Extension Quiver for Lie Superalgebra q(3)

We describe all blocks of the category of finite-dimensional q(3)-supermodules by providing their extension quivers. We also obtain two general results about the representation of q(n): we show that the Ext quiver of the standard block of q(n) is obtained from the principal block of q(n-1) by identifying certain vertices of the quiver and prove a"virtual"BGG-reciprocity for q(n). The latter result is used to compute the radical filtrations of q(3) projective covers.


Introduction
The "queer" Lie superalgebra q(n) is an interesting super analogue of the Lie algebra gl(n). Other related queer-type Lie superalgebras include the subsuperalgebra sq(n) obtained by taking odd trace 0, and for n ≥ 3, the simple Lie superalgebra psq(n) obtained by taking the quotient of the commutator [q(n), q(n)] by the center. These queer superalgebras have a rich representation theory, partly due to the Cartan subsuperalgebra h not being abelian and hence having nontrivial representations, called Clifford modules.
Finite-dimensional representation theory of q(n) was initiated in [Kac] and developed in [P]. Algorithms for computing characters of irreducible finite-dimensional representations were obtained in [PS1], [PS2] using methods of supergeometry and in [B1], [B2] using a categorification approach. Finite-dimensional representations of half-integer weights were studied in detail in [CK,CKW,B3]. In [Maz], the blocks in the category of finite-dimensional q(2)-modules semisimple over the even part were classified and described using quivers and relations. A general classification of blocks was obtained in [Ser-ICM] using translation functors and supergeometry.
In this paper, we describe the blocks in the category of finite-dimensional q(3) and sq(3) modules semisimple over the even part in terms of quiver and relations. We found that to describe blocks of q(n) in general, it remains to consider the principal block. For n = 3, this is the first example of a wild block in q. Our main tools are relative Lie superalgebra cohomology and geometric induction.
In section 1, we describe some background information for q(n) and quivers, and we formulate our main theorems, Theorem 1.5 and Theorem 1.6. In section 2, we introduce geometric induction and prove a "virtual" BGG reciprocity law, Theorem 2.2, that generalizes [GS2] to the queer Lie superalgebras. This result allows us to describe radical filtrations of all finite-dimensional indecomposable projective modules for sq(3) and q(3). Diagrams of these are provided in section 6, the appendix. In section 3, we prove a result on self extensions of simples for g = q(n), Theorem 3.1, and for g = sq(n), Theorem 3.2. In section 4, we show the standard block for q(n) is closely related to the principal block of q(n − 1), Proposition 4.1.1, and in particular deduce the quiver for sq(3), q(3) standard block. Finally in section 5, we compute the quiver for principal block of sq(3), q(3).
0.1. Acknowledgements. The authors would like to thank Dimitar Grantcharov for numerous helpful discussions. The second author was supported by NSF grant 1701532.

Preliminaries and Main Theorem
1.1. General Definitions. Throughout we work with C as the ground field. We set Z 2 = Z/2Z. Recall that a vector superspace V = V0 ⊕ V1 is a Z 2 -graded vector space. Elements of V 0 and V 1 are called even and odd, respectively. If V, V ′ are superspaces, then the space Hom C (V, V ′ ) is naturally Z 2 graded with grading f ∈ Hom C (V, V ′ ) s if f (V r ) ⊂ V ′ r+s for all r ∈ Z 2 . A superalgebra is a Z 2 -graded, unital, associative algebra A = A 0 ⊕ A 1 which satisfies A r A s ⊂ A r+s . A Lie superalgebra is a superspace g = g0 ⊕ g1 with bracket operation [, ] : g ⊗ g → g which preserves the graded version of the usual Lie bracket axioms. The universal enveloping algebra U (g) is Z 2 -graded and satisfies a PBW type theorem [Kac]. A g-module is a left Z 2 -graded U (g)-module. A morphism of g-modules M → M ′ is an element of Hom C (M, M ′ )0 satisfying f (xm) = xf (m) for all m ∈ M, x ∈ U (g). We denote by g-mod the category of g-modules. This is a symmetric monoidal category. The primary category of interest F consists of finite-dimensional g-modules which are semisimple over g0. We stress that we only allow for parity preserving morphisms in F. In this way, F is an abelian rigid symmetric monoidal category: for V, W ∈ F, define V ⊗ W and V * using the coproduct and antipode of U (g), respectively: For V ∈ g-mod, we denote by S(V ) the symmetric superalgebra. As a g0-module, S(V ) is isomorphic to S(V ) = S(V0) ⊗ Λ(V1), where Λ(V1) is the exterior algebra of V1 in the category of vector spaces. For V a g 1 -module and W a g 2 -module W , we define the outer tensor product V ⊠ W to be the g 1 ⊕ g 2 -module with the action for (q 1 , q 2 ) ∈ g 1 ⊕ g 2 given by (q 1 , q 2 )(v ⊠ w) := (−1) q 2 v (q 1 v ⊠ q 2 w).
