K-linear Map Research Articles

Dimension vectors with the equal kernels property

Let r∈N, Kr be the generalized Kronecker quiver with r arrows γ1,…,γr:1→2 and δ∈Δ+(Kr) be a positive root of Kr. We say that δ has the equal kernels property if for all α∈kr∖{0} and every indecomposable representation M with dimension vector dim_M=δ the k-linear map Mα:=∑i=1rαiM(γi):M1→M2 is injective. We show that δ has the equal kernels property if and only if qKr(δ)+δ2−δ1≥1, where qKr:Z2→Z,(x,y)↦x2+y2−rxy denotes the Tits quadratic form of Kr.

A Radon–Nikodým theorem for local completely positive invariant multilinear maps

In this article, we introduce local completely positive k-linear maps between locally -algebras and obtain Stinespring type representation by adopting the notion of ‘invariance’ defined by J. Heo for k-linear maps between -algebras. Also, we supply the minimality condition to make certain that minimal representation is unique up to unitary equivalence. As a consequence, we prove Radon–Nikodým theorem for unbounded operator-valued local completely positive invariant k-linear maps. The obtained Radon–Nikodým derivative is a positive contraction on some Hilbert space with several reducing subspaces.

Congruence of matrix spaces, matrix tuples, and multilinear maps

Two matrix vector spaces V,W⊂Cn×n are said to be equivalent if SVR=W for some nonsingular S and R. These spaces are congruent if R=ST. We prove that if all matrices in V and W are symmetric, or all matrices in V and W are skew-symmetric, then V and W are congruent if and only if they are equivalent.Let F:U×…×U→V and G:U′×…×U′→V′ be symmetric or skew-symmetric k-linear maps over C. If there exists a set of linear bijections φ1,…,φk:U→U′ and ψ:V→V′ that transforms F to G, then there exists such a set with φ1=…=φk.

A Stinespring type theorem for completely positive multilinear maps on Hilbert C*-modules

We extend the Stinespring’s representation theorem for two k-linear maps on a Hilbert -module and a -algebra. We prove a covariant representation theorem for two covariant k-linear maps and construct the extension of two k-linear maps to the crossed product Hilbert -module and the crossed product -algebra.

The LNED and LFED conjectures for algebraic algebras

Let K be a field of characteristic zero and A a K-algebra such that all the K-subalgebras generated by finitely many elements of A are finite dimensional over K. A K-E-derivation of A is a K-linear map of the form I−ϕ for some K-algebra endomorphism ϕ of A, where I denotes the identity map of A. In this paper we first show that for all locally finite K-derivations D and locally finite K-algebra automorphisms ϕ of A, the images of D and I−ϕ do not contain any nonzero idempotent of A. We then use this result to show some cases of the LFED and LNED conjectures proposed in [21]. More precisely, we show the LNED conjecture for A, and the LFED conjecture for all locally finite K-derivations of A and all locally finite K-E-derivations of the form δ=I−ϕ with ϕ being surjective. In particular, both conjectures are proved for all finite dimensional K-algebras. Furthermore, some finite extensions of derivations and automorphism to inner derivations and inner automorphisms, respectively, have also been established. This result is not only crucial in the proofs of the results above, but also interesting on its own right.

Radford’s Biproducts for Hopf Group-Coalgebras and Its Quasitriangular Structures

Let π be a group and H={Hα}α∈π be a semi-Hopf π-coalgebra in the sense of Virelizier (J. Pure Appl. Algebra 171:75–122, 2002). Let H coact weakly on a coalgebra B and λ={λα,β:B⟶Hα⊗Hβ} be a family of k-linear maps. Then in this paper we first introduce the notion of a π-crossed coproduct \(B\times_{\lambda }^{\pi}H=\{B\times_{\lambda }H_{\alpha }\}_{\alpha \in \pi }\) and find some sufficient and necessary conditions making it into a π-coalgebra, generalizing the main construction in Lin (Commun. Algebra 10:1–17, 1982). Secondly, we find a sufficient and necessary condition for \(B_{\#^{\pi}}^{\times_{\lambda }^{\pi}} H\), with the π-crossed coproduct \(B\times_{\lambda }^{\pi}H\) and π-smash product B#πH to form a semi-Hopf π-coalgebra, if λ is convolution invertible dual 2-cocycle, which generalizes the well-known Radford’s biproduct in Radford (J. Algebra 92:322–347, 1985). Furthermore, we derive some sufficient conditions for \(B_{\#^{\pi}}^{\times_{\lambda }^{\pi}} H\) to be a Hopf π-coalgebra. Finally, we construct a quasitriangular structure on the Hopf π-coalgebra \(B\times_{\lambda }^{\pi}H\) (with the usual tensor product).

