Explicit Isomorphism Research Articles

Quantized nilradicals of parabolic subalgebras of and algebras of coinvariants

Let PJ be the standard parabolic subgroup of SLn associated to a subset J of simple roots, and let be the standard Levi decomposition. We let and denote the quantized algebras of regular functions on PJ and LJ , respectively. Following work of the first author, we study the quantum analogue of an induced coaction and the corresponding subalgebra of coinvariants. It was previously shown that the smash product algebra is isomorphic to In view of this, – while it is not a Hopf algebra – can be viewed as a quantum analogue of the coordinate ring In this article we prove that when is nonzero and not a root of unity, is isomorphic to a quantum Schubert cell algebra associated to a certain parabolic element w in the Weyl group of In this setting, the quantum Schubert cell algebra is a q-deformation of the universal enveloping algebra of the nilradical of a parabolic subalgebra of We also compute explicit commutation relations among the Lusztig root vectors for these particular quantized nilradicals and we give explicit algebra isomorphisms from to

Modular plethystic isomorphisms for two-dimensional linear groups

Let E be the natural representation of the special linear group SL2(K) over an arbitrary field K. We use the two dual constructions of the symmetric power when K has prime characteristic to construct an explicit isomorphism SymmSymℓE≅SymℓSymmE. This generalises Hermite reciprocity to arbitrary fields. We prove a similar explicit generalisation of the classical Wronskian isomorphism, namely SymmSymℓE≅⋀mSymℓ+m−1E. We also generalise a result first proved by King, by showing that if ∇λ is the Schur functor for the partition λ and λ∘ is the complement of λ in a rectangle with ℓ+1 rows, then ∇λSymℓE≅∇λ∘SymℓE. To illustrate that the existence of such ‘plethystic isomorphisms’ is far from obvious, we end by proving that the generalisation ∇λSymℓE≅∇λ′Symℓ+ℓ(λ′)−ℓ(λ)E of the Wronskian isomorphism, known to hold for a large class of partitions over the complex field, does not generalise to fields of prime characteristic, even after considering all possible dualities.

Nakajima quiver varieties, affine crystals and combinatorics of Auslander-Reiten quivers

We obtain an explicit crystal isomorphism between two realizations of crystal bases of finite dimensional irreducible representations of simple Lie algebras of type A and D. The first realization we consider is a geometric construction in terms of irreducible components of certain Nakajima quiver varieties established by Saito and the second is a realization in terms of isomorphism classes of quiver representations obtained by Reineke. We give a homological description of the irreducible components of Lusztig's quiver varieties which correspond to the crystal of a finite dimensional representation and describe the promotion operator in type A to obtain a geometric realization of Kirillov-Reshetikhin crystals.

Computation of lattice isomorphisms and the integral matrix similarity problem

AbstractLetKbe a number field, letAbe a finite-dimensionalK-algebra, let$\operatorname {\mathrm {J}}(A)$denote the Jacobson radical ofAand let$\Lambda $be an$\mathcal {O}_{K}$-order inA. Suppose that each simple component of the semisimpleK-algebra$A/{\operatorname {\mathrm {J}}(A)}$is isomorphic to a matrix ring over a field. Under this hypothesis onA, we give an algorithm that, given two$\Lambda $-latticesXandY, determines whetherXandYare isomorphic and, if so, computes an explicit isomorphism$X \rightarrow Y$. This algorithm reduces the problem to standard problems in computational algebra and algorithmic algebraic number theory in polynomial time. As an application, we give an algorithm for the following long-standing problem: Given a number fieldK, a positive integernand two matrices$A,B \in \mathrm {Mat}_{n}(\mathcal {O}_{K})$, determine whetherAandBare similar over$\mathcal {O}_{K}$, and if so, return a matrix$C \in \mathrm {GL}_{n}(\mathcal {O}_{K})$such that$B= CAC^{-1}$. We give explicit examples that show that the implementation of the latter algorithm for$\mathcal {O}_{K}=\mathbb {Z}$vastly outperforms implementations of all previous algorithms, as predicted by our complexity analysis.

