Limits Of Algebras Research Articles

Classifying large N limits of multiscalar theories by algebra

We develop a new approach to RG flows and show that one-loop flows in multiscalar theories can be described by commutative but non-associative algebras. As an example related to D-brane field theories and tensor models, we study the algebra of a theory with M SU(N) adjoint scalars and its large N limits. The algebraic concepts of idempotents and Peirce numbers/Kowalevski exponents are used to characterise the RG flows. We classify and describe all large N limits of algebras of multiadjoint scalar models: the standard ‘t Hooft matrix theory limit, a ‘multi-matrix’ limit, each with one free parameter, and an intermediate case with extra symmetry and no free parameter of the algebra, but an emergent free parameter from a line of one-loop fixed points. The algebra identifies these limits without diagrammatic or combinatorial analysis.

Simple representations of BPS algebras: the case of Y(gl^2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Y(\\widehat{\\mathfrak {gl}}_2)$$\\end{document}

BPS algebras are the symmetries of a wide class of brane-inspired models. They are closely related to Yangians – the peculiar and somewhat sophisticated limit of DIM algebras. Still they possess some simple and explicit representations. We explain here that for Y(gl^r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Y(\\widehat{\\mathfrak {gl}}_r)$$\\end{document} these representations are related to Uglov polynomials, whose families are also labeled by natural r. They arise in the limit ħ⟶0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbar {\\longrightarrow } 0$$\\end{document} from Macdonald polynomials, and generalize the well-known Jack polynomials (β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}-deformation of Schur functions), associated with r=1.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r=1.$$\\end{document} For r=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r=2$$\\end{document} they approximate Macdonald polynomials with the accuracy O(ħ2),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(\\hbar ^2),$$\\end{document} so that they are eigenfunctions of two immediately available commuting operators, arising from the ħ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbar $$\\end{document}-expansion of the first Macdonald Hamiltonian. These operators have a clear structure, which is easily generalizable, – what provides a technically simple way to build an explicit representation of Yangian Y(gl^2),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Y(\\widehat{\\mathfrak {gl}}_2),$$\\end{document} where U(2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U^{(2)}$$\\end{document} are associated with the states |λ⟩,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|\\lambda {\\rangle },$$\\end{document} parametrized by chess-colored Young diagrams. An interesting feature of this representation is that the odd time-variables p2n+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_{2n+1}$$\\end{document} can be expressed through mutually commuting operators from Yangian, however even time-variables p2n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_{2n}$$\\end{document} are inexpressible. Implications to higher r become now straightforward, yet we describe them only in a sketchy way.

Free limits of free algebras

Consider a diagram ⋯→F3→F2→F1 of algebraic systems, where Fn denotes the free object on n generators and the connecting maps send the extra generator to some distinguished trivial element. We prove that (a) if the Fi are free associative algebras over a fixed field then the limit in the category of graded algebras is again free on a set of homogeneous generators; (b) on the other hand, the limit in the category of associative (ungraded) algebras is a free formal power series algebra on a set of homogeneous elements, and (c) if the Fi are free Lie algebras then the limit in the category of graded Lie algebras is again free.

Algebraic valuation ring extensions as limits of complete intersection algebras

We show that an algebraic immediate valuation ring extension of characteristic $$p>0$$ is a filtered union of complete intersection algebras of finite type.

The small index property of the Fraïssé limit of finite Heyting algebras

We show that if a subgroup of the automorphism group of the Fraïssé limit of finite Heyting algebras has a countable index, then it lies between the pointwise and setwise stabilizer of some finite set.

On ultragraph Leavitt path algebras with finite Gelfand-Kirillov dimension

In this article, we prove that every ultragraph Leavitt path algebra is a direct limit of Leavitt path algebras of finite graphs and determine the Gelfand-Kirillov dimension of an ultragraph Leavitt path algebra. We also characterize ultragraph Leavitt path algebras whose simple modules are finitely presented, and show that these algebras have finite Gelfand-Kirillov dimension. Moreover, we construct new classes of simple modules over ultragraph Leavitt path algebras associated with minimal infinite emitters and minimal sinks, which have not appeared in the context of Leavitt path algebras of graphs.

