Quantum trilogy: discrete Toda, Y-system and chaos

This chapter contains well-known facts about semisimple finite-dimensional Lie algebras. For proofs the reader is referred to the standard text (Humphreys (1972)).

We associate to an arbitrary positive root [Formula: see text] of a complex semisimple finite-dimensional Lie algebra [Formula: see text] a twisting endofunctor [Formula: see text] of the category of [Formula: see text]-modules. We apply this functor to generalized Verma modules in the category [Formula: see text] and construct a family of [Formula: see text]-Gelfand–Tsetlin modules with finite [Formula: see text]-multiplicities, where [Formula: see text] is a commutative [Formula: see text]-subalgebra of the universal enveloping algebra of [Formula: see text] generated by a Cartan subalgebra of [Formula: see text] and by the Casimir element of the [Formula: see text]-subalgebra corresponding to the root [Formula: see text]. This covers classical results of Andersen and Stroppel when [Formula: see text] is a simple root and previous results of the authors in the case when [Formula: see text] is a complex simple Lie algebra and [Formula: see text] is the maximal root of [Formula: see text]. The significance of constructed modules is that they are Gelfand–Tsetlin modules with respect to any commutative [Formula: see text]-subalgebra of the universal enveloping algebra of [Formula: see text] containing [Formula: see text]. Using the Beilinson–Bernstein correspondence we give a geometric realization of these modules together with their explicit description. We also identify a tensor subcategory of the category of [Formula: see text]-Gelfand–Tsetlin modules which contains constructed modules as well as the category [Formula: see text].

We consider associative algebras with an action by derivations by some finite dimensional and semisimple Lie algebra. We prove that if a differential variety has almost polynomial growth, then it is generated by one of the algebras U T 2 ( W λ ) UT_2(W_\lambda ) or E n d ( W μ ) End(W_\mu ) for some integral dominant weight λ , μ \lambda ,\mu with μ ≠ 0 \mu \neq 0 . In the special case L = s l 2 L=\mathfrak {sl}_2 we prove that this is a sufficient condition too.

If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German g"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">g</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathfrak g}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a Lie algebra of derivations of an associative algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , then the subalgebra of invariants is the set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Superscript German g Baseline equals left-brace r element-of upper R bar delta left-parenthesis r right-parenthesis equals 0 for all delta element-of German g right-brace period"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>R</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">g</mml:mi> </mml:mrow> </mml:mrow> </mml:msup> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi>r</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mi>R</mml:mi> <mml:mo> ∣ </mml:mo> <mml:mi> δ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>r</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mtext> </mml:mtext> <mml:mtext> for all </mml:mtext> <mml:mi> δ </mml:mi> <mml:mo> ∈ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">g</mml:mi> </mml:mrow> </mml:mrow> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">R^{\mathfrak g} = \{ r \in R \mid \delta (r) = 0 \ \text { for all } \delta \in {\mathfrak g}\}.</mml:annotation> </mml:semantics> </mml:math> </inline-formula> In this paper, we study the relationship between the structure of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Superscript German g"> <mml:semantics> <mml:msup> <mml:mi>R</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">g</mml:mi> </mml:mrow> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">R^{\mathfrak g}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the structure of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German g"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">g</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a finite dimensional semisimple Lie algebra over a field of characteristic zero acting finitely on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is semiprime.

In an earlier paper [On modules induced from Whittaker modules, J. Algebra 96 (1985)] we constructed a class of induced modules, over a finite-dimensional semisimple Lie algebra, which includes the Verma modules of Verma [Structure of certain induced representations of complex semisimple Lie algebras, Bull. Amer. Math. Soc. 74 (1968)] and the irreducible Whittaker modules of Kostant [On Whittaker vectors and representation theory, Invent. Math. 48 (1978)]. We proved that every module in this class has finite length and is irreducible most of the time. In this article we present a concrete example of this construction, over sl ⁡ ( 3 , C ) \operatorname {sl} (3,C) , showing that proper submodules can exist when the induced module is not a Verma module.

We propose a new realization, using Harish-Chandra bimodules, of the Serre functor for the BGG category 𝒪 associated to a semi-simple complex finite dimensional Lie algebra. We further show that our realization carries over to classical Lie superalgebras in many cases. Along the way we prove that category 𝒪 and its parabolic generalizations for classical Lie superalgebras are categories with full projective functors. As an application we prove that in many cases the endomorphism algebra of the basic projective-injective module in (parabolic) category 𝒪 for classical Lie superalgebras is symmetric. As a special case we obtain that in these cases the algebras describing blocks of the category of finite dimensional modules are symmetric. We also compute the latter algebras for the superalgebra 𝔮(2).

