@article{bakalov_villarreal_2023, title={Logarithmic Vertex Algebras and Non-local Poisson Vertex Algebras}, ISSN={["1432-0916"]}, DOI={10.1007/s00220-023-04839}, journal={COMMUNICATIONS IN MATHEMATICAL PHYSICS}, author={Bakalov, Bojko and Villarreal, Juan J.}, year={2023}, month={Sep} }
@article{bakalov_elsinger_kac_todorov_2023, title={Orbifolds of lattice vertex algebras}, ISSN={["1861-3624"]}, DOI={10.1007/s11537-023-2249-7}, journal={JAPANESE JOURNAL OF MATHEMATICS}, author={Bakalov, Bojko and Elsinger, Jason and Kac, Victor G. and Todorov, Ivan}, year={2023}, month={Jun} }
@article{bakalov_villarreal_2022, title={Logarithmic Vertex Algebras}, ISSN={["1531-586X"]}, DOI={10.1007/s00031-022-09759-z}, abstractNote={We introduce and study the notion of a logarithmic vertex algebra, which is a vertex algebra with logarithmic singularities in the operator product expansion of quantum fields; thus providing a rigorous formulation of the algebraic properties of quantum fields in logarithmic conformal field theory. We develop a framework that allows many results about vertex algebras to be extended to logarithmic vertex algebras, including in particular the Borcherds identity and Kac Existence Theorem. Several examples are investigated in detail, and they exhibit some unexpected new features that are peculiar to the logarithmic case.}, journal={TRANSFORMATION GROUPS}, author={Bakalov, Bojko N. and Villarreal, Juan J.}, year={2022}, month={Sep} }
@article{bakalov_de sole_heluani_kac_vignoli_2021, title={Classical and variational Poisson cohomology}, ISSN={["1861-3624"]}, DOI={10.1007/s11537-021-2109-2}, abstractNote={We prove that, for a Poisson vertex algebra $${\cal V}$$ , the canonical injective homomorphism of the variational cohomology of $${\cal V}$$ to its classical cohomology is an isomorphism, provided that $${\cal V}$$ , viewed as a differential algebra, is an algebra of differential polynomials in finitely many differential variables. This theorem is one of the key ingredients in the computation of vertex algebra cohomology. For its proof, we introduce the sesquilinear Hochschild and Harrison cohomology complexes and prove a vanishing theorem for the symmetric sesquilinear Harrison cohomology of the algebra of differential polynomials in finitely many differential variables.}, journal={JAPANESE JOURNAL OF MATHEMATICS}, author={Bakalov, Bojko and De Sole, Alberto and Heluani, Reimundo and Kac, Victor G. and Vignoli, Veronica}, year={2021}, month={Aug} }
@article{bakalov_d'andrea_kac_2021, title={Irreducible modules over finite simple Lie pseudoalgebras III. Primitive pseudoalgebras of type H}, volume={392}, ISSN={["1090-2082"]}, DOI={10.1016/j.aim.2021.107963}, abstractNote={A Lie conformal algebra is an algebraic structure that encodes the singular part of the operator product expansion of chiral fields in conformal field theory. A Lie pseudoalgebra is a generalization of this structure, for which the algebra of polynomials k[∂] in the indeterminate ∂ is replaced by the universal enveloping algebra U(d) of a finite-dimensional Lie algebra d over the base field k. The finite (i.e., finitely generated over U(d)) simple Lie pseudoalgebras were classified in our 2001 paper [1]. The complete list consists of primitive Lie pseudoalgebras of type W,S,H, and K, and of current Lie pseudoalgebras over them or over simple finite-dimensional Lie algebras. The present paper is the third in our series on representation theory of simple Lie pseudoalgebras. In the first paper, we showed that any finite irreducible module over a primitive Lie pseudoalgebra of type W or S is either an irreducible tensor module or the image of the differential in a member of the pseudo de Rham complex. In the second paper, we established a similar result for primitive Lie pseudoalgebras of type K, with the pseudo de Rham complex replaced by a certain reduction, called the contact pseudo de Rham complex. This reduction in the context of contact geometry was discovered by M. Rumin [11]. In the present paper, we show that for primitive Lie pseudoalgebras of type H, a similar to type K result holds with the contact pseudo de Rham complex replaced by a suitable complex. However, the type H case in more involved, since the annihilation algebra is not the corresponding Lie–Cartan algebra, as in other cases, but an irreducible central extension. When the action of the center of the annihilation algebra is trivial, this complex is related to work by M. Eastwood [6] on conformally symplectic geometry, and we call it conformally symplectic pseudo de Rham complex.