Explicit Isomorphism Research Articles

Isomorphism Between Twisted q-Yangians and Affine ι Quantum Groups: Type AI

Abstract By employing Gauss decomposition, we establish a direct and explicit isomorphism between the twisted $q$-Yangians (in R-matrix presentation) and affine $\imath $quantum groups (in current presentation) associated to symmetric pair of type AI introduced by Molev–Ragoucy–Sorba and Lu–Wang, respectively. As a corollary, we obtain a PBW-type basis for affine $\imath $quantum groups of type AI.

Explicit inverse Shapiro isomorphism and its application

We find an explicit form of the inverse isomorphism from Shapiro’s lemma in terms of inhomogeneous cocycles and apply it to construct special nonsplit coverings of groups with a unique conjugacy class of involutions.

Duality and stacking of bosonic and fermionic SPT phases

We study the interplay of duality and stacking of bosonic and fermionic symmetry-protected topological phases in one spatial dimension. In general the classifications of bosonic and fermionic phases have different group structures under the operation of stacking, but we argue that they are often isomorphic and give an explicit isomorphism when it exists. This occurs for all unitary symmetry groups and many groups with antiunitary symmetries, which we characterize. We find that this isomorphism is typically not implemented by the Jordan-Wigner transformation, nor is it a consequence of any other duality transformation that falls within the framework of topological holography. Along the way to this conclusion, we recover the fermionic stacking rule in terms of G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{G} $$\\end{document}-pin partition functions, give a gauge-invariant characterization of the twisted group cohomology invariant, and state a procedure for stacking gapped phases in the formalism of symmetry topological field theory.

GIT quotient of holomorphic foliations on ℂℙ2 of degree 2 and quartic plane curves

Abstract We study the quotient variety of the space of foliations on ℂ ⁢ ℙ 2 {\mathbb{CP}^{2}} of degree 2 up to change of coordinates. We find the intersection Betti numbers of this variety. As a corollary, we have that these intersection Betti numbers coincide with the intersection Betti numbers of the quotient variety of quartic plane curves. Finally, we give an explicit isomorphism between the space of foliations of degree 2 with different singular points, without invariant lines and the space of smooth quartic plane curves.

A comparison of endomorphism algebras

Let F be a non-archimedean local field and G be a connected reductive group over F. For a Bernstein block in the category of smooth complex representations of G(F), we have two kinds of progenerators: the compactly induced representation indKG(F)(ρ) of a type (K,ρ), and the parabolically induced representation IPG(ΠM) of a progenerator ΠM of a Bernstein block for a Levi subgroup M of G. In this paper, we construct an explicit isomorphism of these two progenerators. Moreover, we compare the description of the endomorphism algebra EndG(F)(indKG(F)(ρ)) for a depth-zero type (K,ρ) in [20] with the description of the endomorphism algebra EndG(F)(IPG(ΠM)) in [33], that are described in terms of affine Hecke algebras.

On the isomorphism between four-dimensional Sklyanin and elliptic algebras

Let [Formula: see text] be an elliptic curve, and let [Formula: see text] be a point on [Formula: see text]. In his groundbreaking paper [E. K. Sklyanin, Some algebraic structures connected with the Yang-Baxter equation. Representations of a quantum algebra, Funktsional. Anal. i Prilozhen. 17(4) (1983) 34–48], Sklyanin introduced a family of algebras [Formula: see text], consisting of four generators and six relations. These algebras hold profound connections with his work on the Quantum Scattering Method. Subsequently, Feigin and Odesskii defined algebras [Formula: see text] for any number of generators [Formula: see text] [Sklyanin’s algebras associated to elliptic curves, Inst. Theor. Phys., Akad. Nauk Ukrain. SSR, Kiev (1988)] and asserted that, for [Formula: see text], they coincide with Sklyanin’s algebras. In this paper, we provide an explicit isomorphism between the algebras [Formula: see text] and [Formula: see text]. As an application, we present a new proof of the Calabi–Yau property for [Formula: see text].

Quantization of Lorentzian free BV theories: factorization algebra vs algebraic quantum field theory

We construct and compare two alternative quantizations, as a time-orderable prefactorization algebra and as an algebraic quantum field theory valued in cochain complexes, of a natural collection of free BV theories on the category of m-dimensional globally hyperbolic Lorentzian manifolds. Our comparison is realized as an explicit isomorphism of time-orderable prefactorization algebras. The key ingredients of our approach are the retarded and advanced Green’s homotopies associated with free BV theories, which generalize retarded and advanced Green’s operators to cochain complexes of linear differential operators.

