Algebra Morphism Research Articles

Naturality and innerness for morphisms of compact groups and (restricted) Lie algebras

An extended derivation (endomorphism) of a (restricted) Lie algebra L L is an assignment of a derivation (respectively) of L ′ L’ for any (restricted) Lie morphism f : L → L ′ f:L\to L’ , functorial in f f in the obvious sense. We show that (a) the only extended endomorphisms of a restricted Lie algebra are the two obvious ones, assigning either the identity or the zero map of L ′ L’ to every f f ; and (b) if L L is a Lie algebra in characteristic zero or a restricted Lie algebra in positive characteristic, then L L is in canonical bijection with its space of extended derivations (so the latter are all, in a sense, inner). These results answer a number of questions of G. Bergman. In a similar vein, we show that the individual components of an extended endomorphism of a compact connected group are either all trivial or all inner automorphisms.

Deformation cohomology of morphisms of Lie-Yamaguti algebras

We study cohomology of morphisms of Lie-Yamaguti algebras. As an application, we establish that this cohomology ‘controls’ the formal deformations. Additionally, we demonstrate its connection to the abelian extensions of morphisms of Lie-Yamaguti algebras.

On deformations of Rota–Baxter algebra morphisms

Rota–Baxter algebras are important in probability, combinatorics, associative Yang–Baxter equation and splitting of algebras. This paper studies the formal deformations of Rota–Baxter algebra morphisms. As a consequence, we develop a cohomology theory of Rota–Baxter algebra morphisms to interpret the lower degree cohomology groups as formal deformations. Finally, we prove the cohomology comparison theorem of Rota–Baxter algebra morphisms, i.e. the cohomology of a morphism of Rota–Baxter algebras is isomorphic to the cohomology of an auxiliary Rota–Baxter algebra.

Weak Hopf algebras, smash products and applications to adjoint-stable algebras

For a semisimple quasi-triangular Hopf algebra [Formula: see text] over a field [Formula: see text] of characteristic zero, and a strongly separable quantum commutative [Formula: see text]-module algebra [Formula: see text], we show that [Formula: see text] is a weak Hopf algebra, and it can be embedded into a weak Hopf algebra [Formula: see text]. With these structures, [Formula: see text] is the monoidal category introduced by Cohen and Westreich, and [Formula: see text] is tensor equivalent to [Formula: see text]. If [Formula: see text] is in the Müger center of [Formula: see text], then the embedding is a quasi-triangular weak Hopf algebra morphism. This explains the presence of a subgroup inclusion in the characterization of irreducible Yetter–Drinfeld modules for a finite group algebra.

Cohomology of Lie Algebra Morphism Triples and Some Applications

Cohomology of Lie Algebra Morphism Triples and Some Applications

Triangular decomposition of $\mathrm{SL}_3$ skein algebras

We give an \mathrm{SL}_3 analogue of the triangular decomposition of the Kauffman bracket stated skein algebras described by Lê. To any punctured bordered surface, we associate an \mathrm{SL}_3 stated skein algebra which contains the \mathrm{SL}_3 skein algebra of closed webs. These algebras admit natural algebra morphisms associated to the splitting of surfaces along ideal arcs. We give an explicit basis for the \mathrm{SL}_3 stated skein algebra and show that the splitting morphisms are injective and describe their images. By splitting a surface along the edges of an ideal triangulation, we see that the \mathrm{SL}_3 stated skein algebra of any ideal triangulable surface embeds into a tensor product of stated skein algebras of triangles. As applications, we prove that the stated skein algebras do not have zero divisors, we construct Frobenius maps at roots of unity, and we obtain a new proof that Kuperberg's web relations generate the kernel of the Reshetikhin–Turaev functor.

Dominating real algebraic morphisms

Let X and Y be nonsingular real algebraic varieties, dim⁡X≥dim⁡Y. Assume that the variety Y is malleable, compact and connected. Our main result implies that each regular map from X to Y is homotopic to a surjective regular map. The class of malleable varieties includes all homogeneous spaces for linear real algebraic groups.

