Interest in combinatorial interpretations of mathematical entities stems from the convenience of the concrete models they provide. Finding a bijective proof of a seemingly obscure identity can reveal unsuspected significance to it. Finding a combinatorial model for some mathematical entity is a particular instance of the process called ``categorification''. Examples include the interpretation of N as the Burnside rig of the category of finite sets with product and coproduct, and the interpretation of N[x] as the category of combinatorial species. This has interesting applications to quantum mechanics, and in particular the quantum harmonic oscillator, via Joyal's ``species'', a new generalization called ``stuff types'', and operators between these, which can be represented as rudimentary Feynman diagrams for the oscillator. In quantum mechanics, we want to represent states in an algebra over the complex numbers, and also want our Feynman diagrams to carry more structure than these ``stuff operators'' can do, and these turn out to be closely related. We will show how to construct a combinatorial model for the quantum harmonic oscillator in which the group of phases, U(1), plays a special role. We describe a general notion of ``M-Stuff Types'' for any monoid M, and see that the case M=U(1) provides an interpretation of time evolution in the combinatorial setting, as well as other quantum mechanical features of the harmonic oscillator.
We define Lie 3-algebras and prove that these are in 1-to-1 correspondence with the 3-term Lie infinity algebras whose bilinear and trilinear maps vanish in degree (1,1) and in total degree 1, respectively. Further, we give an answer to a question of [Roy07] pertaining to the use of the nerve and normalization functors in the study of the relationship between categorified algebras and truncated sh algebras.
This expository article explains how planar diagrammatics naturally arise in the study of categorified quantum groups with a focus on the categorification of quantum sl2. We derive the definition of categorified quantum sl2 and highlight some of the new structure that arises in categorified quantum groups. The expert will find a discussion of rescalling isomorphisms for categorified quantum sl2, a proof that cyclotomic quotients of the nilHecke algebra are isomorphic to matrix rings over the cohomology ring of Grassmannians, and an interpretation of `fake bubbles' using symmetric functions.
We give an introductory account of Khovanov's categorification of the Heisenberg algebra, and construct a combinatorial model for it in a 2-category of spans of groupoids. We also treat a categorification of $U(sl_n)$ in a similar way. These give rise to standard representations on vector spaces through a linearization process.
Categorified quantum groups play an increasing role in quantum topology and representation theory. The Steenrod algebra is a fundamental component of algebraic topology. In this paper we show that categorified quantum groups can be extended to module categories over the Steenrod algebra in a natural way. This yields an intepretation of the small quantum group by Khovanov and Qi.
We identify the trace, or 0th Hochschild homology, of type ADE categorified quantum groups with the corresponding current algebra of the same type. To prove this, we show that 2-representations defined using categories of modules over cyclotomic (or deformed cyclotomic) quotients of KLR-algebras correspond to local (or global) Weyl modules. We also investigate the implications for centers of categories in 2-representations of categorified quantum groups.
In this note we give explicit isomorphisms of 2-categories between various versions of the categorified quantum group associated to a simply-laced Kac-Moody algebra. These isomorphisms are convenient when working with the categorified quantum group. They make it possible to translate results from the gl(n) variant of the 2-category to the sl(n) variant and transfer results between various conventions in the literature. We also extend isomorphisms of finite type KLR algebras for different choices of parameters to the level of 2-categories.
We describe how certain cyclotomic Nazarov-Wenzl algebras occur as endomorphism rings of projective modules in a parabolic version of BGG category O of type $D$. Furthermore we study a family of subalgebras of these endomorphism rings which exhibit similar behaviour to the family of Brauer algebras even when they are not semisimple. The translation functors on this parabolic category O are studied and proven to yield a categorification of a coideal subalgebra of the general linear Lie algebra. Finally this is put into the context of categorifying skew Howe duality for these subalgebras.