An explicit isomorphism of different representations of the Ext functor using residue currents

An explicit isomorphism of different representations of the Ext functor using residue currents

Geometric Hodge filtered complex cobordism

We construct a geometric cycle model for a Hodge filtered extension of complex cobordism for every smooth manifold with a filtration on its de Rham complex with complex coefficients. Using a refinement of the Pontryagin–Thom construction, we construct an explicit isomorphism between our geometric model and the abstract model of Hodge filtered complex cobordism of Hopkins–Quick for every complex manifold with the Hodge filtration.

Path isomorphisms between quiver Hecke and diagrammatic Bott–Samelson endomorphism algebras

We construct an explicit isomorphism between (truncations of) quiver Hecke algebras and Elias–Williamson's diagrammatic endomorphism algebras of Bott–Samelson bimodules. As a corollary, we deduce that the decomposition numbers of these algebras (including as examples the symmetric groups and generalised blob algebras) are tautologically equal to the associated p-Kazhdan–Lusztig polynomials, provided that the characteristic is greater than the Coxeter number. We hence give an elementary and more explicit proof of the main theorem of Riche–Williamson's recent monograph and extend their categorical equivalence to cyclotomic quiver Hecke algebras, thus solving Libedinsky–Plaza's categorical blob conjecture.

Eigenfunctions of Transfer Operators and Automorphic Forms for Hecke Triangle Groups of Infinite Covolume

We develop cohomological interpretations for several types of automorphic forms for Hecke triangle groups of infinite covolume. We then use these interpretations to establish explicit isomorphisms between spaces of automorphic forms, cohomology spaces and spaces of eigenfunctions of transfer operators. These results show a deep relation between spectral entities of Hecke surfaces of infinite volume and the dynamics of their geodesic flows.

On the functoriality of 𝔰𝔩2 tangle homology

We construct an explicit equivalence between the (bi)category of gl(2) webs and foams and the Bar-Natan (bi)category of Temperley-Lieb diagrams and cobordisms. With this equivalence we can fix functoriality of every link homology theory that factors through the Bar-Natan category. To achieve this, we define web versions of arc algebras and their quasi-hereditary covers, which provide strictly functorial tangle homologies. Furthermore, we construct explicit isomorphisms between these algebras and the original ones based on Temperley-Lieb cup diagrams. The immediate application is a strictly functorial version of the Beliakova-Putyra-Wehrli quantization of the annular link homology.

On the joins of group rings

Given a collection {Gi}i=1d of finite groups and a ring R, we define a subring of the ring Mn(R) (n=∑i=1d|Gi|) that encompasses all the individual group rings R[Gi] along the diagonal blocks as Gi-circulant matrices. The precise definition of this ring was inspired by a construction in graph theory known as the joined union of graphs. We call this ring the join of group rings and denote it by JG1,…,Gd(R). In this paper, we present a systematic study of the algebraic structure of JG1,…,Gd(R). We show that it has a ring structure and characterize its center, group of units, and Jacobson radical. When R=k is an algebraically closed field, we derive a formula for the number of irreducible modules over JG1,…,Gd(k). We also show how a blockwise extension of the Fourier transform provides both a generalization of the Circulant Diagonalization Theorem to joins of circulant matrices and an explicit isomorphism between the join algebra and its Wedderburn components.

Criticism of the Philosophy of Dual Isomorphism of Explicit and Implicit and Its Evolution Law: An Interpretation of Yang Jiyong's Thesis and Works from Literary and Artistic Viewpoints

To evaluate whether the book "Literary Aesthetics from the Perspective of Explicit and Implicit Philosophy" and its related papers have academic rationality and theoretical value, this article relates to theoretical foundations of Liu Xie's "Wen Xin Diao Long", Taoist thought in the pre-Qin period, classics in the Han Dynasty and Wei and Jin Dynasties, and The Six Dynasties Metaphysics and so on. Adopting specific methods such as induction and deduction, analogy analysis and synthesis, and the combination of theory and history, it is discovered that the theory of explicit and implicit binary isomorphism (as well as its evolution and hidden theoretical composition), the view that it emphasizes the way things are spoken and the pursuit of clarity must also be based on the hidden as the source, the aesthetic diagram of the relationship between life vitality and the hidden, the research methods that emphasizes the metamorphosis of metaphoric light and dark, embodying the real possibility of poetry, and confirming the invisible origin of all things are related to modern natural science do have certain theoretical and practical enlightenment value. For example, it shows that it has theoretical significance for exploring, arguing, summarizing, and inheriting the clues of Chinese traditional culture and the essential characteristics of history, and it has the highlight and enlightenment of the essence of historical philosophy; so, it can be seen that the academic circles highly value this monograph, indicating that attention to and in-depth criticism are both very necessary.

