Computation of cohomology of Lie conformal and Poisson vertex algebras
- PDF / 660,703 Bytes
- 51 Pages / 439.37 x 666.142 pts Page_size
- 86 Downloads / 204 Views
Selecta Mathematica New Series
Computation of cohomology of Lie conformal and Poisson vertex algebras Bojko Bakalov1 · Alberto De Sole2
· Victor G. Kac3
© Springer Nature Switzerland AG 2020
Abstract We develop methods for computation of Poisson vertex algebra cohomology. This cohomology is computed for the free bosonic and fermionic Poisson vertex (super)algebras, as well as for the universal affine and Virasoro Poisson vertex algebras. We establish finite dimensionality of this cohomology for conformal Poisson vertex (super)algebras that are finitely and freely generated by elements of positive conformal weight. Keywords Lie conformal (super)algebras · Poisson vertex (super)algebras · Affine Lie algebras · Virasoro algebra · Basic cohomology · LCA cohomology · Variational PVA cohomology · Energy operator Mathematics Subject Classification Primary 17B69; Secondary 17B63 · 17B56
1 Introduction In the papers [2–4], we laid down, with our collaborators, the foundations of the cohomology theory of vertex algebras. Recall that, to any linear symmetric (super)operad P over a field F, one canonically associates a Z-graded Lie superalgebra
B
Alberto De Sole [email protected] http://www1.mat.uniroma1.it/∼desole Bojko Bakalov [email protected] Victor G. Kac [email protected]
1
Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA
2
Dipartimento di Matematica, Sapienza Università di Roma, P.le Aldo Moro 2, 00185 Rome, Italy
3
Department of Mathematics, MIT, 77 Massachusetts Ave., Cambridge, MA 02139, USA 0123456789().: V,-vol
50
Page 2 of 51
B. Bakalov et al.
WP =
∞
WPk ,
where WPk = P(k + 1) Sk+1 .
(1.1)
k=−1
The Lie bracket of WP is defined via the ◦i -products of the operad P, see [20] or [2] for details. An odd element X ∈ WP1 satisfying [X , X ] = 0 defines a cohomology complex (WP , ad X ), which is a differential graded Lie superalgebra. The most well-known example of this construction is the Lie (super)algebra cohomology. In this case one takes the operad Hom(V ), for which Hom(V )(n) = Hom(V ⊗n , V ), where V is a fixed vector superspace, with the action of Sn permuting the factors of V ⊗n , and the well-known ◦i -products, see e.g. [2]. Then WHom(V ) is the Lie superalgebra of polynomial vector fields on V . Furthermore, odd elements 1 X ∈ WH om(V ) , where stands for reversing the parity, such that [X , X ] = 0, correspond bijectively to Lie superalgebra structures on V , by letting [a, b] = (−1) p(a) X (a ⊗ b),
a, b ∈ V .
(1.2)
The complex (WHom(V ) , ad X ) is then the Chevalley–Eilenberg cohomology complex of the Lie superalgebra (1.2) with coefficients in the adjoint module. Moreover, given a V -module M, we extend the Lie superalgebra structure on V to V ⊕ M by abelian ideal. Then the natural reduction of the complex making M to be an WHom((V ⊕M)) , ad X produces the Chevalley–Eilenberg cohomology complex of V with coefficients in M, see e.g. [14]. Note that, although the cohomology of V with coefficients in its adjoint module inherit
Data Loading...
