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Generalized Syzygies for Commutative Koszul Algebras Joint Work in Progress with Vassily Gorbounov (Aberdeen) Zain Shaikh (Cologne) and Andrew Tonks (Londonmet)

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Generalized Syzygies for Commutative Koszul Algebras Joint Work in Progress with Vassily Gorbounov (Aberdeen) Zain Shaikh (Cologne) and Andrew Tonks (Londonmet) Koszul Duality and Cohomology: Generalized Syzygies for Commutative Koszul Algebras Joint Work in Progress with Vassily Gorbounov (Aberdeen) Zain Shaikh (Cologne) and Andrew Tonks (Londonmet) Imma G´alvezCarrillo UPC-EET, Terrassa UAB Algebra and Combinatorics Seminar 2011 CRM May 26, 2011 Imma G´alvezCarrillo UPC-EET, Terrassa Koszul Duality and Cohomology: Generalized Syzygies for Commutative Koszul Algebras Towards generalized syzygies Definition: Koszul homology Let fa1;:::; ang be a sequence of elements in a commutative C-algebra A. Let W be an n-dimensional complex vector space with basis fθ1; : : : ; θng. The Koszul homology of A with respect to the sequence fa1;:::; ang is the homology of the complex ^ A ⊗ W ; where A has homological degree zero, each θi has homological degree one, and the Koszul differential is given by the formula n X @ dK = ai : @θi i=1 Imma G´alvezCarrillo UPC-EET, Terrassa Koszul Duality and Cohomology: Generalized Syzygies for Commutative Koszul Algebras Koszul Algebras (I) Definition: Quadratic algebra Let A be a positively graded connected algebra, locally finite-dimensional. A is called quadratic if it is determined by a vector space of generators V = A1 and subspace of quadratic relations I ⊂ A1 ⊗ A1 Definition: Koszul quadratic dual The Koszul dual algebra A! associated with a quadratic algebra A is A! = T (V ∗)=I ?; where I ? ⊂ V ∗ ⊗ V ∗ is the annihilator of I . Clearly A!! = A. Imma G´alvezCarrillo UPC-EET, Terrassa Koszul Duality and Cohomology: Generalized Syzygies for Commutative Koszul Algebras Koszul algebras (II) Definition: Koszul algebra A quadratic as above is a Koszul algebra iff ! ∼ ∗ A = ExtA(|; |) Imma G´alvezCarrillo UPC-EET, Terrassa Koszul Duality and Cohomology: Generalized Syzygies for Commutative Koszul Algebras Lie superalgebras (I) Definition: Lie superalgebra A Lie superalgebra over C is a Z=2-graded vector space (over C) L = L(0) ⊕ L(1) with a map [·; ·]: L ⊗ L ! L of Z=2-graded spaces, satisfying: 1 (anti-symmetry) [x; y] = −(−1)jxjjyj[y; x] for all homogeneous x; y 2 L, 2 (Jacobi identity) (−1)jxjjzj[x; [y; z]]+(−1)jyjjxj[y; [z; x]]+(−1)jzjjyj[z; [x; y]] = 0 for all homogeneous x; y; z 2 L. Imma G´alvezCarrillo UPC-EET, Terrassa Koszul Duality and Cohomology: Generalized Syzygies for Commutative Koszul Algebras Lie superalgebras (II) Definition: Lie superalgebra (continued) Here jxj is the parity of x: jxj = i when x 2 L(i) for i = 0; 1. An element x in L(0) or L(1) is termed even or odd respectively. We recover the familiar definition of a Lie algebra (over C) in the case L = L(0). Definition: graded Lie superalgebra A graded Lie superalgebra is a Lie superalgebra L together with a grading compatible with the bracket and supergrading. That is, L L L = m≥1 Lm such that [Li ; Lj ] ⊂ Li+j and L(i) = m≥1 L2m−i for i = 0; 1. Imma G´alvezCarrillo UPC-EET, Terrassa Koszul Duality and Cohomology: Generalized Syzygies for Commutative Koszul Algebras Koszul dual as universal envelope Koszul dual as universal envelope Assume that A = T (V )=I is commutative, hence V2 V ⊂ I . Therefore I ? is contained in S2(V ∗), and so is generated by certain ∗ ∗ ∗ ∗ ∗ ∗ linear combinations of anti-commutators [ai ; aj ] = ai aj + aj ai . As a consequence, the Koszul quadratic dual of A can be described as the universal envelope of a graded Lie superalgebra, ! M ∗ A = U(L); L = Lm = L(V )=J; (1) m≥1 where L is the free Lie superalgebra functor, the space of (odd) generators V ∗ is concentrated in degree 1, and J is the Lie ideal with the same generators as I ? but viewed as linear combinations of supercommutators. Imma G´alvezCarrillo UPC-EET, Terrassa Koszul Duality and Cohomology: Generalized Syzygies for Commutative Koszul Algebras Some Lie ideals The following Lie ideals are main characters of our work. Definition For k ≥ 2, we define the graded Lie superalgebras M L≥k = Lm: m≥k Imma G´alvezCarrillo UPC-EET, Terrassa Koszul Duality and Cohomology: Generalized Syzygies for Commutative Koszul Algebras The algebra of syzygies (I) Definition: the algebra of syzygies Let A be a commutative C-algebra which is a module over S(V ), and suppose we have a minimal free resolution of A, ···! F2 ! F1 ! F0 ! A ! 