Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userZeta functions for tensor products of locally coprime integral adjacency algebras of association schemes
247
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The zeta function of an integral lattice $Lambda$ is the generating function $zeta_{Lambda}(s) = sumlimits_{n=0}^{infty} a_n n^{-s}$, whose coefficients count the number of left ideals of $Lambda$ of index $n$. We derive a formula for the zeta function of $Lambda_1 otimes Lambda_2$, where $Lambda_1$ and $Lambda_2$ are $mathbb{Z}$-orders contained in finite-dimensional semisimple $mathbb{Q}$-algebras that satisfy a locally coprime condition. We apply the formula obtained above to $mathbb{Z}S otimes mathbb{Z}T$ and obtain the zeta function of the adjacency algebra of the direct product of two finite association schemes $(X,S)$ and $(Y,T)$ in several cases where the $mathbb{Z}$-orders $mathbb{Z}S$ and $mathbb{Z}T$ are locally coprime and their zeta functions are known.
rate research
Read More
The extending structures and unified products for Malcev algebras are developed. Some special cases of unified products such as crossed products and matched pair of Malcev algebras are studied. It is proved that the extending structures can be classified by some non-abelian cohomology theory. One dimensional flag extending structures of Malcev algebras are also investigated.
In this paper we aim to characterize association schemes all of whose symmetric fusion schemes have only integral eigenvalues, and classify those obtained from a regular action of a finite group by taking its orbitals.
We show that the tensor product of two cyclic $A_infty$-algebras is, in general, not a cyclic $A_infty$-algebra, but an $A_infty$-algebra with homotopy inner product. More precisely, we construct an explicit combinatorial diagonal on the pairahedra, which are contractible polytopes controlling the combinatorial structure of an $A_infty$-algebra with homotopy inner products, and use it to define a categorically closed tensor product. A cyclic $A_infty$-algebra can be thought of as an $A_infty$-algebra with homotopy inner products whose higher inner products are trivial. However, the higher inner products on the tensor product of cyclic $A_infty$-algebras are not necessarily trivial.
In [19] it was explained how one can naturally associate a Deitmar scheme (which is a scheme defined over the field with one element, $mathbb{F}_1$) to a so-called loose graph (which is a generalization of a graph). Several properties of the Deitmar scheme can be proven easily from the combinatorics of the (loose) graph, and known realizations of objects over $mathbb{F}_1$ such as combinatorial $mathbb{F}_1$-projective and $mathbb{F}_1$-affine spaces exactly depict the loose graph which corresponds to the associated Deitmar scheme. In this paper, we first modify the construction of loc. cit., and show that Deitmar schemes which are defined by finite trees (with possible end points) are defined over $mathbb{F}_1$ in Kurokawas sense; we then derive a precise formula for the Kurokawa zeta function for such schemes (and so also for the counting polynomial of all associated $mathbb{F}_q$-schemes). As a corollary, we find a zeta function for all such trees which contains information such as the number of inner points and the spectrum of degrees, and which is thus very different than Iharas zeta function (which is trivial in this case). Using a process called surgery, we show that one can determine the zeta function of a general loose graph and its associated { Deitmar, Grothendieck }-schemes in a number of steps, eventually reducing the calculation essentially to trees. We study a number of classes of examples of loose graphs, and introduce the Grothendieck ring of $mathbb{F}_1$-schemes along the way in order to perform the calculations. Finally, we compare the new zeta function to Iharas zeta function for graphs in a number of examples, and include a computer program for performing more tedious calculations.
In this paper, we establish Composition-Diamond lemma for tensor product $k< X> otimes k< Y>$ of two free algebras over a field. As an application, we construct a Groebner-Shirshov basis in $k< X> otimes k< Y>$ by lifting a Groebner-Shirshov basis in $k[X] otimes k< Y>$, where $k[X]$ is a commutative algebra.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
