Throughout this paper, let k be a field of characteristic 2. Let G be a classical group over k and g be its Lie algebra. Let g be the dual vector space of g. Fix a Borel subgroup B and a maximal torus T ⊂ B of G. Let B = TU be the Levi decomposition of B. Let b, t and n be the Lie algebra of B, T and U respectively. We have a natural action of G on g, g.ξ(x) = ξ(Ad(g)x) for g ∈ G, ξ ∈ g and x ∈...

Algebraic properties of the Becchi-Rouet-Stora-Tyutin (BRST) symmetry associated to twisted gauge occurring in $\ensuremath{\kappa}$-Poincar\'e invariant theories on $\ensuremath{\kappa}$-Minkowski space are investigated. We find that BRST operation invariance action functional can be continuously deformed together with its corresponding Leibniz rule, into a nilpotent operation, leading algebra...

A normal complex algebraic variety X is called a symplectic variety (cf. [Be]) if its regular locus Xreg admits a holomorphic symplectic 2-form ω such that it extends to a holomorphic 2-form on a resolution f : X̃ → X. Affine symplectic varieties are constructed in various ways such as nilpotent orbit closures of a semisimple complex Lie algebra (cf. [CM]), Slodowy slices to nilpotent orbits (cf...

A relation between $\frac{1}{2}$-derivations of Lie algebras and transposed Poisson was established. Some non-trivial with a certain algebra (Witt algebra, $\mathcal{W}(a,-1)$, thin solvable abelian nilpotent radical) were constructed. In particular, we constructed an example the associative parts isomorphic to Laurent polynomials Witt algebra. On other side, it proven that there are no part se...

For the Lie algebras L\(H(2)) and L\(W(2)), we study their infinitesimal deformations and the corresponding global ones. We show that, as in the case of L\{W(\)), each integrable infinitesimal deformation of L\(H(2)) and L1(W/(2)) can be represented by a 2-cocycle that defines a global deformation by means of a trivial extension. We also illustrate that all deformations of L\{H{2)) arise as res...

A finite W-algebra U (g, e) is a certain finitely generated algebra that can be viewed as the enveloping algebra of the Slodowy slice to the adjoint orbit of a nilpotent element e of a complex reductive Lie algebra g. It is possible to give the tensor product of a U (g, e)-module with a finite dimensional U (g)-module the structure of a U (g, e)-module; we refer to such tensor products as trans...

Fix a prime p, and let A be the polynomial part of the dual Steenrod algebra. The Frobenius map on A induces the Steenrod operation P̃0 on cohomology, and in this paper, we investigate this operation. We point out that if p = 2, then for any element in the cohomology of A, if one applies P̃0 enough times, the resulting element is nilpotent. We conjecture that the same is true at odd primes, and t...

This paper presents an operator calculus approach to computing with non-commutative variables. First, we recall the product formulation of formal exponential series. Then we show how to formulate canonical boson calculus on formal series. This calculus is used to represent the action of a Lie algebra on its universal enveloping algebra. As applications, Hamilton's equations for a general Hamilt...