}, journal={ADVANCES IN MATHEMATICS}, author={Bakalov, Bojko and D'Andrea, Alessandro and Kac, Victor G.}, year={2021}, month={Dec} }
@article{bakalov_kirk_2021, title={Representations of twisted toroidal Lie algebras from twisted modules over vertex algebras}, volume={62}, ISSN={["1089-7658"]}, DOI={10.1063/5.0028122}, abstractNote={Given a simple finite-dimensional Lie algebra and an automorphism of finite order, one defines the notion of a twisted toroidal Lie algebra. In this paper, we construct representations of twisted toroidal Lie algebras from twisted modules over affine and lattice vertex algebras.}, number={3}, journal={JOURNAL OF MATHEMATICAL PHYSICS}, author={Bakalov, Bojko and Kirk, Samantha}, year={2021}, month={Mar} }
@article{bakalov_de sole_heluani_kac_2020, title={Chiral Versus Classical Operad}, volume={2020}, ISSN={["1687-0247"]}, DOI={10.1093/imrn/rnz373}, abstractNote={We establish an explicit isomorphism between the associated graded of the filtered chiral operad and the classical operad, which is useful for computing the cohomology of vertex algebras.}, number={19}, journal={INTERNATIONAL MATHEMATICS RESEARCH NOTICES}, author={Bakalov, Bojko and De Sole, Alberto and Heluani, Reimundo and Kac, Victor G.}, year={2020}, month={Oct}, pages={6463–6488} }
@article{bakalov_de sole_kac_2020, title={Computation of cohomology of Lie conformal and Poisson vertex algebras}, volume={26}, ISSN={["1420-9020"]}, DOI={10.1007/s00029-020-00578-2}, abstractNote={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.}, number={4}, journal={SELECTA MATHEMATICA-NEW SERIES}, author={Bakalov, Bojko and De Sole, Alberto and Kac, Victor G.}, year={2020}, month={Jul} }
@article{bakalov_yadavalli_2020, title={Darboux transformations and Fay identities for the extended bigraded Toda hierarchy*}, volume={53}, ISSN={["1751-8121"]}, DOI={10.1088/1751-8121/ab604d}, abstractNote={The extended bigraded Toda hierarchy (EBTH) is an integrable system satisfied by the Gromov-Witten total descendant potential of $\mathbb{CP}^1$ with two orbifold points. We write a bilinear equation for the tau-function of the EBTH and derive Fay identities from it. We show that the action of Darboux transformations on the tau-function is given by vertex operators. As a consequence, we obtain generalized Fay identities.}, number={6}, journal={JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL}, author={Bakalov, Bojko and Yadavalli, Anila}, year={2020}, month={Feb} }
@article{bakalov_de sole_heluani_kac_2019, title={An operadic approach to vertex algebra and Poisson vertex algebra cohomology}, volume={14}, ISBN={1861-3624}, DOI={10.1007/s11537-019-1825-3}, abstractNote={We translate the construction of the chiral operad by Beilinson and Drinfeld to the purely algebraic language of vertex algebras. Consequently, the general construction of a cohomology complex associated to a linear operad produces the vertex algebra cohomology complex. Likewise, the associated graded of the chiral operad leads to the classical operad, which produces a Poisson vertex algebra cohomology complex. The latter is closely related to the variational Poisson cohomology studied by two of the authors.}, number={2}, journal={JAPANESE JOURNAL OF MATHEMATICS}, author={Bakalov, Bojko and De Sole, Alberto and Heluani, Reimundo and Kac, Victor G.}, year={2019}, month={Sep}, pages={249–342} }