Representations of weakly triangular categories

A new class of locally unital and locally finite dimensional algebras A over an arbitrary algebraically closed field is discovered. Each of them admits an upper finite weakly triangular decomposition, a generalization of a split triangular decomposition. It is established that the category A-lfdmod of locally finite dimensional left A-modules is an upper finite fully stratified category in the sense of Brundan-Stroppel. Moreover, A is semisimple if and only if its centralizer subalgebras associated to certain idempotent elements are semisimple. Furthermore, certain endofunctors are defined so as to give categorical actions of some Lie algebras on the subcategory of A-lfdmod consisting of all objects which have a finite standard filtration.As an application, we study representations of A associated to either cyclotomic Brauer categories or cyclotomic Kauffman categories in details, including explicit criteria on the semisimplicity of A over an arbitrary field, and on A-lfdmod being upper finite highest weight category in the sense of Brundan-Stroppel, and on Morita equivalence between A and direct sum of infinitely many (degenerate) cyclotomic Hecke algebras. Finally, we obtain categorifications of representations of the classical limits of coideal algebras, which come from all integrable highest weight modules of sl∞ or slˆe.

THE AUTOMORPHISM GROUP OF THE FRAÏSSÉ LIMIT OF FINITE HEYTING ALGEBRAS

AbstractRoelcke non-precompactness, simplicity, and non-amenability of the automorphism group of the Fraïssé limit of finite Heyting algebras are proved among others.

Constructing equivalence bimodules between noncommutative solenoids: A two-pronged approach

We revisit and generalize the application of a method introduced by Latrémolière and Packer for constructing finitely generated projective modules over the noncommutative solenoid C*-algebras. By realizing them as direct limits of rotation algebras, the method constructs directed systems of equivalence bimodules between rotation algebras that satisfy the necessary compatibility conditions to build Morita equivalence bimodules between the direct limit C*-algebras. In the irrational case, we use a fixed projection in a matrix algebra over the rotation algebra satisfying a key condition to build an equivalence bimodule at each stage following a construction of Rieffel. From this, our main result shows that two irrational noncommutative solenoids are Morita equivalent if and only if such a projection exists. We also make additional observations about the Heisenberg bimodules construction studied by the aforementioned two authors and connect the two constructions.

The large N limit of orbifold vertex operator algebras

We investigate the large N limit of permutation orbifolds of vertex operator algebras. To this end, we introduce the notion of nested oligomorphic permutation orbifolds and discuss under which conditions their fixed point VOAs converge. We show that if this limit exists, then it has the structure of a vertex algebra. Finally, we give an example based on mathrm {GL}(N,q) for which the fixed point VOA limit is also the limit of the full permutation orbifold VOA.

ON A GENERALIZED FRAÏSSÉ LIMIT CONSTRUCTION AND ITS APPLICATION TO THE JIANG–SU ALGEBRA

AbstractIn this paper, we present a version of Fraïssé theory for categories of metric structures. Using this version, we show that every UHF algebra can be recognized as a Fraïssé limit of a class of C*-algebras of matrix-valued continuous functions on cubes with distinguished traces. We also give an alternative proof of the fact that the Jiang–Su algebra is the unique simple monotracial C*-algebra among all the inductive limits of prime dimension drop algebras.

The classification of simple separable KK-contractible C*-algebras with finite nuclear dimension

The class of simple separable KK-contractible (KK-equivalent to {0}) C*-algebra s which have finite nuclear dimension is shown to be classified by the Elliott invariant. In particular, the class of C*-algebra s A⊗W is classifiable, where A is a simple separable C*-algebra with finite nuclear dimension and W is the simple inductive limit of Razak algebras with unique trace, which is bounded (see Razak (2002) and Jacelon (2013)).

Blaschke $$\boldsymbol{C}^{\boldsymbol{*}}$$-algebras

In the paper we introduce the notion of the Blaschke $$C^{*}$$ -algebra and we consider the isometric representations of uniform Blaschke algebra. We extend the Coburn’s Theorem for a family of non-unitary isometries connected by a family of finite Blaschke products. We show that the Blaschke $$C^{*}$$ -algebra is isomorphic to the inductive limit of Toeplitz algebras, as well as the (non commutative) $$C^{*}$$ -algebra generated by a unital isometric representation of the uniform Blaschke algebra by multiplications in the respective Hardy space.