Let $\mathfrak{g}$ be a finite-dimensional semisimple complex Lie algebra and $\theta$ an involutive automorphism of $\mathfrak{g}$. According to G. Letzter, S. Kolb and M. Balagovi\'c the fixed-point subalgebra $\mathfrak{k} = \mathfrak{g}^\theta$ has a quantum counterpart $B$, a coideal subalgebra of the Drinfeld-Jimbo quantum group $U_q(\mathfrak{g})$ possessing a universal K-matrix $\mathcal{K}$. The objects $\theta$, $\mathfrak{k}$, $B$ and $\mathcal{K}$ can all be described in terms of Satake diagrams. In the present work we extend this construction to generalized Satake diagrams, combinatorial data first considered by A. Heck. A generalized Satake diagram naturally defines a semisimple automorphism $\theta$ of $\mathfrak{g}$ restricting to the standard Cartan subalgebra $\mathfrak{h}$ as an involution. It also defines a subalgebra $\mathfrak{k}\subset \mathfrak{g}$ satisfying $\mathfrak{k} \cap \mathfrak{h} = \mathfrak{h}^\theta$, but not necessarily a fixed-point subalgebra. The subalgebra $\mathfrak{k}$ can be quantized to a coideal subalgebra of $U_q(\mathfrak{g})$ endowed with a universal K-matrix in the sense of Kolb and Balagovi\'c. We conjecture that all such coideal subalgebras of $U_q(\mathfrak{g})$ arise from generalized Satake diagrams in this way.

We generalize the notion of coherent states toarbitrary Lie algebras by making an analogy with the GNSconstruction in C*-algebras. The method is illustratedwith examples of semisimple and nonsemisimple finite-dimensional Lie algebras as well as loopand Kac–Moody algebras. A deformed addition on theparameter space is also introduced simplifying someexpressions and some applications to conformal field theory are pointed out, e.g., differentialoperator and free field realizations found.

We present general symmetry arguments that show the appearance of doubly degenerate states protected from external perturbations in a wide class of Hamiltonians. We construct the simplest spin Hamiltonian belonging to this class and study its properties both analytically and numerically. We find that this model generally has a number of low energy modes which might destroy the protection in the thermodynamic limit. These modes are qualitatively different from the usual gapless excitations as their number scales as the linear size (instead of volume) of the system. We show that the Hamiltonians with this symmetry can be physically implemented in Josephson junction arrays and that in these arrays one can eliminate the low energy modes with a proper boundary condition. We argue that these arrays provide fault tolerant quantum bits. Further we show that the simplest spin model with this symmetry can be mapped to a very special ${\mathbb{Z}}_{2}$ Chern-Simons model on the square lattice. We argue that appearance of the low energy modes and the protected degeneracy is a natural property of lattice Chern-Simons theories. Finally, we discuss a general formalism for the construction of discrete Chern-Simons theories on a lattice.

We apply the principles discussed in an earlier paper to the construction of discrete time field theories. We derive the discrete time field equations of motion and Noether's theorem and apply them to the Schrödinger equation to illustrate the methodology. Stationary solutions to the discrete time Schrödinger wave equation are found to be identical to standard energy eigenvalue solutions except for a fundamental limit on the energy. Then we apply the formalism to the free neutral Klein - Gordon system, deriving the equations of motion and conserved quantities such as the linear momentum and angular momentum. We show that there is an upper bound on the magnitude of linear momentum for physical particle-like solutions. We extend the formalism to the charged scalar field coupled to Maxwell's electrodynamics in a gauge invariant way. We apply the formalism to include the Maxwell and Dirac fields, setting the scene for second quantization of discrete time mechanics and discrete time quantum electrodynamics.

In representation theory, the existence of a ℤ+-grading on a related finite dimensional algebra often plays an important role. For example, such a grading arises from the Koszul structure of the finite dimensional algebra representing the principal block of the BGG category 𝒪 associated to a complex semisimple Lie algebra. But Koszul gradings in positive characteristic have proved elusive. For example, except for small values of a positive integer n, it is not known if the Schur algebra S(n, n) has such a Koszul grading, assuming the characteristic p of the base field satisfies p ≥ n, though this grading would suffice to establish Lusztig's character formula for these algebras. (And even though the character formula is known for p sufficiently large [H. Andersen, J. Jantzen and W. Soergel, Representations of Quantum Groups at a pth Root of Unity and of Semisimple Groups in Characteristic p, Astérique, Vol. 220, 1994], it is not known if the Schur algebra is Koszul for p sufficiently large.) This paper introduces ℤ/2-gradings on quasi-hereditary algebras, and shows that these gradings are almost as useful as a full ℤ+-grading, while being possibly much easier to find. We define the notion of a ℤ/2-based Kazhdan–Lusztig theory, which appears to be more flexible than, and generalizes, the notion of a Kazhdan–Lusztig theory (as first defined in [E. Cline, B. Parshall and L. Scott, Abstract Kazhdan–Lusztig theories, Tôhoku Math. J. 45 (1993), 511–534]). However, its existence suffices, as was the case with the original notion, to establish character formulas in the standard settings, determine Extn-groups, and show that homological duals behave well. Finally, we present some suggestive symmetric group examples involving Schur algebras which were an outgrowth of this work.

