Subregular Representations of {\mathfrak{s}}{\mathfrak{l}}_n and Simple Singularities of Type A n−1

Alexander Premet has stated the following problem: what is a relation between subregular nilpotent representations of a classical semisimple restricted Lie algebra and non-commutative deformations of the corresponding singularities? We solve this problem for type A.     

Combinatorial Hopf algebra of superclass functions of type D

We provide a Hopf algebra structure on the space of superclass functions on the unipotent upper triangular group of type D over a finite field based on a supercharacter theory constructed by André and Neto in J. Algebra 305, 394–429 (2006) and 322 (2009). Also, we make further comments with respect to types B and C. Type A was explored by M. Aguiar et al. in Adv. Math. 229 (2012); thus this paper is a contribution to understand combinatorially the supercharacter theory of the other classical Lie types.     

On ad-nilpotent b-ideals for orthogonal Lie algebras

Krattenthaler, Orsina and Papi provided explicit formulas for the number of ad-nilpotent ideals with fixed class of nilpotence of a Borel subalgebra of a classical Lie algebra. Especially for types A and C they obtained refined results about these ideals with not only fixed class of nilpotence but also fixed dimension. In this paper, we shall follow their algorithm to determine the enumeration of ad-nilpotent b-ideals with fixed class of nilpotence and dimension for orthogonal Lie algebras, i.e., types B and D.     

Crystal energy functions via the charge in types A and C

The Ram–Yip formula for Macdonald polynomials (at t = 0) provides a statistic which we call charge. In types A and C it can be defined on tensor products of Kashiwara–Nakashima single column crystals. In this paper we prove that the charge is equal to the (negative of the) energy function on affine crystals. The algorithm for computing charge is much simpler and can be more efficiently computed than the recursive definition of energy in terms of the combinatorial R-matrix.     

Characters of modular torsion free representations of classical lie algebras

In this paper, we analyze the characters of modular, irreducible rep-resentations of classical Lie algebras g of types A_l-1 and Ci arising from a characteristic 0 construction of torsion free representations. By character, we refer to linear functionals on g identified with algebra homomorphisms from a distinguished central subalgebra O of the universal enveloping algebra of g. If Lie(G') = g, then for each character X standard representatives with respect to a fixed toral subalgebra are found in the (2-orbit containing the character X For many parameters, these characters are nilpotent. Furthermore, modular representations of type A_l-1 and type C_l Lie algebras constructed by induction from these irreducible, torsion free representations are shown to admit characters in a family of both Richardson and non-Richardson nilpotent orbits. Through this explicit induction construction, irreducible representations of minimal p-power dimension under the Kac-Weisfeiler conjecture are realized     

The Newton Iteration on Lie Groups

We define the Newton iteration for solving the equation f(y) = 0, where f is a map from a Lie group to its corresponding Lie algebra. Two versions are presented, which are formulated independently of any metric on the Lie group. Both formulations reduce to the standard method in the Euclidean case, and are related to existing algorithms on certain Riemannian manifolds. In particular, we show that, under classical assumptions on f, the proposed method converges quadratically. We illustrate the techniques by solving a fixed-point problem arising from the numerical integration of a Lie-type initial value problem via implicit Euler.     

Global convergence properties of two modified BFGS-type methods

This article studies a modified BFGS algorithm for solving smooth unconstrained strongly convex minimization problem. The modified BFGS method is based on the new quasi-Newton equation B_k+1s_k=yk where y_k*, =yk + A_ks_k andA _k is a matrix. Wei, Li and Qi [WLQ] have proven that the average performance of two of those algorithms is better than that of the classical one. In this paper, we prove the global convergence of these algorithms associated to a general line search rule.     

Vertex-algebraic structure of the principal subspaces of level one modules for the untwisted affine Lie algebras of types A,D,E

Generalizing some of our earlier work, we prove natural presentations of the principal subspaces of the level one standard modules for the untwisted affine Lie algebras of types A, D and E, and also of certain related spaces. As a consequence, we obtain a canonical complete set of recursions (q-difference equations) for the (multi-)graded dimensions of these spaces, and we derive their graded dimensions. Our methods are based on intertwining operators in vertex operator algebra theory.     

Euler-Bernoulli beams from a symmetry standpoint-characterization of equivalent equations

We completely solve the equivalence problem for Euler-Bernoulli equation using Lie symmetry analysis. We show that the quotient of the symmetry Lie algebra of the Bernoulli equation by the infinite-dimensional Lie algebra spanned by solution symmetries is a representation of one of the following Lie algebras: 2A₁, A₁⊕A₂, 3A₁, or A3,3⊕A₁. Each quotient symmetry Lie algebra determines an equivalence class of Euler-Bernoulli equations. Save for the generic case corresponding to arbitrary lineal mass density and flexural rigidity, we characterize the elements of each class by giving a determined set of differential equations satisfied by physical parameters (lineal mass density and flexural rigidity). For each class, we provide a simple representative and we explicitly construct transformations that maps a class member to its representative. The maximally symmetric class described by the four-dimensional quotient symmetry Lie algebra A3,3⊕A₁ corresponds to Euler-Bernoulli equations homeomorphic to the uniform one (constant lineal mass density and flexural rigidity). We rigorously derive some non-trivial and non-uniform Euler-Bernoulli equations reducible to the uniform unit beam. Our models extend and emphasize the symmetry flavor of Gottlieb's iso-spectral beams [H.P.W. Gottlieb, Isospectral Euler-Bernoulli beam with continuous density and rigidity functions, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 413 (1987) 235-250].     