@article{bakalov_sullivan_2019, title={TWISTED LOGARITHMIC MODULES OF LATTICE VERTEX ALGEBRAS}, volume={371}, ISSN={["1088-6850"]}, DOI={10.1090/tran/7703}, abstractNote={Twisted modules over vertex algebras formalize the relations among twisted vertex operators and have applications to conformal field theory and representation theory. A recent generalization, called a twisted logarithmic module, involves the logarithm of the formal variable and is related to logarithmic conformal field theory. We investigate twisted logarithmic modules of lattice vertex algebras, reducing their classification to the classification of modules over a certain group. This group is a semidirect product of a discrete Heisenberg group and a central extension of the additive group of the lattice.}, number={11}, journal={TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY}, author={Bakalov, Bojko and Sullivan, McKay}, year={2019}, month={Jun}, pages={7995–8027} }
@misc{bakalov_sullivan_2018, title={Inhomogeneous supersymmetric bilinear
forms}, ISBN={9781470436964 9781470448820}, ISSN={0271-4132 1098-3627}, url={http://dx.doi.org/10.1090/conm/713/14311}, DOI={10.1090/conm/713/14311}, abstractNote={We consider inhomogeneous supersymmetric bilinear forms, i.e., forms that are neither even nor odd. We classify such forms up to dimension seven in the case when the restrictions of the form to the even and odd parts of the superspace are nondegenerate. As an application, we introduce a new type of oscillator Lie superalgebra.}, journal={Representations of Lie Algebras, Quantum
Groups and Related Topics}, publisher={American Mathematical
Society}, author={Bakalov, Bojko and Sullivan, McKay}, year={2018}, pages={35–45} }
@article{bakalov_wheeless_2016, title={Additional symmetries of the extended bigraded Toda hierarchy}, volume={49}, ISSN={["1751-8121"]}, DOI={10.1088/1751-8113/49/5/055201}, abstractNote={The extended bigraded Toda hierarchy (EBTH) is an integrable system satisfied by the total descendant potential of $\mathbb{CP}^1$ with two orbifold points. We construct additional symmetries of the EBTH and describe explicitly their action on the Lax operator, wave operators, and tau-function of the hierarchy. In particular, we obtain infinitesimal symmetries of the EBTH that act on the tau-function as a subalgebra of the Virasoro algebra, generalizing those of Dubrovin and Zhang.}, number={5}, journal={JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL}, author={Bakalov, Bojko and Wheeless, William}, year={2016} }
@article{bakalov_2016, title={Twisted Logarithmic Modules of Vertex Algebras}, volume={345}, ISSN={["1432-0916"]}, DOI={10.1007/s00220-015-2503-9}, abstractNote={Motivated by logarithmic conformal field theory and Gromov-Witten theory, we introduce a notion of a twisted module of a vertex algebra under an arbitrary (not necessarily semisimple) automorphism. Its main feature is that the twisted fields involve the logarithm of the formal variable. We develop the theory of such twisted modules and, in particular, derive a Borcherds identity and commutator formula for them. We investigate in detail the examples of affine and Heisenberg vertex algebras.}, number={1}, journal={COMMUNICATIONS IN MATHEMATICAL PHYSICS}, author={Bakalov, Bojko}, year={2016}, month={Jul}, pages={355–383} }
@article{bakalov_sullivan_2016, title={Twisted logarithmic modules of free field algebras}, volume={57}, ISSN={["1089-7658"]}, DOI={10.1063/1.4953249}, abstractNote={Given a non-semisimple automorphism $\varphi$ of a vertex algebra $V$, the fields in a $\varphi$-twisted $V$-module involve the logarithm of the formal variable, and the action of the Virasoro operator $L_0$ on such module is not semisimple. We construct examples of such modules and realize them explicitly as Fock spaces when $V$ is generated by free fields. Specifically, we consider the cases of symplectic fermions (odd superbosons), free fermions, and $\beta\gamma$-system (even superfermions). In each case, we determine the action of the Virasoro algebra.}, number={6}, journal={JOURNAL OF MATHEMATICAL PHYSICS}, author={Bakalov, Bojko and Sullivan, McKay}, year={2016}, month={Jun} }