An identification system based on the explicit isomorphism problem

We propose a new identification system based on algorithmic problems related to computing isomorphisms between central simple algebras. We design a statistical zero knowledge protocol which relies on the hardness of computing isomorphisms between orders in division algebras which generalizes a protocol by Hartung and Schnorr, which relies on the hardness of integral equivalence of quadratic forms.

Poincaré–Birkhoff–Witt isomorphisms and Kapranov dg-manifolds

We prove that to every inclusion A↪L of Lie algebroids over the same base manifold M corresponds a Kapranov dg-manifold structure on A[1]⊕L/A, which is canonical up to isomorphism. As a consequence, Γ(Λ•A∨⊗L/A) carries a canonical L∞[1] algebra structure whose unary bracket is the Chevalley–Eilenberg differential dA∇Bott corresponding to the Bott representation of A on L/A and whose binary bracket is a cocycle representative of the Atiyah class of the Lie pair (L,A). To this end, we construct explicit isomorphisms of C∞(M)-coalgebras Γ(S(L/A))→∼U(L)U(L)Γ(A), which we elect to call Poincaré–Birkhoff–Witt maps. These maps admit a recursive characterization that allows for explicit computations. They generalize both the classical symmetrization map S(g)→U(g) of Lie theory and (the inverse of) the complete symbol map for differential operators. Finally, we prove that the Kapranov dg-manifold A[1]⊕L/A is linearizable if and only if the Atiyah class of the Lie pair (L,A) vanishes.

New examples of Bernoulli algebraic actions

AbstractWe give an example of a principal algebraic action of the non-commutative free group ${\mathbb {F}}$ of rank two by automorphisms of a connected compact abelian group for which there is an explicit measurable isomorphism with the full Bernoulli 3-shift action of ${\mathbb {F}}$ . The isomorphism is defined using homoclinic points, a method that has been used to construct symbolic covers of algebraic actions. To our knowledge, this is the first example of a Bernoulli algebraic action of ${\mathbb {F}}$ without an obvious independent generator. Our methods can be generalized to a large class of acting groups.

The alternating presentation of [formula omitted] from Freidel-Maillet algebras

An infinite dimensional algebra denoted A¯q that is isomorphic to a central extension of Uq+ - the positive part of Uq(sl2ˆ) - has been recently proposed by Paul Terwilliger. It provides an ‘alternating’ Poincaré-Birkhoff-Witt (PBW) basis besides the known Damiani's PBW basis built from positive root vectors. In this paper, a presentation of A¯q in terms of a Freidel-Maillet type algebra is obtained. Using this presentation: (a) finite dimensional tensor product representations for A¯q are constructed; (b) explicit isomorphisms from A¯q to certain Drinfeld type ‘alternating’ subalgebras of Uq(gl2ˆ) are obtained; (c) the image in Uq+ of all the generators of A¯q in terms of Damiani's root vectors is obtained. A new tensor product decomposition for Uq(sl2ˆ) in terms of Drinfeld type ‘alternating’ subalgebras follows. The specialization q→1 of A¯q is also introduced and studied in details. In this case, a presentation is given as a non-standard Yang-Baxter algebra.

Sur une construction algébrique d'isomorphismes génériques d'algèbres de Hecke de type A

We construct explicit homomorphisms between the generic Hecke algebra Hn(q) of type A and the degenerate affine Hecke algebra DH(Sn) by means of a generic element Rqν(X,S) which satisfies the quadratic Hecke relation. For the specialization of DH(Sn) by means of the Jucys-Murphy's elements associated with the symmetric group Sn, we obtain explicit isomorphisms, after extension of the scalars to C(q), between Hn(q) and the group algebra of the symmetric group defined over C(q). Moreover, this isomorphism is compatible with the Hoefsmit's model of irreducible representations of Hn(q) and Sn.