The Integrals of Entwining Structure

In this paper the integrals of entwining structure (A,C,ψ) are discussed, where A is a k-algebra, C a k-coalgebra and a k-linear map. We prove that there exists a normalized integral γ:C→Hom(C,A) of (A,C,ψ) if and only if any representation of (A,C,ψ) is injective in a functorial way as a corepresentation of C. We give the dual results as well.

Homomorphisms of Finitary Incidence Algebras

For any field K and poset P, the incidence space I(P) and the finitary incidence algebra FI(P) were introduced in [5]. The K-vector space I(P) is an FI(P)-bimodule. We investigate K-linear maps from FI(P) to I(P) that preserve submodules. We also consider the idealization FI(P)(+)I(P) of I(P).

Linear maps on kI, and homomorphic images of infinite direct product algebras

Let k be an infinite field, I an infinite set, V a k-vector-space, and g : kI → V a k-linear map. It is shown that if dimk(V) is not too large (under various hypotheses on card(k) and card(I), if it is finite, respectively less than card(k), respectively less than the continuum), then ker(g) must contain elements (ui)i∈I with all but finitely many components ui nonzero.These results are used to prove that every homomorphism from a direct product ∏IAi of not-necessarily-associative algebras Ai onto an algebra B, where dimk(B) is not too large (in the same senses) is the sum of a map factoring through the projection of ∏IAi onto the product of finitely many of the Ai, and a map into the ideal {b∈B|bB=Bb={0}}⊆B.Detailed consequences are noted in the case where the Ai are Lie algebras.A version of the above result is also obtained with the field k replaced by a commutative valuation ring.

Relations between twisted derivations and twisted cyclic homology

For a given endomorphism on a unitary k-algebra, A, with k in the center of A, there are definitions of twisted cyclic and Hochschild homology. This paper will show that the method used to define them can be used to define twisted de Rham homology. The main result is that twisted de Rham homology can be thought of as the kernel of the Connes map from twisted cyclic homology to twisted Hochschild homology. For a noncommutative algebra A, Connes [1] and Karoubi [4] define the module of n-forms by taking the iterated tensor product of the bimodule of 1-forms. Karoubi in [4, Chapter 2] defines noncommutative de Rham homology. For the case where A is an associative unitary algebra over a commutative ring, k, containing Q, he shows that the reduced noncommutative de Rham homology is isomorphic to the kernel of the Connes map B from HCn(A), reduced cyclic homology, to HHn+1(A), reduced Hochschild homology. The bimodule of 1-forms used comes from the derivation d : A → A⊗kA, with d(a) = 1⊗a−a⊗1. In this paper we will generalize that result to the case of twisted cyclic homology based on a given k-algebra endomorphism, h : A → A. Here the bimodule of 1-forms will come from the twisted derivation, d : A → A ⊗k A, with d(a) = 1 ⊗ a − h(a)⊗ 1. If h = id, then we have d = d. A twisted derivation is any k-linear map, d, from A to an A-bimodule such that d(ab) = d(a) · b + h(a) · d(b). The definitions for twisted Hochschild homology and twisted cyclic homology are given in [3]. As pointed out there, in the twisted case C∗(A) itself is a paracyclic object, and we need to take an appropiate quotient of C∗(A) to obtain a cyclic object. It is quite interesting that even though the intrinsic definitions of the h-twisted theories differ considerably from the classical nontwisted theories, still there is an appropriate extension of de Rahm homology to the twisted case and an extension of the maps between all the twisted theories that give us a generalization of [4, Theorem 2.15]. Theorem. Suppose A is a unitary k-algebra with Q ⊂ k. If h is a k-algebra endomorphism then the reduced twisted de Rham homology and the reduced twisted cyclic homology are related by the exact sequence 0 → HDR n(A) → HC n (A) → HH n+1(A). HH ∗ (A) is the reduced twisted Hochschild homology and HC h,λ ∗ (A) is the reduced twisted cyclic homology based on the Connes complex {C n (A), b}. Details of the Connes complex together with the definition of HDR n(A) will be given Received by the editors March 11, 2009 and, in revised form, March 15, 2011. 2010 Mathematics Subject Classification. Primary 16E40; Secondary 16T20. c ©2011 American Mathematical Society Reverts to public domain 28 years from publication