@article{bakalov_fleisher_2015, title={Bosonizations of (sl)over-cap(2) and Integrable Hierarchies}, volume={11}, ISSN={["1815-0659"]}, DOI={10.3842/sigma.2015.005}, abstractNote={We construct embeddings of $\widehat{\mathfrak{sl}}_2$ in lattice vertex algebras by composing the Wakimoto realization with the Friedan-Martinec-Shenker bosonization. The Kac-Wakimoto hierarchy then gives rise to two new hierarchies of integrable, non-autonomous, non-linear partial differential equations. A new feature of our construction is that it works for any value of the central element of $\widehat{\mathfrak{sl}}_2$; that is, the level becomes a parameter in the equations.}, journal={SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS}, author={Bakalov, Bojko and Fleisher, Daniel}, year={2015} }
@article{bakalov_elsinger_2015, title={Orbifolds of lattice vertex algebras under an isometry of order two}, volume={441}, ISSN={["1090-266X"]}, DOI={10.1016/j.jalgebra.2015.06.028}, abstractNote={Every isometry σ of a positive-definite even lattice Q can be lifted to an automorphism of the lattice vertex algebra VQ. An important problem in vertex algebra theory and conformal field theory is to classify the representations of the σ-invariant subalgebra VQσ of VQ, known as an orbifold. In the case when σ is an isometry of Q of order two, we classify the irreducible modules of the orbifold vertex algebra VQσ and identify them as submodules of twisted or untwisted VQ-modules. The examples where Q is a root lattice and σ is a Dynkin diagram automorphism are presented in detail.}, journal={JOURNAL OF ALGEBRA}, author={Bakalov, Bojko and Elsinger, Jason}, year={2015}, month={Nov}, pages={57–83} }
@article{bakalov_d’andrea_kac_2013, title={Irreducible modules over finite simple Lie pseudoalgebras II. Primitive pseudoalgebras of type K}, volume={232}, ISSN={0001-8708}, url={http://dx.doi.org/10.1016/j.aim.2012.09.012}, DOI={10.1016/j.aim.2012.09.012}, abstractNote={One of the algebraic structures that has emerged recently in the study of the operator product expansions of chiral fields in conformal field theory is that of a Lie conformal algebra. A Lie pseudoalgebra is a generalization of the notion of a Lie conformal algebra for which C[∂] is replaced by the universal enveloping algebra H of a finite-dimensional Lie algebra. The finite (i.e., finitely generated over H) simple Lie pseudoalgebras were classified in our previous work (Bakalov et al., 2001) [2]. The present paper is the second in our series on representation theory of simple Lie pseudoalgebras. In the first paper we showed that any finite irreducible module over a simple Lie pseudoalgebra of type W or S is either an irreducible tensor module or the kernel of the differential in a member of the pseudo de Rham complex. In the present paper we establish a similar result for Lie pseudoalgebras of type K, with the pseudo de Rham complex replaced by a certain reduction called the contact pseudo de Rham complex. This reduction in the context of contact geometry was discovered by Rumin.}, number={1}, journal={Advances in Mathematics}, publisher={Elsevier BV}, author={Bakalov, Bojko and D’Andrea, Alessandro and Kac, Victor G.}, year={2013}, month={Jan}, pages={188–237} }
@article{bakalov_milanov_2013, title={W-constraints for the total descendant potential of a simple singularity}, volume={149}, ISSN={["1570-5846"]}, DOI={10.1112/s0010437x12000668}, abstractNote={Abstract Simple, or Kleinian, singularities are classified by Dynkin diagrams of type $ADE$ . Let $\mathfrak {g}$ be the corresponding finite-dimensional Lie algebra, and $W$ its Weyl group. The set of $\mathfrak {g}$ -invariants in the basic representation of the affine Kac–Moody algebra $\hat {\mathfrak {g}}$ is known as a $\mathcal {W}$ -algebra and is a subalgebra of the Heisenberg vertex algebra $\mathcal {F}$ . Using period integrals, we construct an analytic continuation of the twisted representation of $\mathcal {F}$ . Our construction yields a global object, which may be called a $W$ -twisted representation of $\mathcal {F}$ . Our main result is that the total descendant potential of the singularity, introduced by Givental, is a highest-weight vector for the $\mathcal {W}$ -algebra.}, number={5}, journal={COMPOSITIO MATHEMATICA}, author={Bakalov, Bojko and Milanov, Todor}, year={2013}, month={May}, pages={840–888} }
