The article is devoted to a thorough study of new algebraic structures appearing in contemporary physical theories (chiral fields, Poisson brackets, vertex algebras, conformal field theory) and we can present only poor information at this place.
The fundamental concepts are not simple. We begin with a pseudotensorial category \(\mathcal M\) which is a class of objects together with vector spaces \(\text{Lin}(\{L_i\}_{i\in I},M)\) equipped moreover with actions of the symmetric group \(S_I\) and the composition maps \[ \text{Lin}(\{L_i\}_{i\in I},M)\otimes\bigotimes_{i\in I}(\text{Lin}\{N_j\}_{j\in J},L_i)\to\text{Lin}(\{N_j\}_{j\in J},M) \] that satisfy the associativity, the existence of the unity \(\text{id}_{\mathcal M}\in\text{Lin}(\{M\},M)\) and equivariance with respect to \(S_I\). (Then, roughly saying, the pseudotensors equipped with polylinear maps take the common role of vector spaces in classical algebra in order to obtain the pseudoalgebras.) In more detail: A Lie algebra in the category \(\mathcal M\) is an object \(L\) equipped with \(\beta\in\text{Lin}(\{L,L\},L)\) satisfying the skew commutativity \(\beta=-\sigma_{12}\beta\) and the Jacobi identity \(\beta(\beta(\cdot,\cdot),\cdot)=\beta(\cdot,\beta(\cdot,\cdot))-\sigma_{12}\beta(\cdot,\beta(\cdot,\cdot))\), where \(\sigma_{12}=(12)\in S_2\). An associative algebra in \(\mathcal M\) is an object \(A\) and product \(\mu\in\text{Lin}(\{A,A\},A)\) satisfying the associativity \(\mu(\mu(\cdot,\cdot),\cdot)=\mu(\cdot,\mu(\cdot,\cdot))\). In particular let \(H\) be a cocommutative bialgebra. The authors introduce a certain special pseudotensorial category \({\mathcal M}^*(H)\) of left \(H\)-modules equipped with \(H\)-polylinear maps and then the Lie \(H\)-pseudoalgebra (\(H\)-pseudoalgebra) is defined as a Lie algebra (associative algebra, respectively) in the category \({\mathcal M}^*(H)\). (The latter two concepts can be seen equivalent with the so called conformal algebra if certain \(x\)-brackets \([a_xb]\) are introduced.)
The central result provides the classification of finite simple Lie pseudoalgebras over the Hopf algebra \(H=U(\delta)\), the universal enveloping algebra of a Lie algebra \(\delta\). This leads to the structural theory of semi-simple Lie pseudoalgebras. The generalized Lie theorem on solvable Lie pseudoalgebras and an analogue of the Cartan-Jacobson theorem hold true, unlike the Levi theorem. The cohomology of Lie pseudoalgebras describes the module extensions, Abelian pseudoalgebra extensions and pseudoalgebra deformations.