Centralizers in endomorphism rings

We prove that the centralizer Cen ( φ ) ⊆ End R ( M ) of a nilpotent endomorphism φ of a finitely generated semisimple left R-module M R (over an arbitrary ring R) is the homomorphic image of the opposite of a certain Z ( R ) -subalgebra of the full m × m matrix algebra M m ( R [ z ] ) , where m is the dimension of ker ( φ ) . If R is a local ring, then we give a complete characterization of Cen ( φ ) and of the containment Cen ( φ ) ⊆ Cen ( σ ) , where σ is a not necessarily nilpotent element of End R ( M ) . For a K-linear map A of a finite dimensional vector space over a field K we determine the PI-degree of Cen ( A ) .

Lie Algebras with a Coalgebra Splitting

In their recent article [5], the authors endow every finite-dimensional simple complex Lie algebra g with a coalgebra structure such that the composition μ◦δ of the two structure maps δ : g −→ g⊗Cg and μ : g⊗C g −→ g coincides with the identity. Moreover, the dual algebra (g∗, δ∗) associated to the Lie coalgebra is isomorphic to (g, μ). The coalgebra map δ is given explicitly for sl(n), those for the other types are obtained via embeddings g ↪→ sl(n). The purpose of the present short note is to elicit the conceptual sources of [5], starting from the observation that the coalgebra maps defined in [5] are in fact homomorphisms of g-modules. For Lie algebras affording non-degenerate symmetric associative forms, such coalgebra maps naturally arise by dualizing the multiplication. This immediately implies the abovementioned duality, and the formulae displayed in [5, §4] can also be subsumed under our general approach. By demanding that δ be g-linear, we depart from the usual compatibility condition of a Lie bialgebra, which requires δ to be a derivation. In that case, μ ◦ δ is a derivation of g. For fields of characteristic zero, only nilpotent Lie algebras afford invertible derivations (cf. [7]), so that non-zero Lie bialgebras over such fields never satisfy μ ◦ δ = idg. Let k be a field. Given a finite-dimensional k-vector space V , we consider the k-linear maps

Model theory of comodules

The purpose of this paper is to establish some basic points in the model theory of comodules over a coalgebra. It is not even immediately apparent that there is a model theory of comodules since these are not structures in the usual sense of model theory. Let us give the definitions right away so that the reader can see what we mean.Fix a field k. A k-coalgebra C is a k-vector space equipped with a k-linear map Δ: C → C ⊗ C, called the comultiplication (by ⊗ we always mean tensor product over k), and a k-linear map ε: C → k, called the counit, such that Δ⊗1C = 1C ⊗ Δ (coassociativity) and (1C ⊗ ε)Δ = 1C = (ε ⊗ 1C)Δ, where we identify C with both k ⊗ C and C ⊗ k. These definitions are literally the duals of those for a k-algebra: express the axioms for C′ to be a k-algebra in terms of the multiplication map μ: C′ ⊗ C′ → C′ and the “unit” (embedding of k into C′), δ: k → C′ in the form that certain diagrams commute and then just turn round all the arrows. See [5] or more recent references such as [7] for more.

