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Register a new userOrders and Polytropes: Matrix Algebras from Valuations
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We apply tropical geometry to study matrix algebras over a field with valuation. Using the shapes of min-max convexity, known as polytropes, we revisit the graduated orders introduced by Plesken and Zassenhaus. These are classified by the polytrope region. We advance the ideal theory of graduated orders by introducing their ideal class polytropes. This article emphasizes examples and computations. It offers first steps in the geometric combinatorics of endomorphism rings of configurations in affine buildings.
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The question of embedding fields into central simple algebras $B$ over a number field $K$ was the realm of class field theory. The subject of embedding orders contained in the ring of integers of maximal subfields $L$ of such an algebra into orders in that algebra is more nuanced. The first such result along those lines is an elegant result of Chevalley cite{Chevalley-book} which says that with $B = M_n(K)$ the ratio of the number of isomorphism classes of maximal orders in $B$ into which the ring of integers of $L$ can be embedded (to the total number of classes) is $[L cap widetilde K : K]^{-1}$ where $widetilde K$ is the Hilbert class field of $K$. Chinburg and Friedman (cite{Chinburg-Friedman}) consider arbitrary quadratic orders in quaternion algebras satisfying the Eichler condition, and Arenas-Carmona cite{Arenas-Carmona} considers embeddings of the ring of integers into maximal orders in a broad class of higher rank central simple algebras. In this paper, we consider central simple algebras of dimension $p^2$, $p$ an odd prime, and we show that arbitrary commutative orders in a degree $p$ extension of $K$, embed into none, all or exactly one out of $p$ isomorphism classes of maximal orders. Those commutative orders which are selective in this sense are explicitly characterized; class fields play a pivotal role. A crucial ingredient of Chinberg and Friedmans argument was the structure of the tree of maximal orders for $SL_2$ over a local field. In this work, we generalize Chinburg and Friedmans results replacing the tree by the Bruhat-Tits building for $SL_p$.
We show that there is a family of complex semisimple Hopf algebras that do not admit a Hopf order over any number ring. They are Drinfeld twists of certain group algebras. The twist contains a scalar fraction which makes impossible the definability of such Hopf algebras over number rings. We also prove that a complex semisimple Hopf algebra satisfies Kaplanskys sixth conjecture if and only if it admits a weak order, in the sense of Rumynin and Lorenz, over the integers.
In this paper we give the relationship between the connected components of pure generalized Dynkin graphs and Nichols braided Lie algebras.
Algebras generated by strictly positive matrices are described up to similarity, including the commutative, simple, and semisimple cases. We provide sufficient conditions for some block diagonal matrix algebras to be generated by a set of nonnegative matrices up to similarity. Also we find all realizable dimensions of algebras generated by two nonnegative semi-commuting matrices. The last result provides the solution to the problem posed by M. Kandi{c}, K. v{S}ivic (2017).
We study the relation between algebraic structures and Graph Theory. We have defined five different weighted digraphs associated to a finite dimensional algebra over a field in order to tackle important properties of the associated algebras, mainly the nilpotency and solvability in the case of Leibniz algebras.
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