Moduli spaces of vector bundles on a curve and opers

Let X be a compact connected Riemann surface of genus g, with $$g\, \ge \,2$$ , and let $$\xi $$ be a holomorphic line bundle on X with $$\xi ^{\otimes 2}\,=\, {\mathcal O}_X$$ . Fix a theta characteristic $${\mathbb {L}}$$ on X. Let $${\mathcal M}_X(r,\xi )$$ be the moduli space of stable vector bundles E on X of rank r such that $$\bigwedge ^r E\,=\, \xi $$ and $$H^0(X,\, E\otimes {\mathbb L})\,=\, 0$$ . Consider the quotient of $${\mathcal M}_X(r,\xi )$$ by the involution given by $$E\, \longmapsto \, E^*$$ . We construct an algebraic morphism from this quotient to the moduli space of $$\textrm{SL}(r,{\mathbb C})$$ opers on X. Since $$\dim {\mathcal M}_X(r,\xi )$$ coincides with the dimension of the moduli space of $$\textrm{SL}(r,{\mathbb C})$$ opers, it is natural to ask about the injectivity and surjectivity of this map.

English

In this article, we introduce equivariant formal deformation theory of associative algebra morphisms. We introduce an equivariant deformation cohomology of associative algebra morphisms and using this we study the equivariant formal deformation theory of associative algebra morphisms.

The structure group for quasi-linear equations via universal enveloping algebras

We replace trees by multi-indices as an index set of the abstract model space to tackle quasi-linear singular stochastic partial differential equations. We show that this approach is consistent with the postulates of regularity structures when it comes to the structure group, which arises from a Hopf algebra and a comodule. Our approach, where the dual of the abstract model space naturally embeds into a formal power series algebra, allows to interpret the structure group as a Lie group arising from a Lie algebra consisting of derivations on this power series algebra. These derivations in turn are the infinitesimal generators of two actions on the space of pairs (non-linearities, functions of space-time mod constants). We also argue that there exist pre-Lie algebra and Hopf algebra morphisms between our structure and the tree-based one in the cases of branched rough paths (Grossman-Larson, Connes-Kreimer) and of the stochastic heat equation.

Quantum operations on conformal nets

On a conformal net [Formula: see text], one can consider collections of unital completely positive maps on each local algebra [Formula: see text], subject to natural compatibility, vacuum preserving and conformal covariance conditions. We call quantum operations on [Formula: see text] the subset of extreme such maps. The usual automorphisms of [Formula: see text] (the vacuum preserving invertible unital *-algebra morphisms) are examples of quantum operations, and we show that the fixed point subnet of [Formula: see text] under all quantum operations is the Virasoro net generated by the stress-energy tensor of [Formula: see text]. Furthermore, we show that every irreducible conformal subnet [Formula: see text] is the fixed points under a subset of quantum operations. When [Formula: see text] is discrete (or with finite Jones index), we show that the set of quantum operations on [Formula: see text] that leave [Formula: see text] elementwise fixed has naturally the structure of a compact (or finite) hypergroup, thus extending some results of [M. Bischoff, Generalized orbifold construction for conformal nets, Rev. Math. Phys. 29 (2017) 1750002]. Under the same assumptions, we provide a Galois correspondence between intermediate conformal nets and closed subhypergroups. In particular, we show that intermediate conformal nets are in one-to-one correspondence with intermediate subfactors, extending a result of Longo in the finite index/completely rational conformal net setting [R. Longo, Conformal subnets and intermediate subfactors, Comm. Math. Phys. 237 (2003) 7–30].