Quantum field theory allows more general symmetries than groups and Lie algebras. For instance quantum groups, that is Hopf algebras, have been familiar to theoretical physicists for a while now. Nowdays many examples of symmetries of categorical flavor -- categorical groups, groupoids, Lie algebroids and their higher analogues -- appear in physically motivated constructions and faciliate constructions of geometrically sound models and quantization of field theories. Here we consider two flavours of categorified symmetries: one coming from noncommutative algebraic geometry where varieties them…
We equip the categorified quantum group attached to a KLR algebra and an arbitrary choice of scalars with duality functor which is cyclic, that is, such that f=f^** for all 2-morphisms f. This is accomplished via a modified diagrammatic formalism.
We equip Ellis and Brundan's version of the odd categorified quantum group for sl(2) with a differential giving it the structure of a graded dg-2-supercategory. The presence of the super grading gives rise to two possible decategorifications of the associated dg-2-category. One version gives rise to a categorification of quantum sl(2) at a fourth root of unity, while the other version produces a subalgebra of quantum gl(1|1) defined over the integers. Both of these algebras appear in connection with quantum algebraic approaches to the Alexander polynomial.
The trace or the $0$th Hochschild--Mitchell homology of a linear category $\mathcal{C}$ may be regarded as a kind of decategorification of $\mathcal{C}$. We compute traces of the two versions $\dot{\mathcal{U}}$ and $\dot{\mathcal{U}}^*$ of categorified quantum $\mathfrak{sl}_2$ introduced by the third author. One version of the trace coincides with the split Grothendieck group $K_0(\dot{\mathcal{U}})$, which is known to be isomorphic to the the integral idempotented form $\dot{\mathbf{U}}(\mathfrak{sl}_2)$ of quantum $\mathfrak{sl}(2)$. The higher Hochschild--Mitchell homology in this case is…
We introduce a simple diagrammatic 2-category $\mathscr{A}$ that categorifies the image of the Fock space representation of the Heisenberg algebra and the basic representation of $\mathfrak{sl}_\infty$. We show that $\mathscr{A}$ is equivalent to a truncation of the Khovanov--Lauda categorified quantum group $\mathscr{U}$ of type $A_\infty$, and also to a truncation of Khovanov's Heisenberg 2-category $\mathscr{H}$. This equivalence is a categorification of the principal realization of the basic representation of $\mathfrak{sl}_\infty$. As a result of the categorical equivalences described abo…
Categorification is a process of lifting structures to a higher categorical level. The original structure can then be recovered by means of the so-called "decategorification" functor. Algebras are typically categorified to additive categories with additional structure and decategorification is usually given by the (split) Grothendieck group. In this expository article we study an alternative decategorification functor given by the trace or the zeroth Hochschild--Mitchell homology. We show that this form of decategorification endows any 2-representation of the categorified quantum sl(n) with an…
This is a follow-up to the paper in which we categorified the affine quantum Schur algebra S(n,r) for 2 < r < n, using a quotient of Khovanov and Lauda's categorification of the affine quantum sl_n. In this paper we categorify S(n,n) for n > 2, using an extension of the aforementioned quotient.
In this paper, we generalize the categorifical construction of a quantum group and its canonical basis introduced by Lusztig (\cite{Lusztig,Lusztig2}) to the generic form of the whole Ringel-Hall algebra. We clarify the explicit relation between the Green formula in \cite{Green} and the restriction functor in \cite{Lusztig2}. By a geometric way to prove the Green formula, we show that the Hopf structure of a Ringel-Hall algebra can be categorified under Lusztig's framework.
Let ${\mathbf U}^-_q$ be the negative half of the quantum group associated to a Kac-Moody algebra ${\mathfrak g}$, and $\underline{\mathbf U}^-_q$ the quantum group obtained by a folding of ${\mathfrak g}$. Let ${\mathbf A} = {\mathbf Z}[q,q^{-1}]$. McNamara showed that $\underline{\mathbf U}^-_q$ is categorified over a certain extenion ring $\widetilde{\mathbf A}$ of ${\mathbf A}$, by uing the folding theory of KLR algebras. He posed a question whether $\widetilde{\mathbf A}$ coincides with ${\mathbf A}$ or not. In this paper, we give an affirmative answer for this problem.
