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Quantum $L_\infty$ algebras are higher loop generalizations of cyclic $L_\infty$ algebras. Motivated by the problem of defining morphisms between such algebras, we construct a linear category of $(-1)$-shifted symplectic vector spaces and…

In a program to formulate and develop two-dimensional conformal field theory in the framework of algebraic geometry, Beilinson and Drinfeld have recently given a notion of ``chiral algebra'' in terms of D-modules on algebraic curves. This…

We give a construction of homotopy algebras based on ``higher derived brackets''. More precisely, the data include a Lie superalgebra with a projector on an Abelian subalgebra satisfying a certain axiom, and an odd element $\Delta$. Given…

Very recently, Galashin, Postnikov, and Williams introduced the notion of higher secondary polytopes, generalizing the secondary polytope of Gelfand, Kapranov, and Zelevinsky. Given an $n$-point configuration $\mathcal{A}$ in…

We introduce the notion of algebraic higher symmetry, which generalizes higher symmetry and is beyond higher group. We show that an algebraic higher symmetry in a bosonic system in $n$-dimensional space is characterized and classified by a…

We extend Barwick's and Haugseng's construction of the double $\infty$-category of spans in a pullback-complete $\infty$-category $\mathfrak{C}$ to more general shapes: for a large class of algebraic patterns $\mathfrak{P}$, we define a…

In the first part of the paper we give the denominator identity for all simple finite-dimensional Lie super algebras $\frak g\/$ with a non-degenerate invariant bilinear form. We give also a character and (super) dimension formulas for all…

An averaging operator on an associative algebra $A$ is an algebraic abstraction of the time average operator on the space of real-valued functions defined in time-space. In this paper, we consider relative averaging operators on a bimodule…

We introduce a notion of dynamics in the setting of Segal topos, by considering the Segal category of stacks $\mathcal{X} = \text{dAff}_{\mathcal{C}}^{\, \sim, \tau}$ on a Segal category $\text{dAff}_{\mathcal{C}}=$…

In 1998, Georgia Benkart and Tom Roby introduced the down-up algebra $\mathcal A$. The algebra $\mathcal A$ is associative, noncommutative, and infinite-dimensional. It is defined by two generators $A,B$ and two relations called the down-up…

We reformulate superalgebra and supergeometry in completely categorical terms by a consequent use of the functor of points. The increased abstraction of this approach is rewarded by a number of great advantages. First, we show that one can…

Let $S$ be an integral domain with field of fractions $F$ and let $A$ be an $F$-algebra. An $S$-subalgebra $R$ of $A$ is called $S$-nice if $R$ is lying over $S$ and the localization of $R$ with respect to $S \setminus \{ 0 \}$ is $A$. Let…

Dimensional reductions of various higher dimensional (super)gravity theories lead to effectively two-dimensional field theories described by gravity coupled G/H nonlinear sigma-models. We show that a new set of complexified variables can be…

We define a class of $A_\infty$-algebras that are obtained by deformations of higher spin symmetries. While higher spin symmetries of a free CFT form an associative algebra, the slightly broken higher spin symmetries give rise to a minimal…

A new interpretation of Dirac singletons \cite{Dirac:1963ta}, i.e. free conformal fields in $d$ dimensions, as relativistic fields in a $d+1$-dimensional space-time with cosmological constant, that differs from the Flato-Fronsdal dipole…

We define a higher analogue of Dirac structures on a manifold M. Under a regularity assumption, higher Dirac structures can be described by a foliation and a (not necessarily closed, non-unique) differential form on M, and are equivalent to…

We derive sufficient conditions under which the ``second'' Hamiltonian structure of a class of generalized KdV-hierarchies defines one of the classical $\cal W$-algebras obtained through Drinfel'd-Sokolov Hamiltonian reduction. These…

Ideas of Rozansky and Witten, as developed by Kapranov, show that a complex symplectic manifold X gives rise to Vassiliev weight systems. In this paper we study these weight systems by using D(X), the derived category of coherent sheaves on…

In this paper, we study gravitational symmetry algebras that live on 2-dimensional cuts $S$ of asymptotic infinity. We define a notion of wedge algebra $\mathcal{W}(S)$ which depends on the topology of $S$. For the cylinder $S=\mathbb{C}^*$…

We develop a new approach to highest weight categories $\cal{C}$ with good (and cogood) posets of weights via pseudocompact algebras by introducing ascending (and descending) quasi-hereditary pseudocompact algebras. For $\cal{C}$ admitting…