Equivariant multiplicities via representations of quantum affine algebras

For any simply-laced type simple Lie algebra mathfrak {g} and any height function xi adapted to an orientation Q of the Dynkin diagram of mathfrak {g}, Hernandez–Leclerc introduced a certain category mathcal {C}^{le xi } of representations of the quantum affine algebra U_q(widehat{mathfrak {g}}), as well as a subcategory mathcal {C}_Q of mathcal {C}^{le xi } whose complexified Grothendieck ring is isomorphic to the coordinate ring mathbb {C}[textbf{N}] of a maximal unipotent subgroup. In this paper, we define an algebraic morphism {widetilde{D}}_{xi } on a torus mathcal {Y}^{le xi } containing the image of K_0(mathcal {C}^{le xi }) under the truncated q-character morphism. We prove that the restriction of {widetilde{D}}_{xi } to K_0(mathcal {C}_Q) coincides with the morphism overline{D} recently introduced by Baumann–Kamnitzer–Knutson in their study of equivariant multiplicities of Mirković–Vilonen cycles. This is achieved using the T-systems satisfied by the characters of Kirillov–Reshetikhin modules in mathcal {C}_Q, as well as certain results by Brundan–Kleshchev–McNamara on the representation theory of quiver Hecke algebras. This alternative description of overline{D} allows us to prove a conjecture by the first author on the distinguished values of overline{D} on the flag minors of mathbb {C}[textbf{N}]. We also provide applications of our results from the perspective of Kang–Kashiwara–Kim–Oh’s generalized Schur–Weyl duality. Finally, we use Kashiwara–Kim–Oh–Park’s recent constructions to define a cluster algebra overline{mathcal {A}}_Q as a subquotient of K_0(mathcal {C}^{le xi }) naturally containing mathbb {C}[textbf{N}], and suggest the existence of an analogue of the Mirković–Vilonen basis in overline{mathcal {A}}_Q on which the values of {widetilde{D}}_{xi } may be interpreted as certain equivariant multiplicities.

Completing simple valuations in -categories

We prove that Keimel and Lawson's K-completion Kc of the simple valuation monad Vs defines a monad Kc∘Vs on each ▪-category K. We also characterise the Eilenberg-Moore algebras of Kc∘Vs as the weakly locally convex K-cones, and its algebra morphisms as the continuous linear maps.In addition, we explicitly describe the distributive law of Vs over Kc, which allows us to show that the K-completion of any locally convex (resp., weakly locally convex, locally linear) topological cone is a locally convex (resp., weakly locally convex, locally linear) K-cone.We also give an example – the Cantor tree with a top – that shows the dcpo-completion of the simple valuations is not the D-completion of the simple valuations in general, where D is the category of monotone convergence spaces and continuous maps.

The quantum trace as a quantum non-abelianization map

We prove that the balanced Chekhov–Fock algebra of a punctured triangulated surface is isomorphic to a skein algebra which is a deformation of the algebra of regular functions of some abelian character variety. We first deduce from this observation a classification of the irreducible representations of the balanced Chekhov–Fock algebra at odd roots of unity, which generalizes to open surfaces the classification of Bonahon, Liu and Wong. We re-interpret Bonahon and Wong’s quantum trace map as a non-commutative deformation of some regular morphism between this abelian character variety and the [Formula: see text]-character variety. This algebraic morphism shares many resemblances with the non-abelianization map of Gaiotto, Moore, Hollands and Neitzke. When the punctured surface is closed, we prove that this algebraic non-abelianization map induces a birational morphism between a smooth torus and the relative [Formula: see text] character variety.

From Hopf Algebra to Braided L∞-Algebra

We show that an L∞-algebra can be extended to a graded Hopf algebra with a codifferential. Then, we twist this extended L∞-algebra with a Drinfel’d twist, simultaneously twisting its modules. Taking the L∞-algebra as its own (Hopf) module, we obtain the recently proposed braided L∞-algebra. The Hopf algebra morphisms are identified with the strict L∞-morphisms, whereas the braided L∞-morphisms define a more general L∞-action of twisted L∞-algebras.

Universal Enveloping Algebras of Lie–Rinehart Algebras as a Left Adjoint Functor

We prove how the universal enveloping algebra constructions for Lie-Rinehart algebras and anchored Lie algebras are naturally left adjoint functors. This provides a conceptual motivation for the universal properties these constructions satisfy. As a supplement, the categorical approach offers new insights into the definitions of Lie-Rinehart algebra morphisms, of modules over Lie-Rinehart algebras and of the infinitesimal gauge algebra of a module.