Maximal prime homomorphic images of mod-p Iwasawa algebras

AbstractLet k be a finite field of characteristic p, and G a compact p-adic analytic group. Write kG for the completed group ring of G over k. In this paper, we describe the structure of the ring kG/P, where P is a minimal prime ideal of kG. We give an explicit isomorphism between kG/P and a matrix ring with coefficients in the ring ${(k'G')_\alpha }$ , where $k'/k$ is a finite field extension, $G'$ is a large subquotient of G with no finite normal subgroups, and (–)α is a “twisting” operation that preserves many desirable properties of the ring structure. We demonstrate the usefulness of this isomorphism by studying the correspondence induced between certain ideals of kG and those of ${(k'G')_\alpha }$ , and showing that this preserves many useful “group-theoretic” properties of ideals, in particular almost-faithfulness and control by a closed normal subgroup.

Congruence subgroups and orthogonal groups

We derive explicit isomorphisms between certain congruence subgroups of the Siegel modular group, the Hermitian modular group over an arbitrary imaginary-quadratic number field and the modular group over the Hurwitz quaternions of degree 2 and the discriminant kernels of special orthogonal groups SO0(2,n), n=3,4,6. The proof is based on an application of linear algebra adapted to the number theoretical needs.

Onn-dependent groups and fields II

AbstractWe continue the study ofn-dependent groups, fields and related structures, largely motivated by the conjecture that everyn-dependent field is dependent. We provide evidence toward this conjecture by showing that every infiniten-dependent valued field of positive characteristic is henselian, obtaining a variant of Shelah’s Henselianity Conjecture in this case and generalizing a recent result of Johnson for dependent fields. Additionally, we prove a result on intersections of type-definable connected components over generic sets of parameters inn-dependent groups, generalizing Shelah’s absoluteness of$G^{00}$in dependent theories and relative absoluteness of$G^{00}$in$2$-dependent theories. In an effort to clarify the scope of this conjecture, we provide new examples of strictly$2$-dependent fields with additional structure, showing that Granger’s examples of non-degenerate bilinear forms over dependent fields are$2$-dependent. Along the way, we obtain some purely model-theoretic results of independent interest: we show thatn-dependence is witnessed by formulas with all but one variable singletons; provide a type-counting criterion for$2$-dependence and use it to deduce$2$-dependence for compositions of dependent relations with arbitrary binary functions (the Composition Lemma); and show that an expansion of a geometric theoryTby a generic predicate is dependent if and only if it isn-dependent for somen, if and only if the algebraic closure inTis disintegrated. An appendix by Martin Bays provides an explicit isomorphism in the Kaplan-Scanlon-Wagner theorem.

A note on optimal systems of certain low-dimensional Lie algebras

Abstract Optimal classifications of Lie algebras of some well-known equations under their group of inner automorphism are re-considered. By writing vector fields of some known Lie algebras in the abstract format, we have proved that there exist explicit isomorphism between Lie algebras and sub-algebras which have already been classified. The isomorphism between Lie algebras is useful in the sense that the classifications of sub-algebras of dimension ≤4 have previously been carried out in literature. These already available classifications can be used to write classification of any Lie algebra of dimension ≤4. As an example, the explicit isomorphism between Lie algebra of variant Boussinesq system and sub-algebra A 3,5 1 / 2 ${A}_{3,5}^{1/2}$ is proved, and subsequently, optimal sub-algebras up to dimension four are obtained. Besides this, some other examples of Lie algebras are also considered for explicit isomorphism.

On Morse–Smale–Witten type cohomologies for orbifolds

Let X:=[M/G] be a compact orbifold given by the quotient of a connected compact Lie group G action on an oriented smooth closed manifold M, where the action is assumed to be smooth, effective, locally free and orientation preserving. Let f¯:X→R be a Morse function which lifts to a G-invariant Morse–Bott function f on M. We construct an explicit isomorphism between the Morse–Bott cohomology of f and the orbifold Morse cohomology of f¯ by constructing a new filtered cochain complex out of basic forms to connect the Morse–Bott cochain complex of f and the orbifold Morse–Smale–Witten cochain complex of f¯.

