arXiv subjects
Publications and source records attributed to Mykola Khrypchenko.
We describe totally compatible structures on the Jacobson radical of the incidence algebra of a finite poset over a field. We show that such structures are in general non-proper.
We study the globalization problem for a strong partial action $α$ of a monoid $M$ on a semigroup $X$ via the associated rewriting system $(X_M^+,\to)$. We show that the local confluence of $(X_M^+,\to)$ is sufficient for the globalizability of $α$ but, unlike the group case, it is not necessary. Focusing on the monoid $M=G^0$, where $G$ is a group, we obtain an explicit criterion for the globalizability of $α$ and a criterion for the local confluence of $(X_M^+,\to)$. Several applications to strong partial actions of the monoid $M=\{0,1\}$ on semigroups and algebras, as well as to strong partial actions of an arbitrary monoid $M$ on left zero and null semigroups, are presented.
Following the ideas in~\cite{yM88}, \cite{T90} and inspiration from~\cite{KO24}, we construct a bialgebra $T_q(n)$ and a pointed Hopf algebra $UT_q(n)$ which quantize the coordinate rings of the algebra of upper triangular matrices and of the group of invertible upper triangular matrices of size $n\geq 2$, respectively, where $q$ is a nonzero parameter. The resulting structure on $UT_q(n)$ is neither commutative nor cocommutative and it can be seen as a Hopf quotient of the Takeuchi's two-parameter quantization~\cite{T90} of ${\rm GL}(n)$ corresponding to a specific choice of parameters. The motivation comes from the idea of quantizing the incidence algebra of a finite poset, as the latter can be embedded as a subalgebra of the algebra of upper triangular matrices. We further study and compare the Lie algebras of derivations, the automorphism groups and the low degree Hochschild cohomology of these algebras in case $n=2$.
Given a unital action $θ$ of an inverse monoid $S$ on an algebra $A$ over a filed $K$ we produce (co)homology spectral sequences which converge to the Hochschild (co)homology of the crossed product $A\rtimes_θS$ with values in a bimodule over $A\rtimes_θS$. The spectral sequences involve a new kind of (co)homology of the inverse monoid $S,$ which is based on $KS$-modules. The spectral sequences take especially nice form, when $(A\rtimes_θS)^e $ is flat as a left (homology case) or right (cohomology case) $A^e$-module, involving also the Hochschild (co)homology of $A.$ Same nice spectral sequences are also obtained if $K$ is a commutative ring, over which $A$ is projective, and $S$ is $E$-unitary. We apply our results to the Steinberg algebra $A_K(\mathscr{G})$ over a field $K$ of an ample groupoid $\mathscr{G},$ whose unit space $\mathscr{G} ^{(0)}$ is compact. In the homology case our spectral sequence collapses on the $p$-axis, resulting in an isomorphism between the Hochschild homology of $A_K(\mathscr{G})$ with values in an $A_K(\mathscr{G})$-bimodule $M$ and the homology of the inverse semigroup of the compact open bisections of $\mathscr{G}$ with values in the invariant submodule of $M.$
Let $M$ be a monoid, $\mathscr{C}$ a category with pullbacks and $X$ an object of $\mathscr{C}$. We introduce the notion of a partial action $α$ of $M$ on $X$ and study the globalization question for $α$. If $α$ admits a reflection in the subcategory of global actions, then we reduce the problem to the verification that a certain diagram is a pullback in $\mathscr{C}$. We then give a construction of such a reflection in terms of a colimit of a certain functor with values in $\mathscr{C}$. We specify this construction to the case of categories admitting certain coproducts and coequalizers.
We describe $σ$-matching, interchangeable and, as a consequence, totally compatible products on some classes of associative algebras, including unital algebras, the semigroup algebras of rectangular bands, algebras with enough idempotents, free non-unital associative algebras and free non-unital commutative associative algebras.
We study symmetric continuous bilinear maps $V$ on a C$^*$-algebra $A$ that have the Jordan product property at a fixed element $z\in A$. We show that, whenever $A$ is a finite direct sum or a $c_0$-sum of infinite simple von Neumann algebras, such a map $V$ has the square-zero property. Then, it is proved that $V(a,b)=T(a\circ b)$ for some bounded linear map $T$ on $A$. As a consequence, Jordan homomorphisms and derivations at $z\in A$ are characterized.
We describe $σ$-matching, interchangeable and, as a consequence, totally compatible structures on the strictly upper triangular matrix algebra $UT_n(K)$ for all $n\ge 3$.
In this work, we introduce the notion of a partial action of a group on a strict monoidal category. We propose, in the context of Monoidal categories, new constructions analogous to those existing for partial group actions over an algebra such as the globalization, the subalgebra of partial invariants, and the partial smash product.
