We establish an explicit criteria (the vanishing of non–degeneracy conditions) for certain noncommutative algebras to have Poincaré–Birkhoff– Witt basis. We study theoretical properties of such G–algebras, concluding they are in some sense ”close to commutative”. We use the non–degeneracy conditions for practical study of certain deformations of Weyl algebras, quadratic and diffusion algebras. ...

The Hodge theorem also asserts that these two spaces are complex conjugates and hence the equality of the two dimensions. In this note we give a p-adic proof of (1.1) when X/C is a smooth projective threefold. Our approach is based on the following observation: one first notes a purely p-adic assertion that (1.1) holds when Hodge numbers are replaced by more delicate p-adic invariants introduce...

The theory of special groups is an axiomatization of the algebraic theory of quadratic forms, introduced by Dickmann and Miraglia (see [4]). The class of special groups, together with its morphisms, forms a category. As for other such axiomatisations, the main examples of special groups are provided by fields, in this case by applying the special group functor, which associates to each field F ...

let $r$ be an associative ring with identity and $z^*(r)$ be its set of non-zero zero divisors.  the zero-divisor graph of $r$, denoted by $gamma(r)$, is the graph whose vertices are the non-zero  zero-divisors of  $r$, and two distinct vertices $r$ and $s$ are adjacent if and only if $rs=0$ or $sr=0$.  in this paper, we bring some results about undirected zero-divisor graph of a monoid ring ov...

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 a smooth and proper scheme over an Artinian local ring with ordinary reduction the perfect residue field, we prove — under some general assumptions that relative de Rham–Witt spectral sequence degenerates crystalline cohomology, equipped its display structure arising from Nygaard complexes, has Hodge–Witt decomposition into direct sum of (suitably Tate-Twisted) multiplicative displays. As e...

A λ-ring is, roughly speaking, a commutative ring R with unit together with operations λi, i ≥ 0, on it that act like the exterior power operations. It is widely used in algebraic topology, algebra, and representation theory. For example, the complex representation ring R(G) of a group G is a λ-ring, where λi is induced by the map that sends a representation to its ith exterior power. Another e...

Recently, a number of new classes of infinite-dimensional simple Lie algebras over a Field of characteristic 0 were discovered by several authors (see the references at the end of this paper). Among those algebras, are the generalized Witt algebras. The higher rank Virasoro algebras was introduced by Patera and Zassenhaus [PZ], which are 1-dimensional universal central extensions of some genera...