomposable bimodule T at m ,n , see 5.3. Detailed investigation of X m ,n action on these bimodules are beyound the scope of this paper. See paper [12], where the spin chain over U q s ℓp2q is investigated for comparison.
In each vertex of the graph there is some subquotient D pλq b s Z p t ,r . The meaning of the arrows is the same as in 4.3.5. On the figures the action of algebra U q s ℓp2|1q is denoted by solid lines, and the action of X m ,n is denoted by dash lines.
The subquotients connected by dash lines have the same U q s ℓp2|1q module as a tensor multiplier. The subquotients connected by solid lines have the same X m ,n module as a tensor multiplier. To simplify the figures we omit X m ,n multiplier where it does not cause inconsistency. We also do not write symbol D each time, and write only λ for simple module D pλq.
For example, the bimodule for T at 3,2 is (A.1) We use shorthand notation for T at 3,2 : We mark in red the subquotient where the figure has irregular form. The structure of T at m ,n for the case 1 ď n ď m 2 is shown in figure 1. The case m 2`1 ď n ď m´2 is shown in figure 2. The case n " m`1 2 , n ě 2 is shown in figure 3. The case n " m´1, n ě 1 is shown in figure 4. The case n " m, n ě 2 is shown in figure 5.