Categorification of (induced) cell modules and the rough structure of generalised Verma modules

This paper presents categorifications of (right) cell modules and induced cell modules for Hecke algebras of finite Weyl groups. In type A we show that these categorifications depend only on the isomorphism class of the cell module, not on the cell itself. Our main application is multiplicity formulas for parabolically induced modules over a reductive Lie algebra of type A, which finally determines the so-called rough structure of generalised Verma modules. On the way we present several categorification results and give a positive answer to Kostant's problem from [A. Joseph, Kostant's problem, Goldie rank and the Gelfand-Kirillov conjecture, Invent. Math. 56 (3) (1980) 191-213] in many cases. We also present a general setup of decategorification, precategorification and categorification.     

Two expanding integrable systems and quasi-Hamiltonian function associated with an equation hierarchy

A Lie algebra sl(2) which is isomorphic to the known Lie algebra A₁ is introduced for which an isospectral Lax pair is presented, whose compatibility condition leads to a soliton-equation hierarchy. By using the trace identity, its Hamiltonian structure is obtained. Especially, as its reduction cases, a Sine equation and a complex modified KdV(cmKdV) equation are obtained,respectively. Then we enlarge the sl(2) into a bigger Lie algebra sl(4) so that a type of expanding integrable model of the hierarchy is worked out. However, the soliton-equation hierarchy is not integrable couplings. In order to generate the integrable couplings, an isospectral Lax pair is introduced. Under the frame of the zero curvature equation, we generate an integrable coupling whose quasi-Hamiltonian function is derived by employing the variational identity. Finally, two types of computing formulas of the constant γ are obtained, respectively.     

Smooth and palindromic Schubert varieties in affine Grassmannians

We completely determine the smooth and palindromic Schubert varieties in affine Grassmannians, in all Lie types. We show that an affine Schubert variety is smooth if and only if it is a closed parabolic orbit. In particular, there are only finitely many smooth affine Schubert varieties in a given Lie type. An affine Schubert variety is palindromic if and only if it is a closed parabolic orbit, a chain, one of an infinite family of “spiral” varieties in type A, or a certain 9-dimensional singular variety in type B ₃. In particular, except in type A there are only finitely many palindromic affine Schubert varieties in a fixed Lie type. Moreover, in types D and E an affine Schubert variety is smooth if and only if it is palindromic; in all other types there are singular palindromics. The proofs are for the most part combinatorial. The main tool is a variant of Mozes’ numbers game, which we use to analyze the Bruhat order on the coroot lattice. In the proof of the smoothness theorem we also use Chevalley’s cup product formula.     

On transitive Lie bialgebroids and Poisson groupoids

We prove that, for any transitive Lie bialgebroid (A, A^∗), the differential associated to the Lie algebroid structure on A^∗ has the form d_∗=A[Λ,⋅]+Ω, where Λ is a section of ^∧2A and Ω is a Lie algebroid 1-cocycle for the adjoint representation of A. Globally, for any transitive Poisson groupoid (Γ,Π), the Poisson structure has the form , where ΠF is a bivector field on Γ associated to a Lie groupoid 1-cocycle.     

THE BANACH-LIE GROUP OF LIE TRIPLE AUTOMORPHISMS OF AN H^＊-ALGEBRA^＊

We study the Banach-Lie group Ltaut(A) of Lie triple automorphisms of a complex associative H*-algebra A. Some consequences about its Lie algebra, the algebra of Lie triple derivations of A, Ltder(A), are obtained. For a topologically simple A, in the infinite-dimensional case we have Ltaut(A)0 = Aut(A) implying Ltder(A) = Der(A). In the finite-dimensional case Ltaut(A)0 is a direct product of Aut(A) and a certain subgroup of Lie derivations δ from A to its center, annihilating commutators.     

Degenerate group of type A: Representations and flag varieties

We consider the degeneration of a simple Lie group which is a semidirect product of its Borel subgroup and a normal Abelian unipotent subgroup. We introduce a class of highest weight representations of the degenerate group of type A, generalizing the construction of PBW-graded representations of the classical group (PBW is an abbreviation for “Poincaré-Birkhoff-Witt”). Following the classical construction of flag varieties, we consider the closures of orbits of the Abelian unipotent subgroup in projectivizations of the representations. We show that the degenerate flag varieties F _n ^a and their desingularizations R _n can be obtained via this construction. We prove that the coordinate ring of R _n is isomorphic as a vector space to the direct sum of the duals of the highest weight representations of the degenerate group. At the end we state several conjectures on the structure of the highest weight representations of the degenerate group of type A.     

On cohomology theory for topological groups

We construct some new cohomology theories for topological groups and Lie groups and study some of its basic properties. For example, we introduce a cohomology theory based on measurable cochains which are continuous in a neighbourhood of the identity. We show that if G and A are locally compact and second countable, then the second cohomology group based on locally continuous measurable cochains as above parametrizes the collection of locally split extensions of G by A.     

Lie point symmetries and some group invariant solutions of the quasilinear equation involving the infinity Laplacian

Using the classical Lie method we obtain the full Lie point symmetry group of the Aronsson equation in two independent variables. Some group invariant solutions of this equation are found and a conjecture on the Lie point symmetry group of the Aronsson equation in Rⁿ is presented.