Nontrivial Homomorphism Research Articles

Quasiregular curves and cohomology

AbstractLet be a closed, connected, and oriented Riemannian manifold, which admits a quasiregular ‐curve with infinite energy. We prove that, if the de Rham class of is nonzero and a finite sum of nontrivial products, then there exists a nontrivial graded algebra homomorphism from the de Rham algebra of to the exterior algebra . As an application, we give examples of pairs , where is a closed manifold and is a closed ‐form for , for which every quasiregular ‐curve is constant.

The Three Faces of U(3)

U(n) is a semi-direct product group characterized by nontrivial homomorphisms mapping U(1) into the automorphism group of SU(n). For U(3), there are three nontrivial homomorphisms that induce three separate defining representations. In a toy model of U(3) Yang–Mills (endowed with a suitable inner product) coupled to massive fermions, this renders three distinct covariant derivatives acting on a single matter field. Employing a mod3 permutation induced by a large gauge transformation acting on the defining representation vector space, the three covariant derivatives and one matter field can alternatively be expressed as a single covariant derivative acting on three distinct species of matter fields possessing the same U(3) quantum numbers. One can interpret this as three species of matter fields in the defining representation.

Inverse homomorphisms of finite groups

Abstract Replacing the equality in the condition of the homomorphism of a multi-based universal algebra with inequality leads to the definition of an inverse homomorphism of a multi-based universal algebra. In this paper methods for constructing nontrivial inverse homomorphisms of finite groups are studied.

Guay's affine Yangians and non-rectangular W-algebras

We construct a non-trivial homomorphism from the Guay's affine Yangian to the universal enveloping algebra of non-rectangular W-algebras of type A. In order to construct the homomorphism, we extend the Guay's affine Yangian and its coproduct.

Decomposition of polynomials using the shift-composition operation

Abstract V. I. Solodovnikov had employed the shift-composition operation to investigate homomorphisms of shift registers into linear automata; in his papers, conditions for the absence of nontrivial inner homomorphisms of shift registers were derived. An essential role was played by the condition of linearity of the left component of the shift-composition operation in the corresponding polynomial decomposition. In this paper we consider the case where the left component is a function belonging to a wider class, which includes the class of linear functions.

Determinantal Ideals and the Canonical Commutation Relations: Classically or Quantized

We construct homomorphic images of $$su(n,n)^{{\mathbb {C}}}$$ in Weyl Algebras $${{\mathcal {H}}}_{2nr}$$ . More precisely, and using the Bernstein filtration of $${{\mathcal {H}}}_{2nr}$$ , $$su(n,n)^{{\mathbb {C}}}$$ is mapped into degree 2 elements with the negative non-compact root spaces being mapped into second order creation operators. Using the Fock representation of $${{\mathcal {H}}}_{2nr}$$ , these homomorphisms give all unitary highest weight representations of $$su(n,n)^{{\mathbb {C}}}$$ thus reconstructing the Kashiwara–Vergne List for the Segal–Shale–Weil representation. Using an idea from the derivation of the their list, we construct a homomorphism of $$u(r)^{{\mathbb {C}}}$$ into $${{\mathcal {H}}}_{2nr}$$ whose image commutes with the image of $$su(n,n)^{{\mathbb {C}}}$$ , and vice versa. This gives the multiplicities. The construction also gives an easy proof that the ideal of $$(r+1)\times (r+1)$$ minors is prime. Here, of course, $$r\le n-1$$ and for a fixed such r, the space of any irreducible representation of $$su(n,n)^{{\mathbb {C}}}$$ is annihilated by this ideal. As a consequence, these representations can be realized in spaces of solutions to Maxwell type equations. We actually go one step further and determine exactly for which representations from our list there is a non-trivial homomorphism between generalized Verma modules, thereby revealing, by duality, exactly which covariant differential operators have unitary null spaces. We prove the analogous results for $${{\mathcal {U}}}_q(su(n,n)^{{\mathbb {C}}})$$ . The Weyl Algebras are replaced by the Hayashi–Weyl Algebras $${{\mathcal {H}}}{{\mathcal {W}}}_{2nr}$$ and the Fock space by a q-Fock space. Further, determinants are replaced by q-determinants, and a homomorphism of $${{\mathcal {U}}}_q(u(r)^{{\mathbb {C}}})$$ into $${{\mathcal {H}}}{{\mathcal {W}}}_{2nr}$$ is constructed with analogous properties. For this purpose a Drinfeld Double is used. We mention one difference: The quantized negative non-compact root spaces, while still of degree 2, are no longer given entirely by second order creation operators.

