The relevance of Freiman’s theorem for combinatorial commutative algebra

Mathematische Zeitschrift - Freiman’s theorem gives a lower bound for the cardinality of the doubling of a finite set in $${mathbb R}^n$$ . In this paper we give an interpretation of his...     

Algorithmic complexity of Greenberg’s conjecture

Archiv der Mathematik - Let k be a totally real number field and p a prime. We show that the “complexity” of Greenberg’s conjecture ( $$\lambda = \mu = 0$$ ) is governed (under...     

A commutant realization of Odake’s algebra

The bcβγ-system $ \mathcal{W} $ of rank 3 has an action of the affine vertex algebra $ {V_0}\left( {\mathfrak{s}{{\mathfrak{l}}_2}} \right) $ , and the commutant vertex algebra $ \mathcal{C}=\mathrm{Com}\left( {{V_0}\left( {\mathfrak{s}{{\mathfrak{l}}_2}} \right),\mathcal{W}} \right) $ contains copies of V _?3/2 $ \left( {\mathfrak{s}{{\mathfrak{l}}_2}} \right) $ and Odake’s algebra $ \mathcal{O} $ . Odake’s algebra is an extension of the N = 2 super-conformal algebra with c = 9, and is generated by eight fields which close nonlinearly under operator product expansions. Our main result is that V _?3/2 $ \left( {\mathfrak{s}{{\mathfrak{l}}_2}} \right) $ and $ \mathcal{O} $ form a Howe pair (i.e., a pair of mutual commutants) inside $ \mathcal{C} $ . More generally, any finite-dimensional representation of a Lie algebra $ \mathfrak{g} $ gives rise to a similar Howe pair, and this example corresponds to the adjoint representation of $ \mathfrak{s}{{\mathfrak{l}}_2} $ .     

A weighted algebra analogues of Wiener’s and Lévy’s theorems

We obtain weighted algebra analogues of the classical theorems of Weiner and Lévy on absolutely convergent Fourier series.     

Einstein’s equations and Clifford algebra

Einstein’s equations of the general theory of relativity are rewritten within a Clifford algebra. This algebra is otherwise isomorphic to a direct product of two quaternion algebras. A multivector calculus is developed within this Clifford algebra which differs from the corresponding complexified algebra used in the standard spacetime algebra approach.     

Nonlinear algebra and Bogoliubov’s recursion

We give many examples of applying Bogoliubov’s forest formula to iterative solutions of various nonlinear equations. The same formula describes an extremely wide class of objects, from an ordinary quadratic equation to renormalization in quantum field theory. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 2, pp. 316–343, February, 2008.     

A note on Bridgeland’s Hall algebra of two-periodic complexes

We show that the Hall algebra of two-periodic complexes, which is recently introduced by T. Bridgeland, coincides with the Drinfeld double of the ordinary Hall bialgebra.     

Lyndon’s groupoid generates a small almost Cross variety

Lyndon’s groupoid of order seven is the first published example of a non-finitely based finite algebra. The main objective of the present article is to investigate the variety generated by this groupoid and its subvarieties. It is shown that the subvarieties of form a chain of order five, all elements of which except are Cross varieties. It follows that the variety is also generated by a groupoid of order six and that any groupoid with five or fewer elements does not generate . Consequently, Lyndon’s example of a non-finitely based finite algebra could have been of order six instead of seven. It is also shown that, with respect to some important properties, Lyndon’s groupoid contrasts greatly with several well-known non-finitely based finite groupoids that were discovered shortly after its publication. Presented by R. Freese.     

Subword complexity of the Fibonacci–Thue–Morse sequence: The proof of Dekking’s conjecture

Recently F. M. Dekking conjectured the form of the subword complexity function for the Fibonacci–Thue–Morse sequence. In this note we prove his conjecture by purely computational means, using the free software Walnut.     

Witt’s theorem in abstract geometric algebra

In an earlier paper of the author, a version of the Witt’s theorem was obtained within a specific subcategory of the category of A{mathcal A}-modules: the full subcategory of convenient A{mathcal A}-modules. A further investigation yields two more versions of the Witt’s theorem by revising the notion of convenient A{mathcal A}-modules. For the first version, the A{mathcal A}-bilinear form involved is either symmetric or antisymmetric, and the two isometric free sub-A{mathcal A}-modules, the isometry between which may extend to an isometry of the non-isotropic convenient A{mathcal A}-module concerned onto itself, are assumed pre-hyperbolic. On the other hand, for the second version, the A{mathcal A}-bilinear form defined on the non-isotropic convenient A{mathcal A}-module involved is set to be symmetric, and the two isometric free sub-A{mathcal A}-modules, whose orthogonals are to be proved isometric, are assumed strongly non-isotropic and disjoint.     

Takeuchi’s free Hopf algebra construction revisited

Takeuchi’s famous free Hopf algebra construction is analyzed from a categorical point of view, and so is the construction of the Hopf envelope of a bialgebra. Both constructions in fact can be described as compositions of well known and natural constructions. This way certain partially wrong perceptions of these constructions are clarified and their mutual relation is made precise. The construction of Hopf envelopes finally is shown to provide a construction of a Hopf coreflection of bialgebras by simple dualization. The results provided hold for any commutative von Neumann regular ring, not only for fields.     

Gosset’s figure in a Clifford algebra

This note describes a way to realize a “projective” version of Gosset’s 240-vertex semiregular polytope 421 using the Clifford algebra Cl(8) generated by an 8-dimensional vector space equipped with a non-degenerate quadratic form. The 120 vertices of this projective Gosset figure are also seen to coincide with a particular basis for the Lie algebra      

On the complexity of the closed fragment of Japaridze’s provability logic

We consider the well-known provability logic GLP. We prove that the GLP-provability problem for polymodal formulas without variables is PSPACE-complete. For a number n, let \({L^{n}_0}\) denote the class of all polymodal variable-free formulas without modalities \({\langle n \rangle,\langle n+1\rangle,...}\) . We show that, for every number n, the GLP-provability problem for formulas from \({L^{n}_0}\) is in PTIME.     

The converse of Schur’s theorem

Schur’s theorem states that for a group G finiteness of G/Z(G) implies the finiteness of G′. In this paper, we show the converse is true provided that G/Z(G) is finitely generated and in such case, we have |G/Z(G)| ≤ |G′| ^d(G/Z(G)). In the special case of G being nilpotent, we prove |G/Z(G)| divides |G′| ^d(G/Z(G)).     

The automorphisms of Petit’s algebras

Let σ be an automorphism of a field K with fixed field F. We study the automorphisms of nonassociative unital algebras which are canonical generalizations of the associative quotient algebras K[t;σ]∕fK[t;σ] obtained when the twisted polynomial f∈K[t;σ] is invariant, and were first defined by Petit. We compute all their automorphisms if σ commutes with all automorphisms in Aut_F(K) and n≥m?1, where n is the order of σ and m the degree of f, and obtain partial results for n<m?1. In the case where K∕F is a finite Galois field extension, we obtain more detailed information on the structure of the automorphism groups of these nonassociative unital algebras over F. We also briefly investigate when two such algebras are isomorphic.