0: This is an exact sequence of graded free S(V )-modules, M Fp = Rpq ⊗ S(V ); q≥mp where Rpq is the finite dimensional vector spaces of p-th order syzygies of degree q for A, and mp is the minimum degree among the p-th order syzygies. Imma G´alvezCarrillo UPC-EET, Terrassa Koszul Duality and Cohomology: Generalized Syzygies for Commutative Koszul Algebras The algebra of syzygies (II) Syzygies and the Koszul complex Since the chosen resolution is minimal, the differential vanishes on tensoring this complex with the trivial S(V )-module C. Hence S(V ) M Torp (A; C) = Rp := Rpq: q≥mp Imma G´alvezCarrillo UPC-EET, Terrassa Koszul Duality and Cohomology: Generalized Syzygies for Commutative Koszul Algebras The algebra of syzygies (III) The functor TorS(V ) may also be calculated by resolving the other argument C. The Koszul complex K(S(V )) of the symmetric algebra V S(V ) = C[a1;:::; an], is given by (S(V ) ⊗ W ; dK ), that is, ^ (C[a1;:::; an] ⊗ (θ1; : : : ; θn); dK ): (2) with the Koszul differential dK of the sequence fa1;:::; ang. Imma G´alvezCarrillo UPC-EET, Terrassa Koszul Duality and Cohomology: Generalized Syzygies for Commutative Koszul Algebras The algebra of syzygies (IV) Since this complex is a resolution of C we can calculate the syzygies Rp of the quadratic algebra A = T (V )=I by the homology of the complex ^ ^ A ⊗S(V ) K(S(V )) = A ⊗S(V ) S(V ) ⊗C W = A ⊗ W : (3) This homology inherits a multiplication from K(S(V )) and becomes an associative algebra. Imma G´alvezCarrillo UPC-EET, Terrassa Koszul Duality and Cohomology: Generalized Syzygies for Commutative Koszul Algebras Lie ideals and the algebra of syzygies ∗ H (L≥2; C) gives indeed the algebra of syzygies of A. The interpretation of the algebras L≥k for k > 2 was outlined by Berkovits in [4]. We concentrate on the case k = 3. Imma G´alvezCarrillo UPC-EET, Terrassa Koszul Duality and Cohomology: Generalized Syzygies for Commutative Koszul Algebras Introduction: Berkovits' example In his work on string quantization and string/gauge theory duality [4], Berkovits uses crucially Koszul duality theory for commutative algebras. There, the coordinate algebra of the orthogonal Grassmannian A = OGr(5; 10), related to the spinor representation of the group SO(10; C) is considered. This algebra A is quadratic, and hence we know that its Koszul dual A! is U(L) the universal enveloping algebra of a L certain (graded) Lie superalgebra L = i≥1 Li . [4] Berkovits, N.; Cohomology in the pure spinor formalism for the superstring, J. High Energy Phys. 9 (2000). Imma G´alvezCarrillo UPC-EET, Terrassa Koszul Duality and Cohomology: Generalized Syzygies for Commutative Koszul Algebras Berkovits' first observation, made in the context of string and gauge theories, was that the algebra of syzygies of A is ∗ isomorphic to the cohomology H (L≥2; C) of the Lie L superalgebra L≥2 = i≥2 Li . He proposed an extension of the Koszul complex of the coordinate algebra of OGr(5; 10), relevant for his quantization procedure, and considered the question of calculating its homology. Imma G´alvezCarrillo UPC-EET, Terrassa Koszul Duality and Cohomology: Generalized Syzygies for Commutative Koszul Algebras This extension is worthy of study for more general algebras. Berkovits' construction is an iteration of the Koszul homology of a sequence of elements of an algebra, first applied to the algebra A and then applied to its syzygies. As we recalled there is a notion of Koszul homology with respect to a sequence of elements fa1; ::; ang of a commutative algebra A. And we recalled as well that if A is finitely generated and presented as C[a1; :::; an]=I , the Koszul homology with respect to the sequence fai g gives the algebra of syzygies of A. Imma G´alvezCarrillo UPC-EET, Terrassa Koszul Duality and Cohomology: Generalized Syzygies for Commutative Koszul Algebras The generators Γj of the ideal I represent syzygies in the lowest degree. The Koszul homology of the algebra of syzygies with respect to the sequence Γj is the homology studied by Berkovits. This is what we call the algebra of generalized syzygies. It turns out that if A is Koszul, the generalized syzygies can also be described explicitly in terms of the Lie superalgebra L. This was shown by Movshev and Schwarz in [12] for the case of the coordinate algebra of the OGr(5; 10). There was proved that in that case the homology of the Berkovits complex is isomorphic to the cohomology ∗ L H (L≥3; C) of the Lie superalgebra L≥3 = i≥3 Li . [12] Movshev, M.; Schwarz, A.; On maximally supersymmetric Yang-Mills theories, Nuclear Physics B 681 (2004) Imma G´alvezCarrillo UPC-EET, Terrassa Koszul Duality and Cohomology: Generalized Syzygies for Commutative Koszul Algebras ∗ Our aim is to extend these statements about H (L≥3; C) from the coordinate algebra of the orthogonal Grassmannian to an arbitrary finitely generated commutative Koszul algebra. Further examples are supplied by the orbits of the highest weight vector in an irreducible representation of a semisimple complex Lie group, as studied by Gorodentsev, Khoroshkin and Rudakov [9]. We prove in our Main Theorem that the cohomology of the Lie algebra L≥3 is isomorphic to the Berkovits homology of A. [9] Gorodentsev, A.; Khoroshkin, A.; Rudakov A.; On syzygies of highest weight orbits, Amer.
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