Complexity and Module Varieties for Classical Lie Superalgebras

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Complexity and Module Varieties for Classical Lie Superalgebras

Brian D. Boe¹ Jonathan R. Kujawa² Daniel K. Nakano¹

1University of Georgia

2University of Oklahoma

Algebras, Representations & Applications Federal University of Amazonas

Manaus, Brazil July 6–10, 2009

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Background

The category of finite-dimensional modules for the classical Lie superalgebrag=gl(m|n)which are completely reducible overg¯0is a highest weight category (as observed by Brundan) but, unlike the blocks of (parabolic) categoryOfor a semisimple Lie algebra, there are infinitely many simple modules. Projective resolutions can have infinite length. Cohomology can be non-zero in infinitely many degrees and can grow in dimension.

This motivates studying these categories cohomologically, with ideas and tools from the modular representation theory of finite groups and restricted Lie algebras.

Boe, Kujawa, Nakano (Georgia, Oklahoma) Module Varieties for Superalgebras Manaus, Brazil, July 7, 2009 2 / 19

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At the third conference in this series, at Maresias in 2007, Dan Nakano talked about our work using relative cohomology and supports to measure certain combinatorial invariants, such as thedefectof a classical Lie superalgebragand theatypicalityof its finite-dimensional irreducible representations.

Missing was a connection of the relative support theory to the notion of complexityof a module. Also missing was a connection to the

“associated varieties” of Duflo-Serganova. Those are the topics of my talk today.

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Background

Notation

Letg=g_¯₀⊕g_¯₁be aclassical Lie superalgebraoverC: this means that gis aZ2-graded vector space with a bracket[, ]which respects the grading and satisfies graded versions of the usual Lie algebra

properties; andg_¯₀=Lie(G₀)for a connected reductive algebraic group G₀. (We do not assumegis simple!)

LetF be the full subcategory ofg-modules which are finite-dimensional and completely reducible overg¯0. This is an interesting category because in general it is not semisimple, and indeed blocks often contain infinitely many simple modules; it has enough projectives; and ifgis Type I thenF is a highest weight category.

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Self-injectivity

A key property ofF is that it isself-injective, in the following sense:

Proposition

A module M inF is projective if and only if it is injective.

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Projective resolutions and complexity

Complexity

Definition

LetV ={V_t |t∈N}={V_•}be a sequence of finite-dimensional C-vector spaces. Therate of growthofV,r(V), is the smallest positive integerc such that there exists a constantC >0 with

dim_CVt ≤C·t^c−1for allt. If no such integer exists thenV is said to have infinite rate of growth.

Following Alperin (1977), we make the Definition

LetM∈ F andP• Mbe a minimal projective resolution forM. The complexityofM,cF(M) =r({P_n |n=0,1,2, . . .}).

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Projective Resolutions

Following Kumar, we have:

Proposition

U(g)⊗_U(g_¯

0)Λ^•_sup(g/g¯0)C is a projective resolution of the trivial module inF.

Tensoring by any moduleMinF produces a projective resolution ofM.

We easily deduce:

Theorem

For any module M inF, the complexity c_F(M)≤dim_Cg_¯₁.

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Examples

1. Consider the Lie superalgebra of typeq(1) = _{a b}

b a

:a,b∈C . The projective coverP(C)ofChas two composition factors, both∼=C. Therefore the minimal projective resolution ofCis given by

· · · →P(C)→P(C)→P(C)→C→0.

ThuscF(C) =1=Kr dim H^•(g,g_¯₀;C).

2. Forgl(1|1)the minimal projective resolution ofCcan be written

· · · →P(2| −2)⊕P(0|0)⊕P(−2|2)→P(1| −1)⊕P(−1|1)→P(0|0)→C→0,

where dimP(λ| −λ) =4. Therefore, dim_CP_n=4(n+1)and c_F(C) =2>1=Kr dim H^•(g,g_¯₀;C).

Evidently the relative cohomology ring may not be large enough to measure the complexity of modules inF.

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Examples

1. Consider the Lie superalgebra of typeq(1) = _{a b}

b a

:a,b∈C . The projective coverP(C)ofChas two composition factors, both∼=C. Therefore the minimal projective resolution ofCis given by

· · · →P(C)→P(C)→P(C)→C→0.

ThuscF(C) =1=Kr dim H^•(g,g_¯₀;C).

2. Forgl(1|1)the minimal projective resolution ofCcan be written

· · · →P(2| −2)⊕P(0|0)⊕P(−2|2)→P(1| −1)⊕P(−1|1)→P(0|0)→C→0,

where dimP(λ| −λ) =4. Therefore, dim_CP_n =4(n+1)and cF(C) =2>1=Kr dim H^•(g,g_¯₀;C).

Evidently the relative cohomology ring may not be large enough to measure the complexity of modules inF.

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Projective and Periodic Modules

Proposition

Let M ∈ F. Then M is projective if and only if c_F(M) =0.

Necessity is obvious. Conversely, ifM has a finite projective resolution 0→P_n→P_n−1→ · · · →P₀→M→0 then sinceP_nis also injective, the resolution splits. It follows thatMitself is projective.

