Narrow Results

We study field models for which a quantum action (i.e. the action appearing in the generating functional of Green functions) is invariant under supersymmetric transformations. We derive the Ward identity which is a direct consequence of this invariance. We consider a change of variables in functional integral connected with supersymmetric transformations when its parameter is replaced by a nilpotent functional of fields. Exact form of the corresponding Jacobian is found. We find restrictions on generators of supersymmetric transformations when a consistent quantum description of given field theories exists.

We construct linear bases of free Gelfand–Dorfman–Novikov (GDN) superalgebras. As applications, we prove a Poincaré–Birkhoff–Witt (PBW) type theorem, that is, every GDN superalgebra can be embedded into its universal enveloping associative differential supercommuative algebra. An Engel theorem is given.

We investigate a class of non-involutive solutions of the Yang–Baxter equation which generalize derived (self-distributive) solutions. In particular, we study generalized multipermutation solutions in this class. We show that the Yang–Baxter (permutation) groups of such solutions are nilpotent. We formulate the results in the language of biracks which allows us to apply universal algebra tools.

We consider expansive group actions on a compact metric space containing a special fixed point denoted by $0$, and endomorphisms of such systems whose forward trajectories are attracted toward $0$. Such endomorphisms are called asymptotically nilpotent, and we study the conditions in which they are nilpotent, that is, map the entire space to $0$ in a finite number of iterations. We show that for a large class of discrete groups, this property of nil-rigidity holds for all expansive actions that satisfy a natural specification-like property and have dense homoclinic points. Our main result in particular shows that the class includes all residually finite solvable groups and all groups of polynomial growth. For expansive actions of the group $\mathbb{Z}$, we show that a very weak gluing property suffices for nil-rigidity. For ${\mathbb{Z}}^2$-subshifts of finite type, we show that the block-gluing property suffices. The study of nil-rigidity is motivated by two aspects of the theory of cellular automata and symbolic dynamics: It can be seen as a finiteness property for groups, which is representative of the theory of cellular automata on groups. Nilpotency also plays a prominent role in the theory of cellular automata as dynamical systems. As a technical tool of possible independent interest, the proof involves the construction of tiered dynamical systems where several groups act on nested subsets of the original space.

We show that the right ideal of a Novikov algebra generated by the square of a right nilpotent subalgebra is nilpotent. We also prove that a $G$-graded Novikov algebra $N$ over a field $K$ with solvable $0$-component $N_0$ is solvable, where $G$ is a finite additive abelean group and the characteristic of $K$ does not divide the order of the group $G$. We also show that any Novikov algebra $N$ with a finite solvable group of automorphisms $G$ is solvable if the algebra of invariants $N^G$ is solvable.

The Milnor Problem (modified) in the theory of group growth asks whether any finite presented group of vanishing algebraic entropy has at most polynomial growth. We show that a positive answer to the Milnor Problem (modified) is equivalent to the Nilpotency conjecture in Riemannian geometry: given $n,d>0$, there exists a constant $𝜖(n,d)>0$ such that if a compact Riemannian $n$-manifold $M$ satisfies that Ricci curvature ${\mathrm{\text{Ric}}}_M\ge -(n-1)$, diameter $d\ge \mathrm{\text{diam}}(M)$ and volume entropy $h(M)<𝜖(n,d)$, then the fundamental group ${\pi }_1(M)$ is virtually nilpotent. We will verify the Nilpotency conjecture in some cases, and we will verify the vanishing gap phenomena for more cases i.e. if $h(M)<𝜖(n,d)$, then $h(M)=0$.

In this work we investigate the complex Leibniz superalgebras with characteristic sequence (n - 1, 1|m₁, …, m_k) and with nilindex equal to n + m. We prove that such superalgebras with the condition m₂ ≠ 0 have nilindex less than n + m. Therefore the complete classification of Leibniz algebras with characteristic sequence (n - 1, 1|m₁, …, m_k) and with nilindex equal to n + m is reduced to the classification of filiform Leibniz superalgebras of nilindex equal to n + m, which was provided in [3, 7].

A digraph is attached to any evolution algebra. This graph leads to some new purely algebraic results on this class of algebras and allows for some new natural proofs of known results. Nilpotency of an evolution algebra will be proved to be equivalent to the nonexistence of oriented cycles in the graph. Besides, the automorphism group of any evolution algebra ℰ with ℰ = ℰ² will be shown to be always finite.

We prove that any Novikov algebra over a field of characteristic $\ne 2$ is Lie-solvable if and only if its commutator ideal $[N,N]$ is right nilpotent. We also construct examples of infinite-dimensional Lie-solvable Novikov algebras $N$ with non-nilpotent commutator ideal $[N,N]$.

Skew braces are algebraic structures related to the solutions of the set-theoretic quantum Yang–Baxter equation. We develop the central nilpotency theory for such algebraic structures in the sense of Freese–McKenzie [12] and we compare the universal algebraic notion of central nilpotency with the notion of right and left nilpotency developed in [7].

The goal of this paper is to deepen the study of multiplicative lattices in the sense of Facchini, Finocchiaro and Janelidze. We provide a sort of Prime Ideal Principle that guarantees that maximal implies prime in a variety of cases (among them the case of commutative rings with identity). This result is used to study the lattice theoretic counterpart of multiplicative closed sets, that of $m$-systems. The notion of $m$-system is also studied from the topological point of view.

In this paper, several theorems of Macdonald [On certain varieties of groups, Math. Z.76 (1961) 270–282; On certain varieties of groups II, Math. Z.78 (1962) 175–188] on the varieties of nilpotent groups will be generalized to the case of Lie rings. We consider three varieties of Lie rings of any characteristic associated with some equations (see Eqs. (1.1)–(1.3)). We prove that each Lie ring in variety (1.1) is nilpotent of exponent at most n + 2; if L is a Lie ring in variety (1.2), then L² is nilpotent of exponent at most n + 1; and each Lie ring in variety (1.3) is solvable of length at most n + 1. We also discuss some varieties of solvable Lie rings and the varieties of Lie rings defined by the properties of subrings.