Homomorphisms of commutator subgroups of braid groups

We give a complete classification of homomorphisms from the commutator subgroup of the braid group on $n$ strands to the braid group on $n$ strands when $n$ is at least 7. In particular, we show that each nontrivial homomorphism extends to an automorphism of the braid group on $n$ strands. This answers four questions of Vladimir Lin. Our main new tool is the theory of totally symmetric sets.

On the existence and non-existence of improper homomorphisms of oriented and $2$-edge-coloured graphs to reflexive targets

We consider non-trivial homomorphisms to reflexive oriented graphs in which some pair of adjacent vertices have the same image. Using a notion of convexity for oriented graphs, we study those oriented graphs that do not admit such homomorphisms. We fully classify those oriented graphs with tree-width $2$ that do not admit such homomorphisms and show that it is NP-complete to decide if a graph admits an orientation that does not admit such homomorphisms. We prove analogous results for $2$-edge-coloured graphs. We apply our results on oriented graphs to provide a new tool in the study of chromatic number of orientations of planar graphs -- a long-standing open problem.

The large cardinal strength of weak Vopenka’s principle

We show that Weak Vopěnka’s Principle, which is the statement that the opposite category of ordinals cannot be fully embedded into the category of graphs, is equivalent to the large cardinal principle Ord is Woodin, which says that for every class [Formula: see text] there is a [Formula: see text]-strong cardinal. Weak Vopěnka’s Principle was already known to imply the existence of a proper class of measurable cardinals. We improve this lower bound to the optimal one by defining structures whose nontrivial homomorphisms can be used as extenders, thereby producing elementary embeddings witnessing [Formula: see text]-strongness of some cardinal.

The First Integral Cohomology of Pure Mapping Class Groups

AbstractIt is a classical result that pure mapping class groups of connected, orientable surfaces of finite type and genus at least 3 are perfect. In stark contrast, we construct nontrivial homomorphisms from infinite-genus mapping class groups to the integers. Moreover, we compute the first integral cohomology group associated to the pure mapping class group of any connected orientable surface of genus at least 2 in terms of the surface’s simplicial homology. In order to do this, we show that pure mapping class groups of infinite-genus surfaces split as a semi-direct product.

Quite free complicated abelian groups, pcf and black boxes

We would like to build Abelian groups (or R-modules) which on the one hand are quite free, say ℵω+1-free, and on the other hand are complicated in a suitable sense. We choose as our test problem one having no nontrivial homomorphism to ℤ (known classically for ℵ1-free, recently for ℵn-free). We succeed to prove the existence of even $${\aleph _{{\omega _1} \cdot n}}$$ -free ones. This requires building n-dimensional black boxes, which are quite free. This combinatorics is of self interest and we believe will be useful also for other purposes. On the other hand, modulo suitable large cardinals, we prove that it is consistent that every $${\aleph _{{\omega _1} \cdot \omega }}$$ -free Abelian group has non-trivial homomorphisms to ℤ.

Anticommutativity of symmetric elements under generalized oriented involutions

Let R be a ring with whose unit group is denoted by G a group with involution and a nontrivial group homomorphism, with satisfying for all Let RG be the group ring of G over R and define the involution in RG by In this article, we will classify the group rings RG such that is anticommutative, where is the largest subset of that can satisfy anticommutativity under Besides that, previous results about or will be extended.Communicated by S. Sehgal