A non-projective module isperiodicif it admits a periodic projective resolution. In the context of finite group representations,Mis periodic

⇐⇒ c_kG(M) =1. In our setting, necessity holds but not sufficiency. For example, ifg=gl(1|1), the Kac moduleK(0|0)has minimal projective resolution

· · · →P(2| −2)→P(1| −1)→P(0|0)→K(0|0)→0. Since dim_CP(λ| −λ) =4,cF(K(0|0)) =1 even though the resolution is not periodic.

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Projective and Periodic Modules

Proposition

Let M ∈ F. Then M is projective if and only if c_F(M) =0.

Necessity is obvious. Conversely, ifM has a finite projective resolution 0→P_n→P_n−1→ · · · →P₀→M→0 then sinceP_nis also injective, the resolution splits. It follows thatMitself is projective.

A non-projective module isperiodicif it admits a periodic projective resolution. In the context of finite group representations,Mis periodic

⇐⇒ c_kG(M) =1. In our setting, necessity holds but not sufficiency.

For example, ifg=gl(1|1), the Kac moduleK(0|0)has minimal projective resolution

· · · →P(2| −2)→P(1| −1)→P(0|0)→K(0|0)→0.

Since dim_CP(λ| −λ) =4,cF(K(0|0)) =1 even though the resolution is not periodic.

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Type I Superalgebras

Kac Modules

From now on, assume that our classical superalgebragisType I, meaning it has a consistentZ-grading of the formg=g₋₁⊕g₀⊕g₁. This includesgl(m|n)and the simple Lie superalgebras of types A(m,n), C(n), andP(n).

Putp⁺=g₀⊕g₁andp⁻=g₀⊕g₋₁. IfL₀(λ)is a simple finite-dimensionalg₀-module, let

K(λ) =U(g)⊗_U(p+)L₀(λ) and K⁻(λ) =Hom_U(p⁻₎(U(g),L₀(λ)) be theKac moduleand thedual Kac module, respectively.

A module inF has a (dual)Kac filtrationif it has a filtration with all nonzero subquotients isomorphic to (dual) Kac modules. Atilting moduleis one which admits both a Kac and a dual Kac filtration.

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Kac Modules

nonzero subquotients isomorphic to (dual) Kac modules. Atilting moduleis one which admits both a Kac and a dual Kac filtration.

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Kac Modules

A module inF has a (dual)Kac filtrationif it has a filtration with all nonzero subquotients isomorphic to (dual) Kac modules. Atilting moduleis one which admits both a Kac and a dual Kac filtration.

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Support Varieties

GivenM ∈ F, define theg_±1-support variety ofMas

V_g_±1(M) ={x ∈g_±1|Mis not projective as aU(hxi)-module} ∪ {0}.

These can also be defined cohomologically, as the maximal ideal spectrum of the quotient of the cohomology ring ofg_±1modulo the annihilator of Ext^•(M,M). The cohomological and rank variety definitions agree becauseg_±1are abelian Lie superalgebras.

The operationsV_g_±1 satisfy the usual properties of a support variety theory, such as the tensor product rule and detectingg_±1-projectivity.

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Theorem

Letgbe a Type I classical Lie superalgebra, and M ∈ F. M has a Kac filtration ⇐⇒ V_g₋₁(M) =0

M has a dual Kac filtration ⇐⇒ V_g₁(M) =0

Corollary

M is a tilting module if and only ifV_g₁(M) =V_g₋₁(M) =0.

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Theorem

Letgbe a Type I classical Lie superalgebra, and M ∈ F. M has a Kac filtration ⇐⇒ V_g₋₁(M) =0

M has a dual Kac filtration ⇐⇒ V_g₁(M) =0

Corollary

M is a tilting module if and only ifV_g₁(M) =V_g₋₁(M) =0.

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Projectivity Test

Theorem

Letgbe a Type I classical Lie superalgebra, and M ∈ F. Then M is projective if and only ifV_g₁(M) =V_g₋₁(M) =0.

The proof uses a tensor-product “trick” due to Cline-Parshall-Scott.

Corollary

M is a tilting module if and only if M is projective.

This result is implicit in the work of Jon Brundan (2004).

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Projectivity Test

Theorem

Letgbe a Type I classical Lie superalgebra, and M ∈ F. Then M is projective if and only ifV_g₁(M) =V_g₋₁(M) =0.

The proof uses a tensor-product “trick” due to Cline-Parshall-Scott.

Corollary

M is a tilting module if and only if M is projective.

This result is implicit in the work of Jon Brundan (2004).

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Connection with Duflo-Serganova Associated Varieties

Consider the following subvariety ofg¯1:

X ={x ∈g₁_¯|[x,x] =0}.

ForM∈ F, Duflo and Serganova (2005) define theassociated variety X_M ={x ∈ X |M is not projective as aU(hxi)-module} ∪ {0}.

In Type I,g_±1⊂ X sinceg_±1is abelian. It follows that V_g_±1(M) =X_M∩g_±1.