Explicit Isomorphism between Cartier and Dieudonné Modules

The aim of this paper is to compare the Cartier and covariant Dieudonné modules of connected p-divisible groups over perfect fields of positive characteristic p. In fact, we construct an explicit, canonical and functorial isomorphism and its inverse from the Dieudonné module to the Cartier module. The fact that the Cartier and (covariant) Dieudonné modules of a p-divisible formal Lie group over a perfect field are isomorphic is attributed to W. Messing, but a written proof does not exist in the literature ([1, §5]). We also give an application of this explicit isomorphism.

On the relative K-group in the ETNC, Part II

In a previous paper we showed that, under some assumptions, the relative K-group in the Burns–Flach formulation of the equivariant Tamagawa number conjecture (ETNC) is canonically isomorphic to a K-group of locally compact equivariant modules. Our approach as well as the standard one both involve presentations: One due to Bass–Swan, applied to categories of finitely generated projective modules; and one due to Nenashev, applied to our topological modules without finite generation assumptions. In this paper we provide an explicit isomorphism.

L ∞ -algebras of Einstein–Cartan–Palatini gravity

We give a detailed account of the cyclic L∞-algebra formulation of general relativity with a cosmological constant in the Einstein–Cartan–Palatini formalism on spacetimes of arbitrary dimension and signature, which encompasses all symmetries, field equations, and Noether identities of gravity without matter fields. We present a local formulation as well as a global covariant framework, and an explicit isomorphism between the two L∞-algebras in the case of parallelizable spacetimes. By duality, we show that our L∞-algebras describe the complete Batalin-Vilkovisky-Becchi-Rouet-Stora-Tyutin formulation of Einstein–Cartan–Palatini gravity. We give a general description of how to extend on-shell redundant symmetries in topological gauge theories to off-shell correspondences between symmetries in terms of quasi-isomorphisms of L∞-algebras. We use this to extend the on-shell equivalence between gravity and Chern–Simons theory in three dimensions to an explicit L∞-quasi-isomorphism between differential graded Lie algebras, which applies off-shell and for degenerate dynamical metrics. In contrast, we show that there is no morphism between the L∞-algebra underlying gravity and the differential graded Lie algebra governing BF theory in four dimensions.

Electric-magnetic duality in twisted quantum double model of topological orders

We derive a partial electric-magnetic (PEM) duality transformation of the twisted quantum double (TQD) model TQD(G, α) — discrete Dijkgraaf-Witten model — with a finite gauge group G, which has an Abelian normal subgroup N , and a three-cocycle α ∈ H3(G, U(1)). Any equivalence between two TQD models, say, TQD(G, α) and TQD(G′, α′), can be realized as a PEM duality transformation, which exchanges the N-charges and N-fluxes only. Via the PEM duality, we construct an explicit isomorphism between the corresponding TQD algebras Dα(G) and Dα′(G′) and derive the map between the anyons of one model and those of the other.

Finite W-superalgebras via super Yangians

Let e be an arbitrary even nilpotent element in the general linear Lie superalgebra glM|N and let We be the associated finite W-superalgebra. Let Ym|n be the super Yangian associated to the Lie superalgebra glm|n. A subalgebra of Ym|n, called the shifted super Yangian and denoted by Ym|n(σ), is defined and studied. Moreover, an explicit isomorphism between We and a quotient of Ym|n(σ) is established.

Some ribbon elements for the quasi-Hopf algebra Dω(H)

We construct an explicit isomorphism between the quasitriangular quasi-Hopf algebra [Formula: see text] defined in [D. Bulacu and F. Panaite, A generalization of the quasi-Hopf algebra [Formula: see text], Commun. Algebra 26 (1998) 4125–4141] and a certain quantum double quasi-Hopf algebra. We give also new characterizations for a quasitriangular quasi-Hopf algebra to be ribbon and use them to construct some ribbon elements for [Formula: see text].