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Simple Weight Q2(ℂ)-Supermodules

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Simple Weight Q2(ℂ)-Supermodules U.U.D.M. Project Report 2009:12 Simple weight q2( )-supermodules Ekaterina Orekhova ℂ Examensarbete i matematik, 30 hp Handledare och examinator: Volodymyr Mazorchuk Juni 2009 Department of Mathematics Uppsala University Simple weight q2(C)-supermodules Ekaterina Orekhova June 6, 2009 1 Abstract The first part of this paper gives the definition and basic properties of the queer Lie superalgebra q2(C). This is followed by a complete classification of simple h-supermodules, which leads to a classification of all simple high- est and lowest weight q2(C)-supermodules. The paper also describes the structure of all Verma supermodules for q2(C) and gives a classification of finite-dimensional q2(C)-supermodules. 2 Contents 1 Introduction 4 1.1 General definitions . 4 1.2 The queer Lie superalgebra q2(C)................ 5 2 Classification of simple h-supermodules 6 3 Properties of weight q-supermodules 11 4 Universal enveloping algebra U(q) 13 5 Simple highest weight q-supermodules 15 5.1 Definition and properties of Verma supermodules . 15 5.2 Structure of typical Verma supermodules . 27 5.3 Structure of atypical Verma supermodules . 28 6 Finite-Dimensional simple weight q-supermodules 31 7 Simple lowest weight q-supermodules 31 3 1 Introduction Significant study of the characters and blocks of the category O of the queer Lie superalgebra qn(C) has been done by Brundan [Br], Frisk [Fr], Penkov [Pe], Penkov and Serganova [PS1], and Sergeev [Se1]. This paper focuses on the case n = 2 and gives explicit formulas, diagrams, and classification results for simple lowest and highest weight q2(C) supermodules. 1.1 General definitions Definition 1.0.1. A vector space V over a field k is said to be a super vector space if V is Z/2Z-graded, that is if V = V0 ⊕ V1 where we identify Z/2Z with {0, 1}.A super subspace of V is a super vector space W = W0 + W1 such that Wi is a subspace of Vi for i = 0, 1. A homogeneous vector v ∈ V has degree (parity) 0 or 1, denoted by |v|. Element v is said to be even if |v| = 0 and odd if |v| = 1. The vector space V is said to be even if V = V0 and odd if V = V1. Given two super vector spaces V and W , a linear map f : V → W is said to be homogeneous of degree d ∈ Z/2Z if f(Vi) ⊆ W|i+d| for i = 0, 1. The degree of a homogeneous map f is denoted by |f|. The map f is said to be even if f is homogeneous of degree 0 (that is, |f| = 0) and odd if f is homogeneous of degree 1 (|f| = 1). We define the super vector space Homk(V, W ) as the direct sum of the homogeneous vector spaces Homk(V, W )d = {f : V → W |f is homogeneous of degree d} where d ∈ Z/2Z. Definition 1.0.2. The super vector spaces and homogeneous mappings de- fined above form a category. If we restrict the homomorphisms to even mappings, we obtain an abelian category, denoted by sV ec(k). Definition 1.0.3. The parity change functor Π: sV ec(k) → sV ec(k) is defined as follows: for V ∈ sV ec(k) and f : V → W ∈ sV ec(k), we take (Π(V ))d = V|d+1|, where d ∈ Z/2Z, and Π(f) = f, as mappings. Note that the underlying field k is considered to be even and hence Π(k) is an odd super vector space, of dimension one. Definition 1.0.4. A superalgebra over a field k is a k-algebra A with a direct sum decomposition A = A0 ⊕A1 together with a bilinear multiplication that respects the Z/2Z grading, that is A|i|A|j| ⊆ A|i+j|. Definition 1.0.5. Let A be a superalgebra. A left A-supermodule is a super vector space V which is a left A-module in the usual sense such that AθVτ ⊆ V|θ+τ| for θ, τ ∈ Z/2Z. Right A-supermodules are defined analogously. Definition 1.0.6. A Lie superalgebra g over a field k is a super vector space over k together with the Lie bracket [·, ·]: g × g → g, which satisfies the 4 following: 1. Bilinearity: For all scalars a, b ∈ k and all elements x, y, z ∈ g, [ax + by, z] = a[x, z] + b[y, z] and [z, ax + by] = a[z, x] + b[z, y]. 