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Register a new userDetermining when an algebra is an evolution algebra
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Evolution algebras are non-associative algebras that describe non-Mendelian hereditary processes and have connections with many other areas. In this paper we obtain necessary and sufficient conditions for a given algebra $A$ to be an evolution algebra. We prove that the problem is equivalent to the so-called $SDC$ $problem$, that is, the $simultaneous$ $diagonalisation$ $via$ $congruence$ of a given set of matrices. More precisely we show that an $n$-dimensional algebra $A$ is an evolution algebra if, and only if, a certain set of $n$ symmetric $ntimes n$ matrices ${M_{1}, ldots, M_{n}}$ describing the product of $A$ are $SDC$. We apply this characterisation to show that while certain classical genetic algebras (representing Mendelian and auto-tetraploid inheritance) are not themselves evolution algebras, arbitrarily small perturbations of these are evolution algebras. This is intriguing as evolution algebras model asexual reproduction unlike the classical ones.
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Let $f$ be a polynomial in the free algebra over a field $K$, and let $A$ be a $K$-algebra. We denote by $S_A(f)$, $A_A(f)$ and $I_A(f)$, respectively, the `verbal subspace, subalgebra, and ideal, in $A$, generated by the set of all $f$-values in $A$. We begin by studying the following problem: if $S_A(f)$ is finite-dimensional, is it true that $A_A(f)$ and $I_A(f)$ are also finite-dimensional? We then consider the dual to this problem for `marginal subspaces that are finite-codimensional in $A$. If $f$ is multilinear, the marginal subspace, $widehat{S}_A(f)$, of $f$ in $A$ is the set of all elements $z$ in $A$ such that $f$ evaluates to 0 whenever any of the indeterminates in $f$ is evaluated to $z$. We conclude by discussing the relationship between the finite-dimensionality of $S_A(f)$ and the finite-codimensionality of $widehat{S}_A(f)$.
Let A be an associative algebra with identity over a field k. An atomistic subsemiring R of the lattice of subspaces of A, endowed with the natural product, is a subsemiring which is a closed atomistic sublattice. When R has no zero divisors, the set of atoms of R is endowed with a multivalued product. We introduce an equivalence relation on the set of atoms such that the quotient set with the induced product is a monoid, called the condensation monoid. Under suitable hypotheses on R, we show that this monoid is a group and the class of k1_A is the set of atoms of a subalgebra of A called the focal subalgebra. This construction can be iterated to obtain higher condensation groups and focal subalgebras. We apply these results to G-algebras for G a group; in particular, we use them to define new invariants for finite-dimensional irreducible projective representations.
A paper of U. First & Z. Reichstein proves that if $R$ is a commutative ring of dimension $d$, then any Azumaya algebra $A$ over $R$ can be generated as an algebra by $d+2$ elements, by constructing such a generating set, but they do not prove that this number of generators is required, or even that for an arbitrarily large $r$ that there exists an Azumaya algebra requiring $r$ generators. In this paper, for any given fixed $nge 2$, we produce examples of a base ring $R$ of dimension $d$ and an Azumaya algebra of degree $n$ over $R$ that requires $r(d,n) = lfloor frac{d}{2n-2} rfloor + 2$ generators. While $r(d,n) < d+2$ in general, we at least show that there is no uniform upper bound on the number of generators required for Azumaya algebras. The method of proof is to consider certain varieties $B^r_n$ that are universal varieties for degree-$n$ Azumaya algebras equipped with a set of $r$ generators, and specifically we show that a natural map on Chow group $CH^{(r-1)(n-1)}_{PGL_n} to CH^{(r-1)(n-1)}(B^r_n)$ fails to be injective, which is to say that the map fails to be injective in the first dimension in which it possibly could fail. This implies that for a sufficiently generic rank-$n$ Azumaya algebra, there is a characteristic class obstruction to generation by $r$ elements.
The present article is a part of the study of solvable Leibniz algebras with a given nilradical. In this paper solvable Leibniz algebras, whose nilradicals is naturally graded quasi-filiform algebra and the complemented space to the nilradical has maximal dimension, are described up to isomorphism.
This paper studies commuting matrices in max algebra and nonnegative linear algebra. Our starting point is the existence of a common eigenvector, which directly leads to max analogues of some classical results for complex matrices. We also investigate Frobenius normal forms of commuting matrices, particularly when the Perron roots of the components are distinct. For the case of max algebra, we show how the intersection of eigencones of commuting matrices can be described, and we consider connections with Boolean algebra which enables us to prove that two commuting irreducible matrices in max algebra have a common eigennode.
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