We define the (super)dimension of V ∈ g-mod as follows. Let C[ε] be polynomial algebra with variable ε and denote two-dimensional C-algebra C[ε]/(ε 2 − 1) as C. Then dim(V ) := dim C (V 0 ) + dim C (V1)ε ∈ C. 1.2. The queer Lie superalgebra q(n). By definition, the queer Lie superalgebra q(n) is the Lie subsuperalgbra of gl(n|n) leaving invariant an odd automorphism of the standard representation p with the property p 2 = −1. In matrix form, Let g = q(n). The even (resp. odd) subspace of g consists of block matrices with B = 0 (resp. A = 0). For 1 ≤ i, j ≤ n, we define the standard basis elements as Furthermore, otr(XY ) defines a nondegenerate g-invariant odd bilinear form on g. In particular, we have an isomorphism q(n) * ∼ = Πq(n) of q(n)-modules.
All Borel Lie superalgebras b ⊂ g are conjugate to the "standard" Borel, i.e block matrices where A, B ∈ gl(n) are upper triangular. The nilpotent subsuperalgebra n consists of block matrices where A, B are strictly upper triangular.
In the standard basis, the supercommutator has the form [e σ ij , e τ kl ] = δ jk e σ+τ il − (−1) στ δ il e σ+τ kj , where σ, τ ∈ Z 2 . The Cartan superalgebra h has basis e σ ii for 1 ≤ i ≤ n, σ ∈ Z 2 . The elements There is a root decomposition of g with respect to the Cartan subalgebra h given by n} is the same as the set of roots of gl n (C). For a root α = ε i − ε j we have dim g α = 1 + ε because g α = span{e σ i,j : σ ∈ Z 2 }. The positive roots are The Weyl group for q(n) is W = S n , the symettric group on n letters.
If m is odd, then there exists a unique simple Cliff(m)-module, denoted by v(m), which is invariant under parity change (this follows from existence of an odd automorphism). If m is even, then there exists 2 nonisomorphic simple Cliff(m)-modules v(m) and Πv(m) which are swapped by the parity change functor. Using the surjective homomorphism where ⌊x⌋ denotes the integer part of x ∈ R. Furthermore, this construction provides a complete irredundant collection of all finite-dimensional simple h-supermodules.
Next define the Verma module where the action of n + on v(λ) is trivial.
. Below is the main theorem about irreducible g-modules, first proven by V. Kac.
(2) For each finite dimensional irreducible g-module V , there exists a unique weight λ ∈ Λ + such that V is a homomorphic image of M g (λ).
We will often omit the subscript g in the notation for Verma, simple, and projective modules.
1.4. The Category F. Let g = q(n). Denote by F n , or simply F the category consisting of finite-dimensional g-supermodules semisimple over g0 (so the center of g0 acts semisimply), with morphisms being parity preserving. The full subcategory of F consisting of modules with integral weights is equivalent to the category of finite-dimensional G-modules, where G is the algebraic supergroup with Lie(G) = g and G0 = GL(n).
Let Z(g) be the center of the universal enveloping algebra U (g). A central character is a homomorphism χ : Z(U (g)) → C. We say that a g-module M has central character χ if for any z ∈ Z(g), m ∈ M , there exists a positive integer n such that (z − χ(z)id) n .m = 0. It is well known from linear algebra that any finite-dimensional indecomposable g-module has a central character, hence F n = ⊕F n χ , where F n χ is the subcategory of modules admitting central character χ. In the most cases F n χ is indecomposable, i.e. a block in the category F n . The only exception is F n χ for even n and typical central character χ. In this case F n χ is semisimple and has two non-isomorphic simple objects L(λ) and ΠL(λ).
Finally, it is well known there are enough projective and injective objects in F [Ser1]. Let P g (λ) denote the projective cover of L g (λ).