Category metrics and valuations of concrete categories

We will introduce the notion of a category metric ϱ on some category S , as a system of metrics (ϱ XY; X,Y ∈ Ob S) on the sets of morphisms S(X,Y) , compatible, in some precise sense, with the composition of S -morphisms. Given a category metric on a category S and a locally closed concrete category ( C,U) over S , one can construct a fairly natural valuation ϱ C of C in the sense of Zlatoš (Fuzzy Sets and Systems 82 (1996) 73–96), such that, for any S -morphism f : UA → UB, ϱ C (f) is the distance of f from the set C(A,B) , embedded into S(UA,UB) via the faithful forgetful functor U : C → S . It turns out that most of the valuations constructed in Zlatoš (Fuzzy Sets and Systems 82 (1996) 73–96) arise in this way from the same category metric on the category Set fin of all finite sets and mappings, or from an analogous metric on the category Vect fin (K) of all finite-dimensional vector spaces over some field K and K-linear maps.

Quantum groups and cylinder braiding

Quantum groups and cylinder braiding

Cohomology of Hopf Algebras and the Clifford's Extension Problem

In this paper we continue our previous study of the extension problem for modules in the case of Hopf Galois extensions. Let H be a Hopf algebra over a field k, let ArB be a faithfully flat H-Galois extension, and Ž . let M, m be a right B-module. We suppose that all of them are k-vector spaces of finite dimension. We say that there exists an A-extension of M if Ž . there exists a k-linear map m: M m A a M, such that M, m is a right k Ž . Ž . A-module, and m m m b s m m m b , ;m g M, ;b g B. w x In 8 for an algebraically closed field k, a cocommutative Hopf algebra H, and a simple stable B-module M we have shown that the extension problem is equivalent to the vanishing of a certain 2-cohomology class in 2Ž . H H, k , and moreover, the set of isomorphism classes of A-extensions is 1Ž . in one-to-one correspondence with H H, k . Here, k is an H-module algebra with trivial action, and the cohomology groups are defined as in w x 16 . The purpose of this paper is to prove these results for any field, not only for algebraically closed ones. In the first section we shall prove some results which will help us to reduce our problem to the case of a simple ring B. First of all, we shall prove that the right ideal generated by the annihilator I of M is a two-sided ideal of A and a right coideal, so ArIA is a comodule algebra. Next, we shall prove that its coinvariant subalgebra is isomorphic with BrI, and the extension ArAI : BrI is faithfully flat H-Galois. Moreover, there exists an A-extension for M if and only if there exists an ArAI-ex-

Topological Auantum Field Theories derived from the Kauffman bracket

IN [41], Witten has made the remarkable discovery of an intricate relationship between the Jones polynomial [15, 163 and gauge theory. (See also the prophetical article by Atiyah [2].) Although his approach uses the Feynman path integral of Quantum Field Theory, Witten gave convincing arguments that a viable combinatorial approach could be made rigorous using the method of surgery. His discovery includes new 3-manifold invariants (sometimes called Jones-W&en invariants), whose existence was first proven by Reshetikhin and Turaev [30] using quantum groups and Kirby’s surgery calculus [19] (see also [20]). Other combinatorial approaches for related invariants were developed by Kohno [21], Turaev and Viro 1361, Lickorish [22,23], the authors [lo], Morton and Strickland [29], Wenzl [40], Turaev and Wenzl [35]. According to Witten, his invariants should belong to a topological quantumfield theory (TQFT). This notion was axiomatized by Atiyah et al. [6,3] (see also [38]). In particular, the states of a manifold, C, form a hermitian vector space, V(E) (more generally V(x) is a module over a commutative ring k with unit and involution), and a cobordism M from x1 to & induces a transition (k-linear map), denoted Zy, from V(&) to V(&). One has that V(0) is the ground ring k, so that if aA = C (i.e., A4 is a cobordism from 0 to x), one obtains a vector Z(M) in V(x), given by Z(M) = Z,(l). Thus, M induces a state of 8M. In particular, if M is closed, Z(M) (also denoted by (M) in keeping with the physicists’ expectation value notation) lies in V(0) = k, so that TQFTs, by their very nature, yield manifold inuariants. In this paper, we give a purely topological construction of the TQFTs associated to invariants satisfying the Kauffman bracket relations [17], that is, essentially, of the TQFTs corresponding to Jones’ original v-polynomial [15]. We renormalize the invariants 8, of [lo] to construct a series of invariants ( )p of banded links in closed 3-manifolds, and then use these invariants to define, in a “universal” way, modules V,(C) (p 2 l), associated to surfaces x (which may have banded links, too). Here, for technical reasons, all manifolds are equipped with a p1 -structure (a weak form of framing, see Appendix B). We prove the finiteness and multiplicativity properties of the V,(x), using the language of bimodules over algebroids. It turns out that the ranks of our modules are given by Verlinde’sformula. Thus, we are led to believe that ours is a rigorous construction of Witten’s theory for SU(2) (and in some sense also for SO(3), see Remark 1.17).