This is a survey on the geometric classification of different varieties of algebras (nilpotent, nil-, associative, commutative associative, cyclic associative, Jordan, Kokoris, standard, noncommutative Jordan, commutative power-associative, weakly associative, terminal, Lie, Malcev, binary Lie, Tortkara, dual mock Lie, $\mathfrak{CD}$-, commutative $\mathfrak{CD}$-, anticommutative $\mathfrak{CD}$-, symmetric Leibniz, Leibniz, Zinbiel, Novikov, bicommutative, assosymmetric, antiassociative, left-symmetric, right alternative, and right commutative), $n$-ary algebras (Fillipov ($n$-Lie), Lie triple systems and anticommutative ternary), superalgebras (Lie and Jordan), and Poisson-type algebras (Poisson, transposed Poisson, Leibniz-Poisson, generic Poisson, generic Poisson-Jordan, transposed Leibniz-Poisson, Novikov-Poisson, pre-Lie Poisson, commutative pre-Lie, anti-pre-Lie Poisson, pre-Poisson, compatible commutative associative, compatible associative, compatible Novikov, compatible pre-Lie). We also discuss the degeneration level classification.
Let $X$ be an arbitrary poset and $K$ an arbitrary field. We describe linear unital invertibility preservers of the finitary incidence algebra $FI(X,K)$ in terms of certain maps of the power set algebra $\mathcal{P}(X)$ and linear maps $FI(X,K)\to J(FI(X,K))$.
We study bounded bilinear maps on a C$^*$-algebra $A$ having product property at $c\in A$. This leads us to the question of when a C$^*$-algebra is determined by products at $c.$ In the first part of our paper, we investigate this question for compact C$^*$-algebras, and in the second part, we deal with von Neumann algebras having non-trivial atomic part. Our results are applicable to descriptions of homomorphism-like and derivation-like maps at a fixed point on such algebras.
In the first part of the paper we describe $φ$-derivations of the incidence algebra $I(X,K)$ of a locally finite poset $X$ over a field $K$, where $φ$ is an arbitrary automorphism of $I(X,K)$. We show that they admit decompositions similar to that of usual derivations of $I(X,K)$. In particular, the quotient of the space of $φ$-derivations of $I(X,K)$ by the subspace of inner $φ$-derivations of $I(X,K)$ is isomorphic to the first group of certain cohomology of $X$, which is developed in the second part of the paper.
Let $X$ be a finite connected poset, $K$ a field of characteristic zero and $I(X,K)$ the incidence algebra of $X$ over $K$ seen as a Lie algebra under the commutator product. In the first part of the paper we show that any $\frac{1}{2}$-derivation of $I(X,K)$ decomposes into the sum of a central-valued $\frac 12$-derivation, an inner $\frac{1}{2}$-derivation and a $\frac{1}{2}$-derivation associated with a map $σ:X^2_<\to K$ that is constant on chains and cycles in $X$. In the second part of the paper we use this result to prove that any transposed Poisson structure on $I(X,K)$ is the sum of a structure of Poisson type, a mutational structure and a structure determined by $λ:X^2_e\to K$, where $X^2_e$ is the set of $(x,y)\in X^2$ such that $x<y$ is a maximal chain not contained in a cycle.
Let $FI(X,K)$ be the finitary incidence algebra of a poset $X$ over a field $K$. In this short note we establish when $FI(X,K)$ satisfies a polynomial identity and when its group of units $\mathcal{U}(FI(X,K))$ satisfies a group identity. The Lie derived length of $FI(X,K)$ and the derived length of $\mathcal{U}(FI(X,K))$ are also determined.
We describe transposed Poisson structures on the upper triangular matrix Lie algebra $T_n(F)$, $n>1$, over a field $F$ of characteristic zero. We prove that, for $n>2$, any such structure is either of Poisson type or the orthogonal sum of a fixed non-Poisson structure with a structure of Poisson type, and for $n=2$, there is one more class of transposed Poisson structures on $T_n(F)$. We also show that, up to isomorphism, the full matrix Lie algebra $M_n(F)$ admits only one non-trivial transposed Poisson structure, and it is of Poisson type.
We describe transposed Poisson structures on generalized Witt algebras $W(A,V, \langle \cdot,\cdot \rangle )$ and Block Lie algebras $L(A,g,f)$ over a field $F$ of characteristic zero, where $\langle \cdot,\cdot \rangle$ and $f$ are non-degenerate. More specifically, if $\dim(V)>1$, then all the transposed Poisson algebra structures on $W(A,V,\langle \cdot,\cdot \rangle)$ are trivial; and if $\dim(V)=1$, then such structures are, up to isomorphism, mutations of the group algebra structure on $FA$. The transposed Poisson algebra structures on $L(A,g,f)$ are in a one-to-one correspondence with commutative and associative multiplications defined on a complement of the square of $L(A,g,f)$ with values in the center of $L(A,g,f)$. In particular, all of them are usual Poisson structures on $L(A,g,f)$. This generalizes earlier results about transposed Poisson structures on Block Lie algebras $\mathcal{B}(q)$.
We fully characterize regular Hom-Lie structures on the incidence algebra $I(X,K)$ of a finite connected poset $X$ over a field $K$. We prove that such a structure is the sum of a central-valued linear map annihilating the Jacobson radical of $I(X,K)$ with the composition of certain inner and multiplicative automorphisms of $I(X,K)$.