@article{bakalov_sole_2009, title={Non-linear Lie conformal algebras with three generators}, volume={14}, ISSN={["1420-9020"]}, DOI={10.1007/s00029-008-0058-8}, abstractNote={We classify certain non-linear Lie conformal algebras with three generators, which can be viewed as deformations of the current Lie conformal algebra of sℓ 2. In doing so we discover an interesting 1-parameter family of non-linear Lie conformal algebras $$R^d_{-1} (d \in\mathbb{N})$$ and the corresponding freely generated vertex algebras $$V^d_{-1}$$ , which includes for d = 1 the affine vertex algebra of sℓ 2 at the critical level k = –2. We construct free-field realizations of the algebras $$V^d_{-1}$$ extending the Wakimoto realization of $$\widehat{s\ell}_{2}$$ at the critical level, and we compute their Zhu algebras.}, number={2}, journal={SELECTA MATHEMATICA-NEW SERIES}, author={Bakalov, Bojko and Sole, Alberto De}, year={2009}, month={Jan}, pages={163–198} }
@inproceedings{bakalov_2009, place={Exeter, UK}, title={Vertex (Lie) algebras in higher dimensions}, booktitle={Proceedings of the XXVI International Colloquium on Group Theoretical Methods in Physics}, publisher={Canon Publishing Ltd}, author={Bakalov, B.}, editor={Birman, J.L. and Catto, S. and Nicolescu, B.Editors}, year={2009}, pages={15–20} }
@article{bakalov_nikolov_2008, title={Constructing models of vertex algebras in higher dimensions}, volume={35}, number={s1}, journal={Bulgarian Journal of Physics}, author={Bakalov, B. and Nikolov, N.M.}, year={2008}, pages={36–42} }
@article{bakalov_nikolov_rehren_todorov_2008, title={Infinite-dimensional Lie algebras in 4D conformal quantum field theory}, volume={41}, ISSN={["1751-8121"]}, DOI={10.1088/1751-8113/41/19/194002}, abstractNote={The concept of global conformal invariance (GCI) opens the way of applying algebraic techniques, developed in the context of 2-dimensional chiral conformal field theory, to a higher (even) dimensional space-time. In particular, a system of GCI scalar fields of conformal dimension two gives rise to a Lie algebra of harmonic bilocal fields, V_m(x,y), where the m span a finite dimensional real matrix algebra M closed under transposition. The associative algebra M is irreducible iff its commutant M' coincides with one of the three real division rings. The Lie algebra of (the modes of) the bilocal fields is in each case an infinite dimensional Lie algebra: a central extension of sp(infty,R) corresponding to the field R of reals, of u(infty,infty) associated to the field C of complex numbers, and of so*(4 infty) related to the algebra H of quaternions. They give rise to quantum field theory models with superselection sectors governed by the (global) gauge groups O(N), U(N), and U(N,H)=Sp(2N), respectively.}, number={19}, journal={JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL}, author={Bakalov, Bojko and Nikolov, Nikolay M. and Rehren, Karl-Henning and Todorov, Ivan}, year={2008}, month={May} }
@article{bakalov_nikolov_rehren_todorov_2007, title={Unitary positive-energy representations of scalar bilocal quantum fields}, volume={271}, ISSN={["1432-0916"]}, DOI={10.1007/s00220-006-0182-2}, abstractNote={The superselection sectors of two classes of scalar bilocal quantum fields in D ≥ 4 dimensions are explicitly determined by working out the constraints imposed by unitarity. The resulting classification in terms of the dual of the respective gauge groups U(N) and O(N) confirms the expectations based on general results obtained in the framework of local nets in algebraic quantum field theory, but the approach using standard Lie algebra methods rather than abstract duality theory is complementary. The result indicates that one does not lose interesting models if one postulates the absence of scalar fields of dimension D−2 in models with global conformal invariance. Another remarkable outcome is the observation that, with an appropriate choice of the Hamiltonian, a Lie algebra embedded into the associative algebra of observables completely fixes the representation theory.}, number={1}, journal={COMMUNICATIONS IN MATHEMATICAL PHYSICS}, author={Bakalov, Bojko and Nikolov, Nikolay M. and Rehren, Karl-Henning and Todorov, Ivan}, year={2007}, month={Apr}, pages={223–246} }