Let A be a finite-dimensional algebra which is quasi-hereditary with re- spect to the poset (�,�), with standard modules �(�) for 2 �. Let F(�) be the category of A-modules which have filtrations where the quotients are standard modules. We determine some inductive results on the relative Auslander-Reiten quiver of F(�). Quasi-hereditary algebras were introduced by Cline, Parshall and Scott in the study of highest weight categories arising in the representation theory of semisimple Lie algebras and algebraic groups (2), and they were studied by Dlab and Ringel and others (4, 5). LetA be quasi-hereditary. Then the simple A-modules are labelled as L(�) forin some partially ordered set (�,≤). Of central importance are the standard modules, denoted by �(�), and the categoryF(�) of modules which have a filtration where the quotients are standard modules. In (12) it was proved that this category has relative Auslander-Reiten (AR) sequences. The aim of our paper is to present some inductive results on the relative Auslander-Reiten quiver ofF(�). For any subset of �, letF (�) be the category of modules inF(�) where all quotients are �(�) with �∈ . If � is an ideal in the poset (�,≤) then there is a subalgebra A of A which is quasi-hereditary with poset (, ≤). In this case, the categoryF (�) is equivalent toF(�A ). A main result in this paper relates, for a non-projective indecomposable mod- ule X inF (�), the two relative Auslander-Reiten sequences with cokernel X inF(�) and inF (�) (Theorem 2.5). By Ringel duality, there is an equivalent version for categories of modules filtered by costandard mod- ules. We describe this and some variations in Section 3. We continue with further results on relative Auslander-Reiten sequences. In particular, we determine certain relative Auslander-Reiten sequences where a projective- injective summand occurs; this generalizes the usual almost split sequence where a projective-injective module occurs.

We present a fully automated method that identifies attractors and their basins of attraction without approximations of the dynamics. The method works by defining a finite state machine on top of the dynamical system flow. The input to the method is a dynamical system evolution rule and a grid that partitions the state space. No prior knowledge of the number, location, or nature of the attractors is required. The method works for arbitrarily high-dimensional dynamical systems, both discrete and continuous. It also works for stroboscopic maps, Poincaré maps, and projections of high-dimensional dynamics to a lower-dimensional space. The method is accompanied by a performant open-source implementation in the DynamicalSystems.jl library. The performance of the method outclasses the naïve approach of evolving initial conditions until convergence to an attractor, even when excluding the task of first identifying the attractors from the comparison. We showcase the power of our implementation on several scenarios, including interlaced chaotic attractors, high-dimensional state spaces, fractal basin boundaries, and interlaced attracting periodic orbits, among others. The output of our method can be straightforwardly used to calculate concepts, such as basin stability and final state sensitivity.

Let g \mathfrak {g} be an affine Kac-Moody Lie algebra and let λ , μ \lambda , \mu be two dominant integral weights for g \mathfrak {g} . We prove that under some mild restriction, for any positive root β \beta , V ( λ ) ⊗ V ( μ ) V(\lambda )\otimes V(\mu ) contains V ( λ + μ − β ) V(\lambda +\mu -\beta ) as a component, where V ( λ ) V(\lambda ) denotes the integrable highest weight (irreducible) g \mathfrak {g} -module with highest weight λ \lambda . This extends the corresponding result by Kumar from the case of finite dimensional semisimple Lie algebras to the affine Kac-Moody Lie algebras. One crucial ingredient in the proof is the action of Virasoro algebra via the Goddard-Kent-Olive construction on the tensor product V ( λ ) ⊗ V ( μ ) V(\lambda )\otimes V(\mu ) . Then, we prove the corresponding geometric results including the higher cohomology vanishing on the G \mathcal {G} -Schubert varieties in the product partial flag variety G / P × G / P \mathcal {G}/\mathcal {P}\times \mathcal {G}/\mathcal {P} with coefficients in certain sheaves coming from the ideal sheaves of G \mathcal {G} -sub-Schubert varieties. This allows us to prove the surjectivity of the Gaussian map.

Building on Seidel and Solomon’s fundamental work [Symplectic cohomology and$q$-intersection numbers, Geom. Funct. Anal.22(2012), 443–477], we define the notion of a$\mathfrak{g}$-equivariant Lagrangian brane in an exact symplectic manifold$M$, where$\mathfrak{g}\subset SH^{1}(M)$is a sub-Lie algebra of the symplectic cohomology of$M$. When$M$is a (symplectic) mirror to an (algebraic) homogeneous space$G/P$, homological mirror symmetry predicts that there is an embedding of$\mathfrak{g}$in$SH^{1}(M)$. This allows us to study a mirror theory to classical constructions of Borel, Weil and Bott. We give explicit computations recovering all finite-dimensional irreducible representations of$\mathfrak{sl}_{2}$as representations on the Floer cohomology of an$\mathfrak{sl}_{2}$-equivariant Lagrangian brane and discuss generalizations to arbitrary finite-dimensional semisimple Lie algebras.