On the Maurer-Cartan simplicial set of a complete curved A_infty -algebra

In this paper, we develop the A_infty -analog of the Maurer-Cartan simplicial set associated to an L_infty -algebra and show how we can use this to study the deformation theory of infty -morphisms of algebras over non-symmetric operads. More precisely, we first recall and prove some of the main properties of A_infty -algebras like the Maurer-Cartan equation and twist. One of our main innovations here is the emphasis on the importance of the shuffle product. Then, we define a functor from the category of complete (curved) A_infty -algebras to simplicial sets, which sends a complete curved A_infty -algebra to the associated simplicial set of Maurer-Cartan elements. This functor has the property that it gives a Kan complex. In all of this, we do not require any assumptions on the field we are working over. We also show that this functor can be used to study deformation problems over a field of characteristic greater than or equal to 0. As a specific example of such a deformation problem, we study the deformation theory of infty -morphisms of algebras over non-symmetric operads.

W-types in setoids

We present a construction of W-types in the setoid model of extensional Martin-L\"of type theory using dependent W-types in the underlying intensional theory. More precisely, we prove that the internal category of setoids has initial algebras for polynomial endofunctors. In particular, we characterise the setoid of algebra morphisms from the initial algebra to a given algebra as a setoid on a dependent W-type. We conclude by discussing the case of free setoids. We work in a fully intensional theory and, in fact, we assume identity types only when discussing free setoids. By using dependent W-types we can also avoid elimination into a type universe. The results have been verified in Coq and a formalisation is available on the author's GitHub page.

Cocyclic complexes of Hopf algebras with special antipodes

We show that for any bimodule M and bicomodule C of a Hopf algebra H with involutive antipode, Hochschild complex and are cocyclic -modules. Their para-cocyclic operators are compatible in Gerstenhaber-Schack bicomplex of a Hopf algebra morphism which is a combination of Hochschild complex and coHochschild complex for any and make the bicomplex into a cylindrical -module under appropriate conditions. From this bicomplex, we get a cocyclic structure of the diagonal complex. With these cocyclic operators, the operadic structure of the diagonal complex is cyclic, so that the Gerstenhaber algebra structure on the diagonal cohomology evolves into a Batalin-Vilkovisky algebra. Hence we obtain the structure of Gerstenhaber-Schack cohomology of which is isomorphic to the diagonal cohomology. Our results can be applied to group algebra and arbitrary G-graded Hopf algebra A. Let be ϵA (resp. uB , idA ). Then we can get the Gerstenhaber structure on Hochschild cohomology (resp. Adams Cobar construction of B, Gerstenhaber-Schack cohomology of A). Moreover, the Gerstenhaber structure operators of ϵA and uB commute with composition.

Algebras with representable representations

AbstractJust like group actions are represented by group automorphisms, Lie algebra actions are represented by derivations: up to isomorphism, a split extension of a Lie algebra $B$ by a Lie algebra $X$ corresponds to a Lie algebra morphism $B\to {\mathit {Der}}(X)$ from $B$ to the Lie algebra ${\mathit {Der}}(X)$ of derivations on $X$. In this article, we study the question whether the concept of a derivation can be extended to other types of non-associative algebras over a field ${\mathbb {K}}$, in such a way that these generalized derivations characterize the ${\mathbb {K}}$-algebra actions. We prove that the answer is no, as soon as the field ${\mathbb {K}}$ is infinite. In fact, we prove a stronger result: already the representability of all abelian actions – which are usually called representations or Beck modules – suffices for this to be true. Thus, we characterize the variety of Lie algebras over an infinite field of characteristic different from $2$ as the only variety of non-associative algebras which is a non-abelian category with representable representations. This emphasizes the unique role played by the Lie algebra of linear endomorphisms $\mathfrak {gl}(V)$ as a representing object for the representations on a vector space $V$.