@inbook{bakalov_kac_2006, place={Sofia}, title={Generalized vertex algebras}, booktitle={Lie theory and its applications in physics VI}, publisher={Heron Press}, author={Bakalov, B. and Kac, V.G.}, editor={Doebner, H.-D. and Dobrev, V.K.Editors}, year={2006}, pages={3–25} }
@article{bakalov_d'andrea_kac_2006, title={Irreducible modules over finite simple Lie pseudoalgebras I. Primitive pseudoalgebras of type W and S}, volume={204}, ISSN={["1090-2082"]}, DOI={10.1016/j.aim.2005.07.003}, abstractNote={One of the algebraic structures that has emerged recently in the study of the operator product expansions of chiral fields in conformal field theory is that of a Lie conformal algebra. A Lie pseudoalgebra is a generalization of the notion of a Lie conformal algebra for which C[∂] is replaced by the universal enveloping algebra H of a finite-dimensional Lie algebra. The finite (i.e., finitely generated over H) simple Lie pseudoalgebras were classified in our previous work [B.Bakalov, A.D’Andrea, V.G. Kac, Theory of finite pseudoalgebras, Adv. Math. 162 (2001) 1–140]. In a series of papers, starting with the present one, we classify all irreducible finite modules over finite simple Lie pseudoalgebras.}, number={1}, journal={ADVANCES IN MATHEMATICS}, author={Bakalov, Bojko and D'Andrea, Alessandro and Kac, Victor G.}, year={2006}, month={Aug}, pages={278–346} }
@article{bakalov_nikolov_2006, title={Jacobi identity for vertex algebras in higher dimensions}, volume={47}, ISSN={["1089-7658"]}, DOI={10.1063/1.2197687}, abstractNote={Vertex algebras in higher dimensions, introduced previously by Nikolov, provide an algebraic framework for investigating axiomatic quantum field theory with global conformal invariance. We develop further the theory of such vertex algebras by introducing formal calculus techniques and investigating the notion of polylocal fields. We derive a Jacobi identity which together with the vacuum axiom can be taken as an equivalent definition of vertex algebra.}, number={5}, journal={JOURNAL OF MATHEMATICAL PHYSICS}, author={Bakalov, Bojko and Nikolov, Nikolay M.}, year={2006}, month={May} }
@inproceedings{bakalov_kac_2004, place={River Edge, NJ}, title={Twisted modules over lattice vertex algebras}, ISBN={9789812389367 9789812702562}, url={http://dx.doi.org/10.1142/9789812702562_0001}, DOI={10.1142/9789812702562_0001}, abstractNote={For any integral lattice $Q$, one can construct a vertex algebra $V_Q$ called a lattice vertex algebra. If $\sigma$ is an automorphism of $Q$ of finite order, it can be lifted to an automorphism of $V_Q$. In this paper we classify the irreducible $\sigma$-twisted $V_Q$-modules. We show that the category of $\sigma$-twisted $V_Q$-modules is a semisimple abelian category with finitely many isomorphism classes of simple objects.}, booktitle={Lie Theory and Its Applications in Physics V}, publisher={World Scientific Publishing Company}, author={Bakalov, Bojko and Kac, Victor G.}, editor={Doebner, H.-D. and Dobrev, V.K.Editors}, year={2004}, month={Jul}, pages={3–26} }
@article{bakalov_kac_2003, volume={2003}, ISSN={1073-7928}, url={http://dx.doi.org/10.1155/s1073792803204232}, DOI={10.1155/s1073792803204232}, abstractNote={A field algebra is a noncommutative generalization of a vertex algebra. In this paper, we develop foundations of the theory of field algebras.}, number={3}, journal={International Mathematics Research Notices}, publisher={Oxford University Press (OUP)}, author={Bakalov, Bojko and Kac, Victor}, year={2003}, pages={123} }