Diffeomorphism groups of critical regularity

Let M be a circle or a compact interval, and let alpha =k+tau ge 1 be a real number such that k=lfloor alpha rfloor . We write {{,mathrm{Diff},}}_+^{alpha }(M) for the group of orientation preserving C^k diffeomorphisms of M whose kth derivatives are Hölder continuous with exponent tau . We prove that there exists a continuum of isomorphism types of finitely generated subgroups Gle {{,mathrm{Diff},}}_+^alpha (M) with the property that G admits no injective homomorphisms into bigcup _{beta >alpha }{{,mathrm{Diff},}}_+^beta (M). We also show the dual result: there exists a continuum of isomorphism types of finitely generated subgroups G of bigcap _{beta <alpha }{{,mathrm{Diff},}}_+^beta (M) with the property that G admits no injective homomorphisms into {{,mathrm{Diff},}}_+^alpha (M). The groups G are constructed so that their commutator groups are simple. We give some applications to smoothability of codimension one foliations and to homomorphisms between certain continuous groups of diffeomorphisms. For example, we show that if alpha ge 1 is a real number not equal to 2, then there is no nontrivial homomorphism {{,mathrm{Diff},}}_+^alpha (S^1)rightarrow bigcup _{beta >alpha }{{,mathrm{Diff},}}_+^{beta }(S^1). Finally, we obtain an independent result that the class of finitely generated subgroups of {{,mathrm{Diff},}}_+^1(M) is not closed under taking finite free products.

On Vector-Valued Characters for Noncommutative Function Algebras

Let A be a closed subalgebra of a $$C^*$$-algebra, that is a norm-closed algebra of Hilbert space operators. We generalize to such operator algebras several key theorems and concepts from the theory of classical function algebras. In particular we consider several problems that arise when generalizing classical function algebra results involving characters (nontrivial homomorphisms from the algebra into the scalars). For example, the Jensen inequality, the related Bishop–Ito–Schreiber theorem, and the theory of Gleason parts. Inspired by Arveson’s work on noncommutative Hardy spaces, we replace characters (classical function algebra case) by D-characters; certain completely contractive homomorphisms $$\Phi : A \rightarrow D$$, where D is a $$C^*$$-subalgebra of A. Using Brown’s measure and a potential theoretic balayage argument we prove a partial noncommutative Jensen inequality appropriate for $$C^*$$-algebras with a tracial state. We also show that this Jensen inequality characterizes D-characters among the module maps. Other advances include a theory of noncommutative Gleason parts appropriate for D-characters, which uses Harris’ noncommutative hyperbolic metric and Schwarz–Pick inequality, and other ingredients. As an application of Gleason parts we show that in the antisymmetric case, one is guaranteed the existence of a ‘quantum’ Wermer embedding function, and also of non-trivial compact Hankel operators, whenever the Gleason part of the canonical trace is rich in tracial states.

Knot Groups and Residual Nilpotence

We study groups of classical links, welded links, and virtual links. For classical braids, it is proved that the closures of a braid and its automorphic image are weakly equivalent. This implies the affirmative answer to the question of the coincidence of the groups constructed from a braid and from its automorphic image. We also study the problem of residual nilpotence of groups of virtual knots. It is known that, in a classical knot group, the commutator subgroup coincides with the third term of the lower central series and, hence, the quotient by the terms of the lower central series yields nothing. We prove that the situation is different for virtual knots. A nontrivial homomorphism of the virtual trefoil group to a nilpotent group of class 4 is constructed. We use the Magnus representation of a free group by power series to construct a homomorphism of the virtual trefoil group to a finite-dimensional algebra. This produces the nontrivial linear representation of the virtual trefoil group by unitriangular matrices of order 8.

Stein fillings and SU(2) representations

We recently defined invariants of contact 3-manifolds using a version of instanton Floer homology for sutured manifolds. In this paper, we prove that if several contact structures on a 3-manifold are induced by Stein structures on a single 4-manifold with distinct Chern classes modulo torsion then their contact invariants in sutured instanton homology are linearly independent. As a corollary, we show that if a 3-manifold bounds a Stein domain that is not an integer homology ball then its fundamental group admits a nontrivial homomorphism to SU(2). We give several new applications of these results, proving the existence of nontrivial and irreducible SU(2) representations for a variety of 3-manifold groups.