Maps between tree and band modules

The indecomposable modules over special biserial algebras are known to be of a specially simple form, so-called “tree” and “band” modules (cf. [6,2]). Crawley-Boevey has shown how to describe the homomorphisms between tree modules (cf. [3]) and our objective in this paper is to extend this result to band modules. Tree and band modules occur as indecomposable representations of arbitrary algebras. Nevertheless we follow Crawley-Boevey and consider zero-relation algebras as the appropriate context. Given two tree or band modules we first study certain quiver homomorphisms between their underlying trees or bands and it turns out that a map between the representations is completely described by k-linear maps which are associated to these quiver homomorphisms. It is interesting to note that in general the classification of maps between representations of a special biserial algebra A is a wild problem. In fact the maps are equivalent to the representations of T,(A). For instance take for A the path algebra of an A,, n 2 6 then T,(A) is wild although the maps between indecomposable representations of A are easily to describe. Throughout, k will be a fixed algebraically closed field. Maps are written on the left. This paper results from a stay at the University of Liverpool supported by the Deutscher Akademischer Austauschdienst. I am very grateful to Sheila Brenner for her suggestions.

On open orbits and their complements

1.1. We fix an algebraically closed field k and a quiver Q without oriented cycles, whose vertex set Q, we identify with { 1, . . . . n}. For a in the set Q, of arrows of Q, we denote by ta and ha the tail and the head of a, respectively. A representation X of Q consists of a family {X(i): in QO} of finite dimensional k-vector spaces and a family {X(a): X(ta) + X(ha): a E Q1 } of k-linear maps. A morphism f: X + Y between two representations of Q is a family {f(i): X(i)+ Y(i): in QO} of k-linear maps satisfying f(ha)oX(a) = Y(a)of(ta) for all arrows a. We denote the category of representations of Q by mod Q. It is an abelian category of global dimension at most 1. We will denote the only possibly non-trivial extension group Ext’ by Ext. The dimension vector of a representation X of Q is the vector dim X= (dim X(l), . . . . dim X(n)) in N”, and the dimension of X is the natural number dim X= dim X( 1) + . . . + dim X(n). The support of X is the full subquiver of Q whose vertices are {i: X(i) ZO}. For a vector d = (d,, . . . . d,) E N”, we define R(Q, d) to be the set of representations X of Q such that X(i) = k” for ie QO. Thus R(Q, d) is the product R(Q, d) = n Hom,(kdru, kdhe).

Polynomials, series and Kronecker modules

A Kronecker module over a field K may be viewed as a quadruple ( V, W; a, b), with a, b K-linear maps from space V to space W. For an indeterminate X let P be the module ( K[ X], K[ X]; identity, multiplication by X). All purely simple Kronecker modules of rank-2, with P as homomorphic image, may be indexed by formal power series α. Homomorphisms and isomorphisms among such P α's are studied by means of operations on polynomials and formal power series. For instance, if P α ≊ P β then ( rα + s) β = tα + u, for some polynomials r, s, t, u not all 0. Also for those α's that are not roots of a linear or quadratic equation over K[ X], End P α = K.