@book{bakalov_kirillov_2001, series={University Lecture Series}, title={Lectures on Tensor Categories and Modular Functors}, ISBN={9780821826867 9781470421687}, ISSN={1047-3998}, url={http://dx.doi.org/10.1090/ulect/021}, DOI={10.1090/ulect/021}, abstractNote={Introduction Braided tensor categories Ribbon categories Modular tensor categories 3-dimensional topological quantum field theory Modular functor Moduli spaces and complex modular functor Wess-Zumino-Witten model Bibliography Index Index of notation.}, publisher={American Mathematical Society}, author={Bakalov, Bojko and Kirillov, Alexander, Jr.}, year={2001}, collection={University Lecture Series} }
@article{bakalov_d'andrea_kac_2001, title={Theory of Finite Pseudoalgebras}, volume={162}, ISSN={0001-8708}, url={http://dx.doi.org/10.1006/aima.2001.1993}, DOI={10.1006/aima.2001.1993}, abstractNote={Conformal algebras, recently introduced by Kac, encode an axiomatic description of the singular part of the operator product expansion in conformal field theory. The objective of this paper is to develop the theory of ``multi-dimensional'' analogues of conformal algebras. They are defined as Lie algebras in a certain ``pseudotensor'' category instead of the category of vector spaces. A pseudotensor category (as introduced by Lambek, and by Beilinson and Drinfeld) is a category equipped with ``polylinear maps'' and a way to compose them. This allows for the definition of Lie algebras, representations, cohomology, etc. An instance of such a category can be constructed starting from any cocommutative (or more generally, quasitriangular) Hopf algebra $H$. The Lie algebras in this category are called Lie $H$-pseudoalgebras.
The main result of this paper is the classification of all simple and all semisimple Lie $H$-pseudoalgebras which are finitely generated as $H$-modules. We also start developing the representation theory of Lie pseudoalgebras; in particular, we prove analogues of the Lie, Engel, and Cartan-Jacobson Theorems. We show that the cohomology theory of Lie pseudoalgebras describes extensions and deformations and is closely related to Gelfand-Fuchs cohomology. Lie pseudoalgebras are closely related to solutions of the classical Yang-Baxter equation, to differential Lie algebras (introduced by Ritt), and to Hamiltonian formalism in the theory of nonlinear evolution equations. As an application of our results, we derive a classification of simple and semisimple linear Poisson brackets in any finite number of indeterminates.}, number={1}, journal={Advances in Mathematics}, publisher={Elsevier BV}, author={Bakalov, Bojko and D'Andrea, Alessandro and Kac, Victor G}, year={2001}, month={Sep}, pages={1–140} }
@article{bakalov_kirillov_2000, title={On the Lego-Teichmüller game}, volume={5}, ISSN={1083-4362 1531-586X}, url={http://dx.doi.org/10.1007/bf01679714}, DOI={10.1007/bf01679714}, number={3}, journal={Transformation Groups}, publisher={Springer Science and Business Media LLC}, author={Bakalov, B. and Kirillov, A., Jr.}, year={2000}, month={Sep}, pages={207–244} }
@article{bakalov_kac_voronov_1999, title={Cohomology of Conformal Algebras}, volume={200}, ISSN={0010-3616 1432-0916}, url={http://dx.doi.org/10.1007/s002200050541}, DOI={10.1007/s002200050541}, abstractNote={The notion of a conformal algebra encodes an axiomatic description of the operator product expansion of chiral fields in conformal field theory. On the other hand, it is an adequate tool for the study of infinite-dimensional Lie algebras satisfying the locality property. The main examples of such Lie algebras are those “based” on the punctured complex plane, such as the Virasoro algebra and loop Lie algebras. In the present paper we develop a cohomology theory of conformal algebras with coefficients in an arbitrary module. It possesses standards properties of cohomology theories; for example, it describes extensions and deformations. We offer explicit computations for the most important examples. To Bertram Kostant on his seventieth birthday}, number={3}, journal={Communications in Mathematical Physics}, publisher={Springer Science and Business Media LLC}, author={Bakalov, Bojko and Kac, Victor G. and Voronov, Alexander A.}, year={1999}, month={Feb}, pages={561–598} }
@inbook{bakalov_horozov_yakimov_1998, title={Automorphisms of the Weyl algebra and