2. Super skew symmetry: For all homogeneous elements x, y ∈ g, [x, y] = −(−1)|x||y|[y, x]. 3. The super Jacobi identity: For all homogeneous elements x, y, z ∈ g, (−1)|z||x|[x, [y, z]] + (−1)|x||y|[y, [z, x]] + (−1)|y||z|[z, [x, y]] = 0. Definition 1.0.7. Let g be a Lie superalgebra. A g-supermodule is a super vector space V together with a map g × V → V such that giVj ⊆ V|i+j| and [x, y]v = (xy − (−1)|x||y|yx)v for all homogeneous elements x, y ∈ g and v ∈ V . Definition 1.0.8. The Lie algebra gl2(C) is the complex vector space a b gl ( ) = : a, b, c, d ∈ 2 C c d C where the Lie bracket [·, ·] is the usual commutator. That is, given two matrices A, B,[A, B] = AB − BA. 1.2 The queer Lie superalgebra q2(C) Definition 1.0.9. The queer Lie superalgebra q2(C) consists of block ma- trices of the form AB BA where A, B ∈ gl2(C), and the supercommutator [·, ·] defined for homoge- neous elements X, Y ∈ q2(C) as follows: [X, Y ] := XY − (−1)|X||Y |Y X. The even subspace of q, denoted q0, consists of the block matrices with B = 0. The odd subspace of q, denoted q1, consists of the block matrices with A = 0. From now on, we will denote the queer Lie superalgebra simply by q. 0 Let eij, 1 ≤ i, j ≤ 2, be the basis elements in q0, defined as Eij 0 0 Eij 5 where Eij ∈ gl2(C) is the matrix with entries (δij). Similarly, we define the 1 basis elements eij, 1 ≤ i, j ≤ 2 in q1. Together, these eight elements form a basis of the whole superalgebra q. In this standard basis, the supercommutator [·, ·] has the following form: σ θ σ+θ σθ σ+θ [eij, ekl] = δjkeil − (−1) δilekj , (1) where σ, θ ∈ Z/2Z and 1 ≤ i, j, k, l ≤ 2. Remark 1.0.1. Note that the supercommutator [A, B] is the usual com- mutator AB − BA when both entries are even, or when one is even and the other is odd. However, in the case where both entries are odd the minus sign is negated, and hence the supercommutator becomes AB + BA. This is known as the anti-commutator. To simplify notation, we will denote the basis elements as follows: 0 0 0 0 e = e12, f = e21, h1 = e11, h2 = e22 1 1 1 1 e = e12, f = e21, h1 = e11, h2 = e22 By a direct calculation, we get the following Cayley table for the super- commutator [·, ·] in this basis: [·, ·] h1 e f h2 h1 e f h2 h1 0 e −f 0 0 e −f 0 e −e 0 h1 − h2 e −e 0 h1 − h2 e f f −(h1 − h2) 0 −f f −(h1 − h2) 0 −f h2 0 −e f 0 0 −e f 0 (2) h1 0 e −f 0 2h1 e f 0 e −e 0 h1 − h2 e e 0 h1 + h2 e f f −(h1 − h2) 0 −f f h1 + h2 0 f h2 0 −e f 0 0 e f 2h2 Remark 1.0.2. Let A, B ∈ {e, f, h1, h2, e, f, h1, h2}. If the parities of A and B are the same, then [A, B] is even. If the parities of A and B are different, then [A, B] is odd. 2 Classification of simple h-supermodules Definition 2.0.10. The Cartan Lie subsuperalgebra of q, denoted by h, is the linear span of h1, h2, h1, and h2. Definition 2.0.11. The Cartan subalgebra of gl2(C), denoted by h0, is the linear span of h1 and h2. 6 ∗ Remark 2.0.3. Let h0 denote the dual space of h0 and let 1, 2 be the dual ∗ basis of h1 and h2. Then any f ∈ h0 can be written as f = a1 + b2 for ∗ ∼ 2 2 some a, b ∈ C. Hence, h0 = C , so we can think of the elements in C as linear functionals on h0, via (λ1, λ2)(ah1 + bh2) = aλ1 + bλ2 2 for (λ1, λ2) ∈ C and a, b ∈ C. From the previous section, we know that the basis elements h1, h2, h1, h2 satisfy the following relations: [·, ·] h1 h2 h1 h2 h1 0 0 0 0 h2 0 0 0 0 (3) h1 0 0 2h1 0 h2 0 0 0 2h2 Remark 2.0.4. Note that h1, h2 commute with all other basis elements 2 and that hi = hi for i = 1, 2. Definition 2.0.12. An h-supermodule is a super vector space V together with two even linear operators H1,H2 and two odd linear operators H1, H2 that satisfy the relations in Table (3). Remark 2.0.5. This explicit definition is equivalent to the general defini- tion of a supermodule given by 1.0.7. Example 2.0.1. Let V (0, 0) = C and let H1, H2, H1, H2 act as zero on V (0, 0).
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