Modified affine Hecke algebras and quiver Hecke algebras of type A

We introduce some modified forms for the degenerate and non-degenerate affine Hecke algebras of type A. These are certain subalgebras living inside the inverse limit of cyclotomic Hecke algebras. We construct faithful representations and standard bases for these algebras and give some explicit description of their centers. We show that there are algebra isomorphisms between some generalized Ore localizations of these modified affine Hecke algebras and of the quiver Hecke algebras of type A. As an application, we show that the center conjecture for the cyclotomic quiver Hecke algebra of type A holds if and only if the center conjecture for the cyclotomic Hecke algebra of type A holds.

Limits of three-dimensional gravity and metric kinematical Lie algebras in any dimension

We extend a recent classification of three-dimensional spatially isotropic homogeneous spacetimes to Chern-Simons theories as three-dimensional gravity theories on these spacetimes. By this we find gravitational theories for all carrollian, galilean, and aristotelian counterparts of the lorentzian theories. In order to define a nondegenerate bilinear form for each of the theories, we introduce (not necessarily central) extensions of the original kinematical algebras. Using the structure of so-called double extensions, this can be done systematically. For homogeneous spaces that arise as a limit of (anti-)de Sitter spacetime, we show that it is possible to take the limit on the level of the action, after an appropriate extension. We extend our systematic construction of nondegenerate bilinear forms also to all higher-dimensional kinematical algebras.

PROFINITENESS IN FINITELY GENERATED VARIETIES IS UNDECIDABLE

AbstractProfinite algebras are exactly those that are isomorphic to inverse limits of finite algebras. Such algebras are naturally equipped with Boolean topologies. A variety ${\cal V}$ is standard if every Boolean topological algebra with the algebraic reduct in ${\cal V}$ is profinite.We show that there is no algorithm which takes as input a finite algebra A of a finite type and decide whether the variety $V\left( {\bf{A}} \right)$ generated by A is standard. We also show the undecidability of some related properties. In particular, we solve a problem posed by Clark, Davey, Freese, and Jackson.We accomplish this by combining two results. The first one is Moore’s theorem saying that there is no algorithm which takes as input a finite algebra A of a finite type and decides whether $V\left( {\bf{A}} \right)$ has definable principal subcongruences. The second is our result saying that possessing definable principal subcongruences yields possessing finitely determined syntactic congruences for varieties. The latter property is known to yield standardness.

Logarithmic W-algebras and Argyres-Douglas theories at higher rank

Families of vertex algebras associated to nilpotent elements of simply-laced Lie algebras are constructed. These algebras are close cousins of logarithmic W-algebras of Feigin and Tipunin and they are also obtained as modifications of semiclassical limits of vertex algebras appearing in the context of S-duality for four-dimensional gauge theories. In the case of type A and principal nilpotent element the character agrees precisely with the Schur-Index formula for corresponding Argyres-Douglas theories with irregular singularities. For other nilpotent elements they are identified with Schur-indices of type IV Argyres-Douglas theories. Further, there is a conformal embedding pattern of these vertex operator algebras that nicely matches the RG-flow of Argyres-Douglas theories as discussed by Buican and Nishinaka.

Shifted Poisson structures and moduli spaces of complexes

In this paper we study the moduli stack of complexes of vector bundles (with chain isomorphisms) over a smooth projective variety X via derived algebraic geometry. We prove that if X is a Calabi–Yau variety of dimension d then this moduli stack has a (1−d)-shifted Poisson structure. In the case d=1, we construct a natural foliation of the moduli stack by 0-shifted symplectic substacks. We show that our construction recovers various known Poisson structures associated to complex elliptic curves, including the Poisson structure on Hilbert scheme of points on elliptic quantum projective planes studied by Nevins and Stafford, and the Poisson structures on the moduli spaces of stable triples over an elliptic curves considered by one of us. We also relate the latter Poisson structures to the semi-classical limits of the elliptic Sklyanin algebras studied by Feigin and Odesskii.

Spectral triples on the Jiang-Su algebra

We construct spectral triples on a class of particular inductive limits of matrix-valued function algebras. In the special case of the Jiang-Su algebra, we employ a particular AF-embedding.

Cartan Theorems for Stein Manifolds Over a Discrete Valuation Base

Let X be a complex manifold, let A be a topological discrete valuation ring, and write for the sheaf of functions on X with values in A. We prove Cartan theorems A and B for coherent -modules, when X is a Stein manifold and A satisfies some requirements like being a nuclear direct limit of Banach algebras. The result is motivated by questions in the work of the second author with Kashiwara in the proof of the codimension-three conjecture for holonomic microdifferential systems.