bispectral operators}, ISBN={9780821809495 9781470439286}, ISSN={1065-8580 2472-4890}, url={http://dx.doi.org/10.1090/crmp/014/01}, DOI={10.1090/crmp/014/01}, abstractNote={In our previous paper q-alg/9605011 we proposed several algebraic methods for constructing new solutions to the bispectral problem. In the present note the corresponding eigenfunctions are explicitly constructed as multiple Laplace integrals.}, booktitle={The Bispectral Problem}, publisher={American Mathematical
Society}, author={Bakalov, Bojko and Horozov, Emil and Yakimov, Milen}, year={1998}, month={Feb}, pages={3–10} }
@article{bakalov_horozov_yakimov_1998, title={Highest weight modules over the W_(1+∞) algebra and the bispectral problem}, volume={93}, ISSN={0012-7094}, url={http://dx.doi.org/10.1215/s0012-7094-98-09302-4}, DOI={10.1215/s0012-7094-98-09302-4}, number={1}, journal={Duke Mathematical Journal}, publisher={Duke University Press}, author={Bakalov, B. and Horozov, E. and Yakimov, M.}, year={1998}, month={May}, pages={41–72} }
@article{bakalov_horozov_yakimov_1997, title={Bispectral Algebras of Commuting Ordinary Differential Operators}, volume={190}, ISSN={0010-3616 1432-0916}, url={http://dx.doi.org/10.1007/s002200050244}, DOI={10.1007/s002200050244}, abstractNote={We develop a systematic way for constructing bispectral algebras of commuting ordinary differential operators of any rank $N$. It combines and unifies the ideas of Duistermaat-Gr\"unbaum and Wilson. Our construction is completely algorithmic and enables us to obtain all previously known classes or individual examples of bispectral operators. The method also provides new broad families of bispectral algebras which may help to penetrate deeper into the problem.}, number={2}, journal={Communications in Mathematical Physics}, publisher={Springer Science and Business Media LLC}, author={Bakalov, B. and Horozov, E. and Yakimov, M.}, year={1997}, month={Dec}, pages={331–373} }
@article{bakalov_horozov_yakimov_1997, title={Highest weight modules of W_(1+∞), Darboux transformations and the bispectral problem}, volume={23}, number={2}, journal={Serdica Mathematical Journal}, author={Bakalov, B. and Horozov, E. and Yakimov, M.}, year={1997}, pages={95–112} }
@inbook{bakalov_georgiev_todorov_1996, place={Sofia, Bulgaria}, title={A QFT approach to W_(1+∞)}, booktitle={New trends in quantum field theory}, publisher={Heron Press}, author={Bakalov, B.N. and Georgiev, L.S. and Todorov, I.T.}, editor={Ganchev, A.Editor}, year={1996}, pages={147–158} }
@article{bakalov_horozov_yakimov_1996, title={Bäcklund-Darboux transformations in Sato's Grassmannian}, volume={22}, number={4}, journal={Serdica Mathematical Journal}, author={Bakalov, B. and Horozov, E. and Yakimov, M.}, year={1996}, pages={571–586} }
@article{bakalov_horozov_yakimov_1996, title={General methods for constructing bispectral operators}, volume={222}, ISSN={0375-9601}, url={http://dx.doi.org/10.1016/0375-9601(96)00624-x}, DOI={10.1016/0375-9601(96)00624-x}, abstractNote={We present methods for obtaining new solutions to the bispectral problem. We achieve this by giving its abstract algebraic version suitable for generalizations. All methods are illustrated by new classes of bispectral operators.}, number={1-2}, journal={Physics Letters A}, publisher={Elsevier BV}, author={Bakalov, B. and Horozov, E. and Yakimov, M.}, year={1996}, month={Oct}, pages={59–66} }
@article{bakalov_horozov_yakimov_1996, title={Tau-functions as highest weight vectors for W_(1+∞) algebra}, volume={29}, ISSN={0305-4470 1361-6447}, url={http://dx.doi.org/10.1088/0305-4470/29/17/027}, DOI={10.1088/0305-4470/29/17/027}, abstractNote={For each we construct a highest weight module of the Lie algebra The highest weight vectors are specific tau-functions of the Nth Gelfand - Dickey hierarchy. We show that these modules are quasifinite and we give a complete description of the reducible ones together with a formula for the singular vectors.}, number={17}, journal={Journal of Physics A: Mathematical and General}, publisher={IOP Publishing}, author={Bakalov, B and Horozov, E and Yakimov, M}, year={1996}, month={Sep}, pages={5565–5573} }