We prove that the basis of cluster monomials of a skew-symmetric cluster algebra A of finite type is the atomic basis of A. This means that an element of A is positive if and only if it has a non-negative expansion in the basis of cluster monomials. In particular cluster monomials are positive indecomposable, i.e. they cannot be written as a sum of positive elements.
In this paper, we build the unfolding approach from acyclic sign-skew-symmetric matrices of finite rank to skew-symmetric matrices of infinite rank, which can be regard as an improvement of that in the skew-symmetrizable case. Using this approach, we give a positive answer to the problem by Berenstein, Fomin and Zelevinsky in \cite{fz3} which asks whether an acyclic sign-skew-symmetric matrix is always totally sign-skew-symmetric. As applications, the positivity for cluster algebras in acyclic sign-skew-symmetric case is given; further, the $F$-polynomials of cluster algebras are proved to hav…
It has been proved in \cite{LS} that cluster variables in cluster algebras of every skew-symmetric cluster algebra are positive. We prove that any regular generalized cluster variable of an affine quiver is positive. As a corollary, we obtain that generalized cluster variables of affine quivers are positive and construct various positive bases. This generalizes the results in \cite{Dupont} and \cite{DXX}.
In this paper we construct an unfolding for $c-$vectors of acyclic sign-skew symmetric matrices and we also prove that the sign-coherence property holds for acyclic sign-skew-symmetric matrices. Then we prove that every acyclic sign-skew-symmetric matrix admits a maximal green sequence.
The product of a complex skew-symmetric matrix and its conjugate transpose is a positive semi-definite Hermitian matrix with nonnegative eigenvalues, with a property that each distinct positive eigenvalue has even multiplicity. This property plays a key role for Professor Loo-Keng Hua to establish the unitary equivalence theorem for complex skew-symmetric matrices. We show that this property is no longer true for quaternion skew-symmetric matrices. This poses a difficulty for finding the canonical form of a quaternion skew-symmetric matrix under unitary equivalence, which may be crucial for th…
An algebra is called skew-symmetric if its multiplication operation is a skew-symmetric bilinear application. We determine all these algebras in dimension $3$ over a field of characteristic different from $2$. As an application, we determine the subvariety of $3$-dimensional Hom-Lie algebras. For this type of algebras, we study also the dimension $4$.
This paper describes the structure of invariant skew fields for linear actions of finite solvable groups on free skew fields in $d$ generators. These invariant skew fields are always finitely generated, which contrasts with the free algebra case. For abelian groups or solvable groups $G$ with a well-behaved representation theory it is shown that the invariant skew fields are free on $|G|(d-1)+1$ generators. Finally, positivity certificates for invariant rational functions in terms of sums of squares of invariants are presented.
Let $L$ be a restricted Lie algebra over a field of characteristic $p>2$ and denote by $u(L)$ its restricted enveloping algebra. We establish when the symmetric or skew elements of $u(L)$ under the principal involution are Lie metabelian.
We show that the set of $m \times m$ complex skew-symmetric matrix polynomials of odd grade $d$, i.e., of degree at most $d$, and (normal) rank at most $2r$ is the closure of the single set of matrix polynomials with the certain, explicitly described, complete eigenstructure. This complete eigenstructure corresponds to the most generic $m \times m$ complex skew-symmetric matrix polynomials of odd grade $d$ and rank at most $2r$. In particular, this result includes the case of skew-symmetric matrix pencils ($d=1$).
Using the unfolding method given in \cite{HL}, we prove the conjectures on sign-coherence and a recurrence formula respectively of ${\bf g}$-vectors for acyclic sign-skew-symmetric cluster algebras. As a following consequence, the conjecture is affirmed in the same case which states that the ${\bf g}$-vectors of any cluster form a basis of $\mathbb Z^n$. Also, the additive categorification of an acyclic sign-skew-symmetric cluster algebra $\mathcal A(Σ)$ is given, which is realized as $(\mathcal C^{\widetilde Q},Γ)$ for a Frobenius $2$-Calabi-Yau category $\mathcal C^{\widetilde Q}$ constructe…
Let $L$ be a simple anti-commutative algebra. In this paper we prove that a non skew-symmetric solution of the classical Yang-Baxter equation on $L$ with $L$-invariant symmetric part induces on $L$ a Rota-Baxter operator of a non-zero weight.
Given a bilinear form on $\mathbb C^n$, represented by a matrix $A\in\mathbb C^{n\times n}$, the problem of finding the largest dimension of a subspace of $\mathbb C^n$ such that the restriction of $A$ to this subspace is a non-degenerate skew-symmetric bilinear form is equivalent to finding the size of the largest invertible skew-symmetric matrix $B$ such that the equation $X^\top AX=B$ is consistent (here $X^\top$ denotes the transpose of the matrix $X$). In this paper, we provide a characterization, by means of a necessary and sufficient condition, for the matrix equation $X^\top AX=B$ to b…
We study possible connections between Rota-Baxter operators of non-zero weight and non-skew-symmetric solutions of the classical Yang-Baxter equation on finite-dimensional quadratic Lie algebras. The particular attention is made to the case when for a solution $r$ the element $r+τ(r)$ is $L$-invariant.