We show that in ADE type the trace of Webster's categorification of a tensor product of irreducibles for the quantum group is isomorphic to a tensor product of Weyl modules for the current algebra $\dot{U}(\mathfrak{g}[t])$. This extends a result of Beliakova, Habiro, Lauda, and Webster who showed that the trace of the categorified quantum group $\dot{\mathcal{U}}^*(\mathfrak{g})$ is isomorphic to $\dot{U}(\mathfrak{g}[t])$, and the trace of a cyclotomic quotient of $\dot{\mathcal{U}}^*(\mathfrak{g})$, which categorifies a single irreducible for the quantum group, is isomorphic to a Weyl modul…
Building on work by Kontsevich, Soibelman, Nagao and Efimov, we prove the positivity of quantum cluster coefficients for all skew-symmetric quantum cluster algebras, via a proof of a conjecture first suggested by Kontsevich on the purity of mixed Hodge structures arising in the theory of cluster mutation of spherical collections in 3-Calabi-Yau categories. The result implies positivity, as well as the stronger Lefschetz property conjectured by Efimov, and also the classical positivity conjecture of Fomin and Zelevinsky, recently proved by Lee and Schiffler. Closely related to these results is …
We introduce a non-degenerate bilinear form and use it to provide a new characterization of quantum Kac-Moody superalgebras with no isotropic odd simple roots. We show that the spin quiver Hecke algebras introduced by Kang-Kashiwara-Tsuchioka provide a categorification of half the quantum Kac-Moody superalgebras, using the recent work of Ellis-Khovanov-Lauda. A new idea here is that a super sign is categorified as spin (i.e., the parity-shift functor).
We re-examine the appearance of semiheaps and (para-associative) ternary algebras in quantum mechanics. In particular, we review the construction of a semiheap on a Hilbert space and the set of bounded operators on a Hilbert space. The new aspect of this work is a discussion of how symmetries of a quantum system induce homomorphisms of the relevant semiheaps and ternary algebras.
$C_λ$-extended oscillator algebras generalizing the Calogero-Vasiliev algebra, where $C_λ$ is the cyclic group of order $λ$, are studied both from mathematical and applied viewpoints. Casimir operators of the algebras are obtained, and used to provide a complete classification of their unitary irreducible representations under the assumption that the number operator spectrum is nondegenerate. Deformed algebras admitting Casimir operators analogous to those of their undeformed counterparts are looked for, yielding three new algebraic structures. One of them includes the Brzeziński {\em et al.} …
C$_λ$-extended oscillator algebras, where C$_λ$ is the cyclic group of order $λ$, are introduced and realized as generalized deformed oscillator algebras. For $λ=2$, they reduce to the well-known Calogero-Vasiliev algebra. For higher $λ$ values, they are shown to provide in their bosonic Fock space representation some interesting applications to supersymmetric quantum mechanics and some variants thereof: an algebraic realization of supersymmetric quantum mechanics for cyclic shape invariant potentials of period $λ$, a bosonization of parasupersymmetric quantum mechanics of order $p = λ-1$, and…
In this paper we will present an ongoing project which aims to use model theory as a suitable mathematical setting for studying the formalism of quantum mechanics. We will argue that this approach provides a geometric semantics for such formalism by means of establishing a (non-commutative) duality between certain algebraic and geometric objects.
The C_λ-extended oscillator algebra is generated by {1,a,a^{\dagger},N,T}, where T is the generator of the cyclic group C_λ of order λ. It can be realized as a generalized deformed oscillator algebra (GDOA). Its unirreps can thus be exhibited using the representation theory of GDOAs and their carrier space show a Z_λ grading structure. Within its infinite-dimensional Fock space representation, this algebra provides a bosonization of parasupersymmetric quantum mechanics of order p= λ- 1.
Besides offering a friendly introduction to knot homologies and quantum curves, the goal of these lectures is to review some of the concrete predictions that follow from the physical interpretation of knot homologies. In particular, this interpretation allows one to pose questions that would not have been asked otherwise, such as, "Is there a direct relation between Khovanov homology and the A-polynomial of a knot?" We will explain that the answer to this question is "yes," and introduce a certain deformation of the planar algebraic curve defined by the zero locus of the A-polynomial. This nov…