Computational complexity and 3–manifolds and zombies

We show the problem of counting homomorphisms from the fundamental group of a homology $3$-sphere $M$ to a finite, non-abelian simple group $G$ is #P-complete, in the case that $G$ is fixed and $M$ is the computational input. Similarly, deciding if there is a non-trivial homomorphism is NP-complete. In both reductions, we can guarantee that every non-trivial homomorphism is a surjection. As a corollary, for any fixed integer $m \ge 5$, it is NP-complete to decide whether $M$ admits a connected $m$-sheeted covering. Our construction is inspired by universality results in topological quantum computation. Given a classical reversible circuit $C$, we construct $M$ so that evaluations of $C$ with certain initialization and finalization conditions correspond to homomorphisms $\pi_1(M) \to G$. An intermediate state of $C$ likewise corresponds to a homomorphism $\pi_1(\Sigma_g) \to G$, where $\Sigma_g$ is a pointed Heegaard surface of $M$ of genus $g$. We analyze the action on these homomorphisms by the pointed mapping class group $\text{MCG}_*(\Sigma_g)$ and its Torelli subgroup $\text{Tor}_*(\Sigma_g)$. By results of Dunfield-Thurston, the action of $\text{MCG}_*(\Sigma_g)$ is as large as possible when $g$ is sufficiently large; we can pass to the Torelli group using the congruence subgroup property of $\text{Sp}(2g,\mathbb{Z})$. Our results can be interpreted as a sharp classical universality property of an associated combinatorial $(2+1)$-dimensional TQFT.

Automorphism groups of affine varieties and a characterization of affine $n$-space

We show that the automorphism group of affine -space determines up to isomorphism: If is a connected affine variety such that as ind-groups, then as varieties. We also show that every torus appears as for a suitable irreducible affine variety , but that cannot be isomorphic to a semisimple group. In fact, if is finite-dimensional and if , then the connected component is a torus. Concerning the structure of we prove that any homomorphism of ind-groups either factors through where is the Jacobian determinant, or it is a closed immersion. For we show that every nontrivial homomorphism is a closed immersion. Finally, we prove that every nontrivial homomorphism is an automorphism, and that is given by conjugation with an element from .

$K$-theory for the crossed products by certain actions of $\mathbb Z^2$

We investigate the K -theory of crossed product C^* -algebras by actions of \mathbb Z^2 . Given a \mathbb Z^2 -action, we associate to it a homomorphism between certain subquotients of the K -theory of the underlying C^* -algebra, which we call the obstruction homomorphism. This homomorphism together with the K -theory of the underlying algebra and the induced action in K -theory determine the K -theory of the associated crossed product C^* -algebra up to group extension problems. A concrete description of this obstruction homomorphism is provided as well. We give examples of \mathbb Z^2 -actions, where the associated obstruction homomorphisms are non-trivial. One class of examples comprises certain outer \mathbb Z^2 -actions on Kirchberg algebras, which act trivially on KK -theory. This relies on a classification result by Izumi and Matui. A second class of examples consists of certain pointwise inner \mathbb Z^2 -actions. One instance is given as a natural action on the group C^* -algebra of the discrete Heisenberg group. We also compute the K -theory of the corresponding crossed product. A general and concrete construction yields various examples of pointwise inner \mathbb Z^2 -actions on amalgamated free product C^* -algebras with non-trivial obstruction homomorphisms. Among these, there are actions that are universal, in a suitable sense, for pointwise inner \mathbb Z^2 -actions with non-trivial obstruction homomorphisms. We also compute the K -theory of the crossed products associated with these universal C^* -dynamical systems.

Indicable groups and $p_c&lt;1$

A conjecture of Benjamini & Schramm from 1996 states that any finitely generated group that is not a finite extension of $\mathbb{Z} $ has a non-trivial percolation phase. Our main results prove this conjecture for certain groups, and in particular prove that any group with a non-trivial homomorphism into the additive group of real numbers satisfies the conjecture. We use this to reduce the conjecture to the case of hereditary just-infinite groups. The novelty here is mainly in the methods used, combining the methods of EIT and evolving sets, and using the algebraic properties of the group to apply these methods.