Let $RG$ be the group ring of a finite group $G$ over a commutative ring $R$ with $1$. An element $x$ in $RG$ is said to be skew-symmetric with respect to an involution $σ$ of $RG$ if $σ(x)=-x.$ A structure theorem for the skew-symmetric elements of $FG$ is given where $F$ is an algebraic extension of $\mathbb{Q}$ which generalizes some previously known results in this direction.
We prove that for every natural number n there exists a natural number N(n) such that every multilinear skew-symmetric polynomial on N(n) or more variables which vanishes in the free associative algebra vanishes as well in any n-generated alternative algebra over a field of characteristic 0. Before this was proved only for a series of such polynomials constructed by the author in [7].
We introduce a class of rings using which we define the concept of skew regularity for quaternion-valued functions over quaternions. It is shown that the notion of skew regularity coincides with the concept of slice regularity over symmetric slice domains. Known results regarding slice-regular functions over symmetric slice domains are generalized to skew-regular functions over general symmetric domains. Furthermore, we present new results concerning slice-regular functions.
For a field $F$ of characteristic not 2 and a directed row-finite graph $Γ$ let $L(Γ)$ be the Leavitt path algebra with the standard involution $*.$ We study the Lie algebra $K=K(L(Γ),*)$ of $*-$skew-symmetric elements and find necessary and sufficient conditions for the Lie algebra $[K,K]$ to be simple.
For a skew left brace $(G, \cdot, \circ)$, the map $λ: (G, \circ) \to \Aut \,(G, \cdot),~~a \mapsto λ_a,$ where $λ_a(b) = a^{-1} \cdot (a \circ b)$ for all $a, b \in G$, is a group homomorphism. Then $λ$ can also be viewed as a map from $(G, \cdot)$ to $\Aut \, (G, \cdot)$, which, in general, may not be a homomorphism. A skew left brace will be called $λ$-anti-homomorphic ($λ$-homomorphic) if $λ: (G, \cdot) \to \Aut \, (G, \cdot)$ is an anti-homomorphism (a homomorphism). We mainly study such skew left braces. We device a method for constructing a class of binary operations on a given set so t…
We use Knuth's combinatorial approach to Pfaffians to reprove and clarify a century-old formula, due to Brill. It expresses arbitrary minors of a skew symmetric matrix in terms of Pfaffians.
Denote $Σ_n$ and $Q_n$ the set of all $n \times n$ symmetric and skew-symmetric matrices over a field $\mathbb{F}$, respectively, where $char(\mathbb{F})\neq 2$ and $\lvert \mathbb{F} \rvert \geq n^2+1$. A characterization of $ϕ,ψ:Σ_n \rightarrow Σ_n$, for which at least one of them is surjective, satisfying $$\det(ϕ(x)+ψ(y))=\det(x+y)\qquad(x,y\in Σ_n)$$ is given. Furthermore, if $n$ is even and $ϕ,ψ:Q_n \rightarrow Q_n$, for which $ψ$ is surjective and $ψ(0)=0$, satisfy $$\det(ϕ(x)+ψ(y))=\det(x+y)\qquad(x,y\in Q_n),$$ then $ϕ=ψ$ and $ψ$ must be a bijective linear map preserving the determina…
Symmetric rings were introduced by Lambek to extend usual commutative ideal theory in noncommutative rings. In this paper, we study symmetric rings over which Ore extensions are symmetric. A ring R is called strongly σ-symmetric if the skew polynomial ring R[x;σ] is symmetric. We consider some properties and extensions of strongly σ-symmetric rings. Then we show the relationship between strongly σ-symmetric rings and other classes of rings. We next argue the polynomial extensions over strongly σ-symmetric rings. Moreover, we prove that if R is a σ-rigid ring, then R[x]/(x_n) is a strongly σ-sy…
In this paper, which is a follow-up to [A. Borobia, R. Canogar, F. De Terán, Mediterr. J. Math. 18, 40 (2021)], we provide a necessary and sufficient condition for the matrix equation $X^\top AX=B$ to be consistent when $B$ is symmetric. The condition depends on the canonical form for congruence of the matrix $A$, and is proved to be necessary for all matrices $A$, and sufficient for most of them. This result improves the main one in the previous paper, since the condition is stronger than the one in that reference, and the sufficiency is guaranteed for a larger set of matrices (namely, those …