1.5. Quivers. Let F be any abelian C-linear category with enough projectives, finite-dimensional morphism spaces, and finite-length composition series for all objects. For us, F will be as in the previous subsection. The following properties are as stated in [Ger,Sect. 1], which are just slight generalizations of results in [Ben,Sect. 4.1].
An Ext-quiver Q for F is a directed graph with vertex set consisting of isomorphism classes of finite-dimensional simple objects of F. In our case, Q = {L(λ), ΠL(λ)} for λ, Πλ ∈ Λ + . We use the notation Πλ ∈ Λ + to distinguish the simple object ΠL(λ) from L(λ) when they are nonisomorphic. The number of arrows between two objects λ, µ will be d λ,µ := dim Ext 1 F (L(λ), L(µ)). We define a C-linear category CQ with objects being vertices of Q and morphisms Hom CQ (λ, µ) being space of formal linear combinations of paths between the two objects λ, µ. Composition of morphisms is concatenation of paths.
The system of relations is determined up to a choice of R λ,µ which is not canonical in general. But, we may multiply the R λ,µ (φ i λ,µ ) by nonzero scalars to make the relations "look nice." There is then an additional proposition, [Ger,Proposition 1.2.2], which states G op is equivalent to F. This then implies the following important theorem of Ext-quivers we use.
Theorem 1.4. [Ger,Theorem 1.4.1] Let F be as above, Q its Ext-quiver, and R be a system of relations as defined in Proposition 1.5.1. Then there exists an equivalence of categories 1.6. Main Theorem. In the statement of the main theorems, we will provide the Ext-quivers of various blocks. The relations are given by labelling the dim Ext 1 g (L(λ), L(µ)) arrows between L(λ), L(µ) ∈ Q by α ∈ Hom Q (L(λ), L(µ)) which is then identified (by some choice of scalar) with α ∈ Hom g (P (λ), radP (µ)/rad 2 P (µ)) via 1.5.1.
Proof. Observe that all blocks of sq(3) have special biserial quivers and hence are tame [Erd]. The same holds for the two typical and standard blocks of q(3). We show the q(3) principal block is wild by "duplicating the quiver" [GS3,Chap. 9]. Namely, label the vertices of the quiver by . } corresponding to top and bottom row, respectively. Let Q 1 denote the arrows and R the relations. Define . . } and set of arrows as . Then k(Q)/R ′ , R ′ being relation defined by any product of 2 arrows is 0, is a quotient of k(Q)/R. Note that the indecomposable representations of (Q 0 , Q 1 , R ′ ) are in bijection with that of Q ′ . But Q ′ is not a union of affine and Dynkin diagrams of type A, D, E (each vertex i, i > 3 has 3 edges coming out), so it is wild and this implies Q is wild.

Geometric Preliminaries and BGG Reciprocity
2.1. Relative cohomology of Lie superalgebras. Let t ⊂ g be a Lie subsuperalgebra and M a g-module. For p ≥ 0, define C p (g, t; M ) = Hom t (∧ p (g/t), M ). where ∧ p (g) is the super wedge product. The differential maps d p : C p (g, t; M ) → C p+1 (g, t; M ) are defined in the same way as for Lie algebras, see for example [BKN,Section 2.2]. The relative cohomology are defined by H p (g, t; M ) = Ker d p / Im d p−1 . We will be interested in the case when t = g0. Then the relative cohomology describe the extension groups in the category F of finite-dimensional g-modules semisimple over g0. More precisely, we have the following relation: For conciseness, we often write Ext q or Ext sq to denote Ext q(n) or Ext sq(n) .
Proof. Note that g 1 ∼ = Πg 1 as a g0-module and therefore Λ The differential is obviously zero and the statement follows.
Remark 1. One can also use the Z 2 -graded version of relative cohomology like in [BKN]. It is more suitable for the superversion of the category F where odd morphisms are allowed.