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Using our previous theorem, we can recover the following theorem of Duflo-Serganova (in Type I):

Theorem (Duflo-Serganova)

Letgbe a Type I classical Lie superalgebra, and M ∈ F. Then M is projective if and only ifX_M =0.

Our proof uses the fact that a projective moduleM isZ-graded. If there were a nonzero element inX_M, a limiting process produces a nonzero element of eitherX_M ∩g₁=V_g₁(M)orX_M∩g₋₁=V_g₋₁(M),

contradicting the projectivity ofM.

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Refinement of the Projectivity Test

Recall the connected reductive algebraic groupG₀with Lie(G₀) =g¯0. Let{x_i |i ∈I}(resp.{y_j |j ∈J}) be a set of orbit representatives for the minimal orbits¹of the action ofG₀ong₁(resp. ong₋₁).

Theorem

Letgbe a Type I classical Lie superalgebra and M ∈ F. Then M is projective if and only if M is projective on restriction to U(hx_ii)for all i ∈I and to U(hy_ji)for all j ∈J.

A similar test detects the existence of a (dual) Kac filtration onM.

The point is that, sinceMis ag₀_¯-module,V_g_±1(M)are closed G₀-stable varieties.

1By minimal orbit, we mean minimal non-zero orbit with respect to the partial order on orbits given by containment in closures.

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Refinement of the Projectivity Test

Recall the connected reductive algebraic groupG₀with Lie(G₀) =g¯0. Let{x_i |i ∈I}(resp.{y_j |j ∈J}) be a set of orbit representatives for the minimal orbits¹of the action ofG₀ong₁(resp. ong₋₁).

Theorem

Letgbe a Type I classical Lie superalgebra and M ∈ F. Then M is projective if and only if M is projective on restriction to U(hx_ii)for all i ∈I and to U(hy_ji)for all j ∈J.

A similar test detects the existence of a (dual) Kac filtration onM.

The point is that, sinceMis ag₀_¯-module,V_g_±1(M)are closed G₀-stable varieties.

1By minimal orbit, we mean minimal non-zero orbit with respect to the partial order on orbits given by containment in closures.

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Example: gl(m|n)

Considerg=gl(m|n). The action ofG₀∼=GL(m)×GL(n)ong₁is given by(A,B)·x =AxB⁻¹forA∈GL(m), B∈GL(n), x ∈g₁. The orbits are

(g1)r ={x ∈g₁|rank(x) =r}

for 0≤r ≤min(m,n). The closure of(g1)r is thedeterminantal variety (g1)r ={x ∈g₁|rank(x)≤r};

thus(g₁)_r ⊂(g₁)_s if and only ifr ≤s. The situation forg₋₁is analogous. Thus to test for Kac filtrations, dual Kac filtrations, and projectivity, it suffices to consider a single rank one element fromg₁ and/org₋₁.

The same holds true forsl(m|n), psl(n|n), and the simple Lie superalgebras of typeC(n)andP(n).

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Example: gl(m|n)

Considerg=gl(m|n). The action ofG₀∼=GL(m)×GL(n)ong₁is given by(A,B)·x =AxB⁻¹forA∈GL(m), B∈GL(n), x ∈g₁. The orbits are

(g1)r ={x ∈g₁|rank(x) =r}

for 0≤r ≤min(m,n). The closure of(g1)r is thedeterminantal variety (g1)r ={x ∈g₁|rank(x)≤r};

thus(g₁)_r ⊂(g₁)_s if and only ifr ≤s. The situation forg₋₁is analogous. Thus to test for Kac filtrations, dual Kac filtrations, and projectivity, it suffices to consider a single rank one element fromg₁ and/org₋₁.

The same holds true forsl(m|n), psl(n|n), and the simple Lie superalgebras of typeC(n)andP(n).

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In fact, forg=gl(m|n)one can say even more. Because the orbit closures form a simple chain, it follows that for anyM∈ F,

V_g₁(M) = (g₁)r for some 0≤r ≤min(m,n), and similarly forV_g₋₁(M). In particular,V_g_±1(M)is an irreducible variety.

WhenM=L(λ)is an irreducible module, it follows from the work of Duflo-Serganova that

V_g_±1(L(λ)) = (g±1)r wherer =atyp(λ).

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Questions

Further Questions

1. By the self-injectivity result, given a simple moduleS ∈ F, its projective cover is the injective envelope of some simpleT ∈ F. What is the relationship betweenS andT? (For finite-dimensional

cocommutative Hopf algebras, the analogous answer is known.)

dimension equals the complexitycF(M)? Thegl(1|1)example shows that we need something bigger than the relative support variety V_(g,g_¯

0)(M)defined in our earlier paper.

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Questions

Further Questions

1. By the self-injectivity result, given a simple moduleS ∈ F, its projective cover is the injective envelope of some simpleT ∈ F. What is the relationship betweenS andT? (For finite-dimensional

cocommutative Hopf algebras, the analogous answer is known.) 2. Can one construct a “support variety” for a moduleM ∈ F whose dimension equals the complexitycF(M)? Thegl(1|1)example shows that we need something bigger than the relative support variety V_(g,g_¯

0)(M)defined in our earlier paper.