We define and study the category Rep(Q, F1) of representations of a quiver in Vect(F1) the category of vector spaces ”over F1”. Rep(Q, F1) is an F1–linear category possessing kernels, co-kernels, and direct sums. Moreover, Rep(Q, F1) satisfies analogues of the Jordan-Hölder and Krull-Schmidt theorems. We are thus able to define the Hall algebra HQ of Rep(Q, F1), which behaves in some ways like ...

Motivated by construction of isospin generators in particle physics (built form the SU(2) algebra), we find an equivalence between the algebra of these generators and those of the Virasoro algebra. The form of the starting generators is fixed and in order to obtain a full equivalence we introduce a new matrix product. The Cartan structure of the starting algebra is reproduced for the Virasoro-l...

Let a ⊕ b = max(a, b), a ⊗ b = a + b for a, b ∈ R := R ∪ {−∞}. By max-algebra we understand the analogue of linear algebra developed for the pair of operations (⊕,⊗) extended to matrices and vectors. Max-algebra, which has been studied for more than 40 years, is an attractive way of describing a class of nonlinear problems appearing for instance in machine-scheduling, information technology and...

The rank two Heisenberg–Virasoro algebra can be viewed as a generalization of the twisted algebra. Lie bialgebras play an important role in searching for solutions quantum Yang–Baxter equations. It is interesting to study bialgebra structures on Since brackets are different from that and Virasoro-like algebras, there inner derivations (from itself its tensor space) which hidden more deeply inte...

In this paper, we introduce the notions of expansion of ideals in $MV$-algebras, $ (tau,sigma)- $primary, $ (tau,sigma)$-obstinate  and $ (tau,sigma)$-Boolean  in $ MV- $algebras. We investigate the relations of them. For example, we show that every $ (tau,sigma)$-obstinate ideal of an $ MV-$ algebra is $ (tau,sigma)$-primary  and $ (tau,sigma)$-Boolean. In particular, we define an expansion $ ...

The division algebra of real quaternions, as the only noncommutative normed division real algebra up to isomorphism of normed algebras, is of great importance. In this note, first we present a brief introduction to quaternion matrices and quaternion linear algebra. This, among other things, will help us present the counterpart of a theorem of Herman Auerbach in the setting of quaternions. More ...

In this paper we brieƒy discuss Rings — an ecient lightweight library for univariate and multivariate polynomial arithmetic over arbitrary coecient rings. Basic algebra, GCDs and factorization of polynomials are implemented with the use of modern asymptotically fast algorithms. Rings provides a clean API for algebra and a fully typed hierarchy of mathematical structures. Scala API additionall...

After dropping the implication-like operation and reversing the order, an MTL-algebra becomes a partially ordered structure (L;≤,⊕, 0, 1) based on the single addition-like operation ⊕. Furthermore, ⊕ may be restricted to a partial, but cancellative addition + without loss of information. We deal in this paper with the case that the resulting partial algebra (L;≤,+, 0, 1) is embeddable into the ...

1. Geometry of primitive ideals. 1.1. Let G be a complex algebraic group with Lie algebra g. I would like to have a geometric picture for the structure of primitive ideals of the enveloping algebra U(g). The only universal answer known so far is the "orbit method," saying that primitive ideals should correspond to coadjoint orbits in the dual g* of g. Sometimes this principle doesn't work, howe...

This can of course be solved using algebra. The rst treatise on algebra was written by Diophantus of Alexandria in the 3rd century AD. The term derives from the Arabic al-jabr or literally \the reunion of broken parts." As well as its mathematical meaning, the word also means the surgical treatment of fractures. It gained widespread use through the title of a book ilm al-jabr wa'l-mukabala|the ...