Multiplicative preprojective algebras of Dynkin quivers

For a commutative ring $R$ and an ADE Dynkin quiver $Q$, we prove that the multiplicative preprojective algebra of Crawley-Boevey and Shaw, with parameter $q=1$, is isomorphic to the (additive) preprojective algebra as $R$-algebras if and only if the bad primes for $Q$ (2 in type D, 2 and 3 for $Q = E_{6}$, $E_{7}$ and 2, 3 and 5 for $Q= E_{8}$) are invertible in $R$. We construct an explicit isomorphism over $\mathbb{Z}[1/2]$ in type D, over $\mathbb{Z}[1/2, 1/3]$ for $Q = E_{6}$, $E_{7}$ and over $\mathbb{Z}[1/2, 1/3, 1/5]$ for $Q=E_{8}$. Conversely, if some bad prime is not invertible in $R$, we show that the additive and multiplicative preprojective algebras differ in zeroth Hochschild homology, and hence are not isomorphic. In fact, one only needs the vanishing of certain classes in zeroth Hochschild homology of the multiplicative preprojective algebra, utilizing a rigidification argument for isomorphisms that may be of independent interest. In the setting of Ginzburg dg-algebras, our obstructions are new in type E and give a more elementary proof of the negative result of Etg\"{u}--Lekili arXiv:1502.07922 in type D. Moreover, the zeroth Hochschild homology of the multiplicative preprojective algebra can be interpreted as the space of unobstructed deformations of the multiplicative Ginzburg dg-algebra by Van den Bergh duality. Finally, we observe that the multiplicative preprojective algebra is not symmetric Frobenius if $Q \neq A_1$, a departure from the additive preprojective algebra in characteristic 2 for $Q = D_{2n}$, $n \geq 2$ and $Q =E_{7}$, $E_{8}$.

Homological mirror symmetry for log Calabi–Yau surfaces

Given a log Calabi-Yau surface $Y$ with maximal boundary $D$ and distinguished complex structure, we explain how to construct a mirror Lefschetz fibration $w: M \to \mathbb{C}$, where $M$ is a Weinstein four-manifold, such that the directed Fukaya category of $w$ is isomorphic to $D^b \text{Coh}(Y)$, and the wrapped Fukaya category $\mathcal{W} (M)$ is isomorphic to $D^b \text{Coh}(Y \backslash D)$. We construct an explicit isomorphism between $M$ and the total space of the almost-toric fibration arising in the work of Gross-Hacking-Keel; when $D$ is negative definite this is expected to be the Milnor fibre of a smoothing of the dual cusp of $D$. We also match our mirror potential $w$ with existing constructions for a range of special cases of $(Y,D)$, notably in work of Auroux-Katzarkov-Orlov and Abouzaid.

