Representations of E7, equivalent combinatorics and algebraic varieties

Let $\mathfrak {g}\neq \mathfrak {s}\mathfrak {o}_{8}$ be a simple Lie algebra of type $A,D,E$ with $\widehat {\mathfrak {g}}$ the corresponding affine Kac–Moody algebra and $\mathfrak {n}_{+}\subset \widehat {\mathfrak {g}}$ is the standard positive nilpotent subalgebra. Given $\mathfrak {n}_{+}$ as above, we provide an infinite collection of cyclic finite abelian subgroups of $SL_{3}(\mathbb {C})$ with the following properties. Let $G$ be any group in the collection, $Y=G\operatorname {-}\mbox {Hilb}(\mathbb {C}^{3})$, and $\Psi : D^{b}_{G}(Coh(\mathbb {C}^{3}))\rightarrow D^{b}(Coh(Y))$ the derived equivalence of Bridgeland, King, and Reid. We present an (explicitly described) subset of objects in $Coh_{G}(\mathbb {C}^{3})$, s.t. the Hall algebra generated by their images under $\Psi $ is isomorphic to $U(\mathfrak {n}_{+})$. In case the field $\Bbbk $ (in place of $\mathbb {C}$) is finite and $\mbox {char}(\Bbbk )$ is coprime with the order of $G$, we establish isomorphisms of the corresponding “counting” Ringel–Hall algebras and the specializations of quantized universal enveloping algebras $U_{v}(\mathfrak {n}_{+})$ at $v=\sqrt {|\Bbbk |}$.

abstract: We introduce the notion of almost representations of Lie algebras and quantum tori, and establish an Ulam-stability type phenomenon: every irreducible almost representation is close to a genuine irreducible representation. As an application, we prove that geometric quantizations of the two-dimensional sphere and the two-dimensional torus are conjugate in the semi-classical limit up to a small error.

In this paper the identities of the complex affine Kac-Moody algebras are studied. It is proved that the identities of twisted affine algebras coincide with those of the corresponding nontwisted algebras. Moreover, in the class of nontwisted affine Kac-Moody algebras, each of these algebras is uniquely defined by its identities. It is shown that the varieties of affine algebras, as well as the varieties defined by finitely generated three-step solvable Lie algebras, have exponential growth.

Using the central charge of the Virasoro algebra as a clue, we recall the known constructions of theA, D, E algebras and discuss new Bosonic constructions of the non simply laced affine Kac-Moody algebras: the twistedA, D, E and theB, C, F, andG algebras. These involve interacting Fermions and a generalization of the Frenkel-Kac sign operators which do not form a 2-cocycle when the horizontal algebra has more than one short simple root.

Families of operator identities appeared as a consequence of an existence of finite-dimensional representation of (super) Lie algebras of first-order differential operators and $q$-deformed (quantum) algebras of first-order finite-difference operators are presented. It is shown that those identities can be rewritten in terms of creation/annihilation operators and it leads to a simplification of the problem of the normal ordering in the second quantization method.

If \({\mathcal{A}}\) is an infinite primal algebra, then we shall represent any algebra in the variety \({V\,(\mathcal{A}}\)) generated by \({\mathcal{A}}\) as a limit reduced power of \({\mathcal{A}}\). Furthermore, we show that any homomorphism between algebras in \({V\,(\mathcal{A}}\)) can be induced by mappings between underlying sets of the limit reduced powers. With this representation of the morphisms between algebras in \({V\,(\mathcal{A}}\)) at hand, we will construct a category equivalent to the category \({V\,(\mathcal{A}}\)).

We study M-theory and D-brane quantum partition functions for microscopic black hole ensembles within the context of the AdS/CFT correspondence in terms of highest weight representations of infinite-dimensional Lie algebras, elliptic genera, and Hilbert schemes, and describe their relations to elliptic modular forms. The common feature in our examples lies in the modular properties of the characters of certain representations of the pertinent affine Lie algebras, and in the role of spectral functions of hyperbolic three-geometry associated with q-series in the calculation of elliptic genera. We present new calculations of supergravity elliptic genera on local Calabi–Yau threefolds in terms of BPS invariants and spectral functions, and also of equivariant D-brane elliptic genera on generic toric singularities. We use these examples to conjecture a link between the black hole partition functions and elliptic cohomology.