2.2. Geometric induction. We next provide a few fact about geometric induction following the exposition in [GS], [PS1]. Let p be any parabolic subsuperalgebra of g containing b. Let G = Q(n), and P, B be the corresponding Lie supergroups of p, b. For a P -module V , we denote by the calligraphic letter V the vector bundle G × P V over the generalized grassmannian G/P . See [Man] for the construction. Note that the space of sections of V on any open set has a natural structure of g-module; in other words the sheaf of sections of V is a g-sheaf. Therefore the cohomology groups H i (G/P, V) are g-modules. Define the geometric induction functor Γ i from category of p-modules to category of g-modules as It is also possible to define Γ i (G/P, V ) without the need of proving the rather technical question of existence of G/P . Namely, consider the Zuckerman functor from the category of P -modules to G-modules defined by: , where Γ g0 (M ) denotes the set of g0-finite vectors of g-module M . One can show easily that H 0 (G/P, V ) has a unique G-module structure compatible with the g-action. It is also straightforward that H 0 (G/P, V ) is left exact and the right adjoint to the restriction functor G − mod → P − mod. We define H i (G/P, ·) to be its right derived functors. Using this definition we can define Γ i (G/P, V ) for any V whose weights are in Λ.
We state some well known results.
(1) For any short exact sequence of P -modules there is a long exact sequence of g-modules (2) For a P -module V and a g-module M , If G = Q(n) then all parabolic subgroups containing the standard Borel subgroup B are in bijection with those of GL(n). Hence they are enumerated by partitions. The Levi subgroup L of Thm 2] Let P be any parabolic supergroup containing B and suppose λ ∈ Λ + is p-typical, where p := Lie(P ). Then Now, for any parabolic supergroup P containing B, define the multiplicity → 0 is an extension. Then V contains a highest weight vector v λ of weight λ coming from the inverse image of that of L(λ). Since V is indecomposable, V is generated by v λ and since µ < λ, V = U (g).v λ is annihilated by n + . Thus V is a highest weight module of weight λ, so it is a finite-dimensional quotient of M (λ) and consequently by Proposition 2.2.1(3), it is a quotient of Γ 0 (G/B, L b (λ)). Each such isomorphism class of extension V thus gives rise to a distinct subquotient L(µ) Remark 2. : In [PS1], the authors work in g Π -mod consisting of Π-invariant g-modules (and even morphisms) and define m i P Π (λ, µ) accordingly. For g = q(n), the simple g Π -modules are L(λ) when |{i : λ i = 0}| is odd and L(λ) ⊕ ΠL(λ) when |{i : λ i = 0}| is even.
There is a canonical projection G/B → G/P with kernel P/B = Q(2)/B ∩ Q(2). By our assumption, the weights λ is B-typical in P . Thus the Leray spectral sequence collapses by the typical lemma.
2.3. Virtual BGG Reciprocity. We now formulate a "virtual" BGG reciprocity theorem for g = sq(n) or q(n) which will be used to compute composition factors of indecomposable projective covers, P g (λ) of L g (λ). This result is a generalization of Theorem 1 in [GS2] in the case when Cartan subalgebra is not purely even. In this section we consider the quotient K Π (F) of the Grothendieck ring where we put dim X := dim X0+dim X1. Then Ch defines an injective homomorphism K Π (F) → R.
For any λ ∈ Λ we define an Euler characteristic as where Γ i is the dual to geometric induction functor as defined in section 2.2. It is straightforward to check (see e.g [B1,Theorem 4.25]) that for λ ∈ Λ such that wt(λ) = γ, then [E(λ)] ∈ K Π (F γ ). We comment that this Euler characteristic is different to the one defined in [B2]. There, the author considered an induction from the maximal parabolic P λ to which v(λ) extends, i.e The following result is a straightforward generalization of [GS,Lemma 1,2]. (2) For all w ∈ W , (3) Let Λ + 0 denote the set regular dominant weights with respect to g0. The set {Ch(E(λ)), λ ∈ Λ + 0 } is linearly independent in the ring R.
