We prove a noncompact Serre-Swan theorem characterising modules which are sections of vector bundles not necessarily trivial at infinity. We then identify the endomorphism algebras of the resulting modules. The endomorphism results continue to hold for the generalisation of these modules to noncommutative, nonunital algebras. Finally, we apply these results to not necessarily compact noncommuta...

For any equivalence relation ≡ on positive integers such that nk ≡ mk if and only if n ≡ m, there is an abelian group G such that the endomorphism ring of Gn and Gm are isomorphic if and only if n ≡ m. However, Gn and Gm are not isomorphic if n = m.

Here we demonstrate the emergence of Grassmann variables in matrix models based on the exceptional Jordan algebra. The Grassmann algebras are built naturally using the octonion algebra. We argue the appearance of Grassmann variables solidifies the relationship between supersymmetry and triality.

We prove, for each 4 ≤ n < ω, that SRaCAn+1 cannot be defined, using only finitely many first-order axioms, relative to SRaCAn. The construction also shows that for 5 ≤ n < ω, SRaCAn is not finitely axiomatisable over RAn, and that for 3 ≤ m < n < ω, SNrmCAn+1 is not finitely axiomatisable over SNrmCAn. In consequence, for a certain standard n-variable first-order proof system `m,n of m-variabl...

If a vertex operator algebra V = ⊕n=0Vn satisfies dimV0 = 1, V1 = 0, then V2 has a commutative (nonassociative) algebra structure called Griess algebra. One of the typical examples of commutative (nonassociative) algebras is a Jordan algebra. For example, the set Symd(C) of symmetric matrices of degree d becomes a Jordan algebra. On the other hand, in the theory of vertex operator algebras, cen...

The Terwilliger algebra plays an important role in the theory of association schemes. The present paper gives the explicit structures of the Terwilliger algebras of the group association schemes of the nite groups PSL(2, 7), A6, and S6.

In the paper, we shall examine the fuzziness measure (FM) of terms or of complete and linear hedge algebras of a linguistic variable. The notion of semantically quantifying mappings (SQMs) previously examined by the first author will be redefined more generally and a closed relation between the FM of linguistic terms and a family of SQMs with the parameters to be the FM of primary terms and lin...

The simple modules with character height at most one for the restricted Witt algebras are considered. Their classification, construction, and dimension formulas are reduced to the same for the general linear algebra. Results of Chang and Shen are recovered in the process.

We construct Nijenhuis operators from particular bialgebras called dendriformNijenhuis bialgebras. It turns out that such Nijenhuis operators commute with TD-operators, kind of Baxter-Rota operators, and therefore closely related dendriform trialgebras. This allows the construction of associative algebras, called dendriform-Nijenhuis algebras made out with nine operations and presenting an exot...

Casimir W-algebras are shown to be exist in such a way that the conformal spins of primary (generating) fields coincide with the orders of independent Casimir operators. We show here that this coincidence can be extended further to the case that these generating fields have the same eigenvalues with the Casimir operators.