We will be interested in complex representations of both real and complex Lie algebras. There is an important distinction to be made. If g is a real Lie algebra, then a complex representation is an ℝ-linear homomorphism g → End(V), where V is a complex vector space. On the other hand, if g is a complex Lie algebra, we require that the homomorphism be (ℂ-linear. The reader should note that we ask more of a complex representation of a complex Lie algebra than we do of a complex representation of a real Lie algebra.

Motivated by an example of Inönü and Wigner, a general procedure for the contraction of irreducible unitary representations of semisimple compact Lie algebras is given. In this situation, contractions mean generalized Inönü–Wigner contractions. In particular, this means that by a simple limiting process, one obtains representations of nonsemisimple noncompact Lie algebras. As examples, a general treatment of SO(3) is given as well as some results for SU(3).

ABSTRACTAs it is well-known there exist matrix representations of any given finite-dimensional complex Lie algebra. More concretely, such representations can be obtained by means of an isomorphic matrix Lie algebra consisting of upper-triangular square matrices. However, there is no general information about the minimal order for the matrices involved in such representations. In this way, our main goal is to revisit, debug and implement an algorithm which provides the minimal order for matrix representations of any finite-dimensional nilpotent Lie algebra from its law, as well as returning a matrix representative of such an algebra by using the minimal order previously computed. In order to show the applicability of this procedure, we have computed minimal representative for each nilpotent Lie algebra of dimensions 6 and 7 and we have also obtained the representation of some families with an arbitrary dimension.

Representations of twisted multi-loop Lie algebras

Inspired by earlier works on representations of the Temperley-Lieb algebra we introduce a novel family of representations of the algebra. This may be seen as a generalization of the so called asymmetric twin representation. The underlying symmetry algebra is also examined and it is shown that in addition to certain obvious exact quantum symmetries non trivial quantum algebraic realizations that exactly commute with the representation also exist. Non trivial representations of the boundary Temperley-Lieb algebra as well as the related residual symmetries are also discussed. The corresponding novel R and K matrices solutions of the Yang-Baxter and reflection equations are identified, the relevant quantum spin chain is also constructed and its exact symmetries are studied.

These varieties are conjectured to be abelian varieties up to finite etale coverings. This conjecture is derived from an affirmative answer to the abundance conjecture in minimal model theory. In particular, this is true for n=3.

A new (abstract algebraic) approach to the solution of the order conditions for Runge-Kutta methods (RK) and to the corresponding simplifying assumptions was suggested in Khashin (Can. Appl. Math. Q. 17(1), 555–569, 2009, Numer. Algorithm, 61(2), 1–11, 2012). The approach implied natural classification of the simplifying assumptions and allowed to find new RK methods of high orders. Here we further this approach. The new approach is based on the upper and lower Butcher's algebras. Here we introduce auxiliary varieties ℳ D and prove that they are projective algebraic varieties (Theorem 3.2). In some cases they are completely described (Theorem 3.5). On the set of the 2-standard matrices (Definition 4.4) (RK methods with the property b 2 = 0) the one-dimensional symmetries are introduced. These symmetries allow to reduce consideration of the RK methods to the methods with c 2 = 2c 3/3, that is c 2can be removed from the list of unknowns. We formulate a hypothesis on how this method can be generalized to the case b 2 = b 3 = 0 where two-dimensional symmetries appear.

For many of the applications of K-theory, it is useful to have the notion of K-theory for categories and not just for rings. In this more general context, the K-theory of a ring Ris just the K-theory of the category Proj R of finitely generated projective modules over R. Another natural example is the topological K-theory of a compact space X, which is the K-theory of the category Vect X of (locally trivial, real, or complex) vector bundles over X. The identification of this with the K-theory of the ring R= C(X) then follows from an equivalence of categories Proj R ≅ Vect X, But there axe also many examples that don’t come so directly from rings; for instance, if X is a projective algebraic variety, one can consider in a similar way the category Vect Xof algebraic vector bundles over X.We will see many more examples shortlyKeywordsExact SequenceDirect SummandShort Exact SequenceProjective ModuleFree Abelian GroupThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.