We call a simple g-module L(λ) of Type M if ΠL(λ) is not isomorphic to L(λ) and of Type Q if ΠL(λ) ∼ = L(λ). Note that the type of L(λ) is the same as the type of v(λ). Furthermore, for g = q(n) the type depends on the number of non-zero entries in λ: the type is M, if this number is even, and Q if it is odd. For example, L(1, 0, 0) is of type Q and L(0) is of type M. We set Theorem 2.2. Let g = q(n) or sq(n). Let µ ∈ Λ + and b µ,λ be the coefficients occurring in the expansion Then there exists coefficients a λ,µ such that where γ µ = 1 if g = q(n) and µ i = 0, or g = sq(n) and Proof. We follow the proof of [GS2][Theorem 1]. First, we have the Bott reciprocity formula ). Let C i (n, −) stand for the i-th term of the cochain complex computing H • (n, −). Note that P (λ) and hence C i (n; V * ⊗ P (λ)) is projective and injective in the category of h-modules semisimple over h0. Hence H j (h, h0; C i (n, V * ⊗ P (λ))) = 0 for any i and j ≥ 1. Therefore the first term of the spectral sequence for the pair (b, h) implies that Furthermore, we have By application of (2.3.2) and (2.3.3) we obtain For any module M ∈ F projective over h we have the equality is the corresponding indecomposable injective h-module. In other words we get If λ is of type Q we obtain If λ is of type M we obtain Taking alternating sum over i we get On the other hand, we have By S n -invariance of Ch(P (λ)), we get This together with dimv(µ) = dimv(w.µ) implies Therefore we obtain the relation Since µ ∈ Λ + 0 at most one µ i = 0. Therefore, we get: In remaining cases dimv(µ) dim v(µ) = 2. Remark 3. Theorem 2.2 holds for any Lie superalgebra g such that h = h0 and g1 = g * 1 . In this case, we get γ µ = 1.
Let K Π P (F) be the subgroup of K Π (F) generated by the classes of all projective modules. It is an ideal in K Π (F) since tensor product of projective with any finite-dimensional module is projective. Let K Π E (F) be the subgroup of K Π (F) generated by the Euler characteristics. Then K Π P (F) ⊂ K Π E (F) ⊂ K Π (F) and the inclusions are in general strict. The b ν,µ express a basis of K Π E (F) in terms of the basis of K Π (F) and a λ,ν express the basis of K Π P (F) in terms of basis of K Π E (F). Thus for two g-dominant weights λ, µ, we have (2.3.6) [P (λ) : 2.4. General lemma. To study relations between block for sq(n) and q(n) we consider the induction and restriction functors We have Thus, we have proved that Ind(A) commutes with θ. Thus there is an injective homomorphism A ⊗ C[θ]/(θ 2 ) → A. The dimension argument implies that it is an isomorphism.

Self Extensions
The goal of this section is to prove the following theorem.
Proof. Suppose we had a sequence of q(n)-modules 0 → L(λ) → M → L(λ) → 0 such that taking n p invariants results in a split short exact sequence of l-modules From before, we know this sequence is the same as Existence of a splitting maps means there exists an l-module homomorphism δ : where in the last line we use that φ is a q(n)-module homomorphism and U (q(n))L(λ) λ = L(λ). Thus M = φ(L(λ)) + L ′ .
Then we have an exact sequence of q(n)-modules To see that it does not split take p such that λ p = 0. On the weight space (L(λ) ⊗ U ) λ the odd basis element H p acts non-trivially while its action on (L(λ) ⊕ ΠL(λ)) λ is obviously trivial. The proof of Theorem 3.1 is complete.
In both cases, the sequence does not split. ΠLsq(λ) . Finally, case (c) was done in Theorem 3.2.

Standard Block
In this section, we compute the Ext-quiver for the standard block of g = q(3) and g = sq(3).

Induction and Restriction Functors.
Our goal is to establish a connection between the standard block F n 1,0,...,0 and the principal block F n−1 0 . As a first step we use the geometric induction in the case when it is an exact functor.
Consider the parabolic subalgebra Its Levi subalgebra l is isomorphic to q(1) ⊕ q(n − 1). Let Let t be a positive integer. A dominant integral weight λ of q(n) is called t-admissible if λ = (t, λ 2 , . . . , λ n ) such that t + λ i = 0, in other words the first mark of λ is t and λ is p-typical.
Let F l (t) denote the category of finite-dimensional l-modules on which H 1 acts by t and all weights of q(n − 1) have integral marks strictly less than t. Let F n (t) denote the Serre subcategory of F n generated by L(λ) for all t-admissible λ. Define the functors Proposition 4.1.1. The functors Γ t and R t define an equivalence between F l (t) and F n (t).
Proof. By Proposition 2.2.2, Γ i (G/P, M ) = 0 for i > 0 and every M ∈ F l (t). Furthermore, Γ 0 (G/P, M ) is simple if M is simple. On the other hand, R t (N ) = H 0 (p, N ) for any N ∈ F n (t). That implies Γ t is left adjoint to R t , both functors are exact and establish bijection on the sets of isomorphism classes of simple objects in both categories. Hence these functors provide an equivalence between the two categories.

