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1. Twisted de Rham complex on line and singular vectors in sl2ˆ Verma modules.

   https://doi.org/doi.org/10.3842/SIGMA.2019.075  

   Slinkin, Alexey; Varchenko, Alexander (May 2019, Symmetry integrability and geometry methods and applications)

   We consider two complexes. The first complex is the twisted de Rham complex of scalar meromorphic differential forms on projective line, holomorphic on the complement to a finite set of points. The second complex is the chain complex of the Lie algebra of sl2-valued algebraic functions on the same complement, with coefficients in a tensor product of contragradient Verma modules over the affine Lie algebra sl2^. In [Schechtman V., Varchenko A., Mosc. Math. J. 17 (2017), 787-802] a construction of a monomorphism of the first complex to the second was suggested and it was indicated that under this monomorphism the existence of singular vectors in the Verma modules (the Malikov-Feigin-Fuchs singular vectors) is reflected in the relations between the cohomology classes of the de Rham complex. In this paper we prove these results. △ Less 

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2. Contravariant forms on Whittaker modules

   https://doi.org/10.1090/proc/15205  

   Brown, Adam; Romanov, Anna (January 2021, Proceedings of the American Mathematical Society)

   Let g \mathfrak {g} be a complex semisimple Lie algebra. We give a classification of contravariant forms on the nondegenerate Whittaker g \mathfrak {g} -modules Y ( χ , η ) Y(\chi , \eta ) introduced by Kostant. We prove that the set of all contravariant forms on Y ( χ , η ) Y(\chi , \eta ) forms a vector space whose dimension is given by the cardinality of the Weyl group of g \mathfrak {g} . We also describe a procedure for parabolically inducing contravariant forms. As a corollary, we deduce the existence of the Shapovalov form on a Verma module, and provide a formula for the dimension of the space of contravariant forms on the degenerate Whittaker modules M ( χ , η ) M(\chi , \eta ) introduced by McDowell. 

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3. B-Twisted Gaiotto–Witten Theory and Topological Quantum Field Theory

   https://doi.org/10.1007/s00220-024-05211-3  

   Garner, Niklas; Geer, Nathan; Young, Matthew B (January 2025, Communications in Mathematical Physics)

   We develop representation theoretic techniques to construct three dimensional non-semisimple topological quantum field theories which model homologically truncated topological B-twists of abelian Gaiotto--Witten theory with linear matter. Our constructions are based on relative modular structures on the category of weight modules over an unrolled quantization of a Lie superalgebra. The Lie superalgebra, originally defined by Gaiotto and Witten, is associated to a complex symplectic representation of a metric abelian Lie algebra. The physical theories we model admit alternative realizations as Chern--Simons-Rozansky--Witten theories and supergroup Chern--Simons theories and include as particular examples global forms of gl(1,1)-Chern--Simons theory and toral Chern--Simons theory. Fundamental to our approach is the systematic incorporation of non-genuine line operators which source flat connections for the topological flavour symmetry of the theory. 

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4. Conformal Blocks on Smoothings via Mode Transition Algebras

   https://doi.org/10.1007/s00220-025-05237-1  

   Damiolini, Chiara; Gibney, Angela; Krashen, Daniel (June 2025, Communications in Mathematical Physics)

   Here we introduce a series of associative algebras attached to a vertex operator algebra V of CFT type, called mode transition algebras, and show they reflect both algebraic properties of V and geometric constructions on moduli of curves. Pointed and coordinatized curves, labeled by admissible V-modules, give rise to sheaves of coinvariants. We show that if the mode transition algebras admit multiplicative identities satisfying certain natural properties (called strong identity elements), these sheaves deform as wanted on families of curves with nodes. This provides new contexts in which coherent sheaves of coinvariants form vector bundles. We also show that mode transition algebras carry information about higher level Zhu algebras and generalized Verma modules. To illustrate, we explicitly describe the higher level Zhu algebras of the Heisenberg vertex operator algebra, proving a conjecture of Addabbo–Barron. 

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5. On the Lie algebra structure of integrable derivations

   https://doi.org/10.1112/blms.12884  

   Briggs, Benjamin; Rubio_y_Degrassi, Lleonard (December 2023, Bulletin of the London Mathematical Society)

   Abstract Building on work of Gerstenhaber, we show that the space of integrable derivations on an Artin algebra forms a Lie algebra, and a restricted Lie algebra if contains a field of characteristic . We deduce that the space of integrable classes in forms a (restricted) Lie algebra that is invariant under derived equivalences, and under stable equivalences of Morita type between self‐injective algebras. We also provide negative answers to questions about integrable derivations posed by Linckelmann and by Farkas, Geiss and Marcos. Along the way, we compute the first Hochschild cohomology of the group algebra of any symmetric group. 

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