Bigraded Lie Algebras Related to Multiple Zeta Values

We prove that the dihedral Lie coalgebra D_{\bullet \bullet}:=\bigoplus_{k\geq m \geq 1} D_{m,k} corresponding to {\widehat{\mathscr{D}}_{\bullet \bullet}(G)} of Goncharov (Duke Math. J. 110 (2001), 397–487) for G=\{e\} is the bigraded dual of the linearized double shuffle Lie algebra \mathfrak{ls}:=\bigoplus_{k\geq m \geq 1}\mathfrak{ls}_m^k\subset \mathbb{Q}\langle x,z \rangle of Brown (Compos. Math. 157 (2021), 529–572) whose Lie bracket is the Ihara bracket initially defined over \mathbb{Q}\langle x,z \rangle , by constructing an explicit isomorphism of bigraded Lie coalgebras D_{\bullet \bullet} \to \mathfrak{ls}^\vee , where \mathfrak{ls}^\vee is the Lie coalgebra dual (in the bigraded sense) to \mathfrak{ls} . The work leads to the equivalence between the two statements “ D_{\bullet \bullet} is a Lie coalgebra with respect to Goncharov’s cobracket formula in Goncharov (Duke Math. J. 110 (2001), 397–487)” and “ \mathfrak{ls} is preserved by the Ihara bracket''. We also prove folklore results from Brown (Compos. Math. 157 (2021), 529–572) and Ihara et al. (Compos. Math. 142 (2006), 307–338) (which apparently have no written proofs in the literature) stating that for m \geq 2 , D_{m,\bullet}:=\bigoplus_{k\geq m} D_{m,k} is graded isomorphic (dual) to the double shuffle space \mathrm{Dsh}_m:=\bigoplus_{k\geq m} \mathrm{Dsh}_{m}({{k}-m}) \subset \mathbb{Q}[x_1,\dots,x_m] (stated in Ihara et al., Compos. Math. \textbf{142} (2006), 307–338), and that the linear map f_m\colon \mathbb{Q}\langle x,z \rangle_m \to \mathbb{Q}[x_1,\dots,x_m] , where \mathbb{Q}\langle x,z \rangle_m is the space linearly generated by monomials of \mathbb{Q}\langle x,z \rangle of degree m with respect to z , given by x^{n_1}z\cdots x^{n_m}zx^{n_{m+1}}\mapsto \delta_{0,n_{m+1}} x_1^{n_1}\cdots x_{n_m}^{n_m} , with \delta_{a,b} the Kronecker delta, restricts to a graded isomorphism \bar{f}_m\colon \mathfrak{ls}_m:=\bigoplus_{k\geq m} \mathfrak{ls}_m^k \to \mathrm{Dsh}_{m} (stated in Brown (Compos. Math. 157 (2021), 529–572)). Here, we establish three explicit compatible isomorphisms D_{\bullet \bullet} \to \mathfrak{ls}^\vee, D_{m\bullet}\to \mathrm{Dsh}_{m}^\vee and \bar{f}_m\colon \mathfrak{ls}_m \to \mathrm{Dsh}_{m} , where \mathrm{Dsh}_{m}^\vee is the graded dual of \mathrm{Dsh}_{m} .

Pell hyperbolas in DLP–based cryptosystems

We present a study on the use of Pell hyperbolas in cryptosystems with security based on the discrete logarithm problem. Specifically, after introducing the group structure over generalized Pell hyperbolas (and also giving the explicit isomorphisms with the classical Pell hyperbolas), we provide a parameterization with both an algebraic and a geometrical approach. The particular parameterization that we propose appears to be useful from a cryptographic point of view because the product that arises over the set of parameters is connected to the Rédei rational functions, which can be evaluated in a fast way. Thus, we exploit these constructions for defining three different public key cryptosystems based on the ElGamal scheme. We show that the use of our parameterization allows to obtain schemes more efficient than the classical ones based on finite fields.

The Hilbert modular group and orthogonal groups

We derive an explicit isomorphism between the Hilbert modular group and certain congruence subgroups on the one hand and particular subgroups of the special orthogonal group SO(2, 2) on the other hand. The proof is based on an application of linear algebra adapted to number theoretical needs.

Higher Du Bois singularities of hypersurfaces

For a complex algebraic variety X $X$ , we introduce higher p $p$ -Du Bois singularity by imposing canonical isomorphisms between the sheaves of Kähler differential forms Ω X q $\Omega _X^q$ and the shifted graded pieces of the Du Bois complex Ω ̲ X q $\underline{\Omega }_X^q$ for q ⩽ p $q\leqslant p$ . If X $X$ is a reduced hypersurface, we show that higher p $p$ -Du Bois singularity coincides with higher p $p$ -log canonical singularity, generalizing a well-known theorem for p = 0 $p=0$ . The assertion that p $p$ -log canonicity implies p $p$ -Du Bois has been proved by Mustata, Olano, Popa, and Witaszek quite recently as a corollary of two theorems asserting that the sheaves of reflexive differential forms Ω X [ q ] $\Omega _X^{[q]}$ ( q ⩽ p $q\leqslant p$ ) coincide with Ω X q $\Omega _X^q$ and Ω ̲ X q $\underline{\Omega }_X^q$ , respectively, and these are shown by calculating the depth of the latter two sheaves. We construct explicit isomorphisms between Ω X q $\Omega _X^q$ and Ω ̲ X q $\underline{\Omega }_X^q$ applying the acyclicity of a Koszul complex in a certain range. We also improve some non-vanishing assertion shown by them using mixed Hodge modules and the Tjurina subspectrum in the isolated singularity case. This is useful for instance to estimate the lower bound of the maximal root of the reduced Bernstein–Sato polynomial in the case where a quotient singularity is a hypersurface and its singular locus has codimension at most 4.