4.2.
Reduction to q(n − 1). Note that every module in F l (t) is of the form L q(1) ⊠ M for some M ∈ F n−1 .
Since all simple constituents of T L q (1, 0, −1) are isomorphic to L q (1, 0, 0) by the weight argument, the statement follows.
Then we immediately obtainã 1b1 = 0 andã 2ã1 =b 1b2 = 0 from the relations for sq(3). From multiplicities of simple modules in projectives and information about Ext 1 q , one can compute the layers of the radical filtration in projective modules which are listed in Section 6.2.
There is one additional loop arrow h : P q (1, 0, 0) → P q (1, 0, 0). Let us show that we can choose h in such a way that h 2 = 0. We claim that there exists a q(3)-module R with radical filtration L q (1, 0, 0) | L q (2, 1, −2) | L q (1, 0, 0). Indeed, it is proven independently in Section 5 that there exists a q-module R ′ with the radical filtration ΠC | L q (2, 0, −2) | C. Set R = T R ′ where T is the translation functor from the principal to the standard block. Lemma 4.3.1 ensures that the composition factors of R are as desired. Furthermore, R ′ is a quotient of P q (0) and hence R is a quotient of P q (1, 0, 0). That settles the radical filtration of R.
Remark 5. Using a minor improvement of the arguments given in [Ser-ICM], we find there is an equivalence of categories F λ with wt(λ) = δ 3 2 and F λ ′ with wt(λ ′ ) = δ 2t+1 2 for any t > 0 given by a composition of translation functors. Then using the typical lemma (which holds for half-integral typical weights), we find an equivalence of categories between F l (t) and F n (t) just as in Proposition 4.1.1. Thus, we have an analogue of Corollary 4.2.1 for the half-standard block.  Proof. Identify ΠL(1) with the simple Lie superalgebra psq(3), then Der psq(3) = ΠC, [Kac]. This implies (5.1.2) by use of duality and Lemma 5.1.1.
That proves (5.1.4) Note that Lemma 5.1.2 and Lemma 5.1.3 prove that the ext quiver for the principal block for sq(3) coincides with one in Theorem 1.5.

5.2.
Relations for the principal block for g = sq(3). We first compute the radical filtration of all indecomposable projectives. Using the self-duality (up to parity) of P (a) and fact that we know all possible extensions of simples, we automatically know the top 2 and bottom 2 layers. It turns out the other layers are fixed as well, as shown below. Diagrams are in the Appendix Section 6.2.
Finally, for t ≥ 3, we find cycle paths in P (a) are a t b t , b t+1 a t+1 . Both have image L(t), hence a t b t = λ t b t+1 a t+1 . Observe we can sufficiently scale all arrows and hence normalize all λ t , λ ′ t ∈ C * to equal 1. The remaining dim Hom g (P (a), P (b)) calculations shows there are no other relations. 5.3. The principal block of q(3). We start with the following general statement.
Lemma 5.3.1 implies that the Ext quiver for the principal block of q(3) is obtained from that of sq(3) by adding a single arrow between each L(a) and ΠL(a) (Theorem 3.1).
Lemma 5.3.2. Let g = q(3) and θ ∈ Hom q (P (λ), ΠP (λ)) be the unique self extension for each λ in the principal q-block. Let P sq (resp., P q ) denote the direct sum of all indecomposable projectives in the principal block for sq(3) (resp., q(3)); A = End sq (P sq ) denote the algebra defined by the quiver with relations in principal sq(3)-block. Then the algebra defined by quiver with relations in principal q(3)-block is A ′ = End q (P q ) ∼ = A ⊗ C[θ]/(θ 2 ).
We use that P q (0) = pr(Ind q(3) q(3)0 C) where pr denote the projection on the principal block. Let l = q(3)0 ⊕ CH whereH =H 1 +H 2 +H 3 . SinceH 2 acts by zero on the modules of our block we have an exact sequence of l-modules By setting θ = pr(Παβ)pr we obtain the desired claim.
To finish the proof we just use Lemma 2.4.1.
In all radical filtrations, an edge denotes an extension. Observe for P q(3) (a, 0, −a), the "left half" corresponds to ker θ and the "right half" corresponds to imθ. The following radical filtrations are deduced from the fact that θ : P (a) → ΠP (a) corresponds to id : P sq (a) → P sq (a) as seen from Lemma 5.3.2.