Algebra Hn Research Articles

Zero product determined n-th Schrödinger algebra

The n-th Schrödinger algebra schn:=sl2⋉hn is the semi-direct product of the Lie algebra sl2 with the n-th Heisenberg Lie algebra hn. Let K be an algebraically closed field of characteristic zero. A Lie algebra L is zero product determined over K, if for every K-linear space V and every bilinear map ϕ:L×L→V with the property that ϕ(x,y)=0 whenever [x,y]=0, then there exists a linear map f:[L,L]→V such that ϕ(x,y)=f([x,y]) for all x,y∈L. This article shows that the n-th Schrödinger algebra schn is zero product determined. Applying this result, product zero derivations and two-sided commutativity-preserving maps on schn are determined. Furthermore, quasi-derivations, linear anti-commuting maps, quasi-automorphisms and strong commutativity-preserving maps of schn are all obtained.

Quasi-Whittaker modules for the n-th Schrödinger algebra

The n-th Schrödinger algebra schn defined in [14] is the semi-direct product of the Lie algebra sl2 with the n-th Heisenberg Lie algebra hn, which generalizes the Schrödinger algebra sl2⋉h1. Let ϕ:hn→C be a nonzero Lie algebra homomorphism. A schn-module V is called quasi-Whittaker of type ϕ if V=U(schn)v, where U(schn) is the universal enveloping algebra of schn, v is a nonzero vector such that xv=ϕ(x)v for any x∈hn. In this paper, we prove that a simple schn-module V is a quasi-Whittaker module if and only if V is a locally finite hn-module. Then we classify the simple quasi-Whittaker modules of ϕ, according to the rank of ϕ. Furthermore, we characterize arbitrary quasi-Whittaker modules through the rank of ϕ.

Trace forms on the cyclotomic Hecke algebras and cocenters of the cyclotomic Schur algebras

We define a unified trace form τ on the cyclotomic Hecke algebras Hn,K of type A, which generalize both Malle-Mathas' trace form on the non-degenerate version (with Hecke parameter ξ≠1) and Brundan-Kleshchev's trace form on the degenerate version. We use seminormal basis theory to construct an explicit pair of dual bases for Hn,K with respect to the form. We also construct an explicit basis for the cocenter of the corresponding cyclotomic Schur algebra, which shows that the cocenter has dimension independent of the ground field K, the Hecke parameter ξ and the cyclotomic parameters Q1,⋯,Qℓ.

On the denominators of Young's seminormal basis

We study the seminormal basis {ft} for the Specht modules of the Iwahori-Hecke algebra Hn(q) of type An−1. We focus on the base change coefficients between the seminormal basis {ft} and Murphys' standard basis {xt} with emphasis on the denominators of these coefficients. In certain important cases we obtain simple formulas for these coefficients involving hook lengths. Even for general standard tableaux we obtain new formulas. On the way we prove a new result about submodules of the restricted Specht module at root of unity.

Combinatorial interpretation of Kazhdan–Lusztig basis elements indexed by 45312-avoiding permutations in 𝔖6

Abstract Deodhar [Geom. Dedicata 36, no. 1 (1990)] introduced the defect statistic on subexpressions of reduced expressions in the symmetric group 𝔖 n to construct an algorithmic description of the Kazhdan–Lusztig basis of the Hecke algebra H n (q). This led Billey–Warrington [J. Algebraic Combin. 13, no. 2 (2001)] and the second author [J. Pure Appl. Algebra 212 (2008)] to state very explicit combinatorial descriptions of the basis elements indexed by permutations avoiding certain patterns. We extend the above work by producing an exhaustive list of graphical representations of Kazhdan–Lusztig basis elements indexed by 45312-avoiding permutations w ∈𝔖5, 𝔖6.

Crystal of affine type [formula omitted] and Hecke algebras at a primitive 2ℓth root of unity

Let ℓ∈N with ℓ>1. In this paper we give a new realization of the crystal of affine type Aˆℓ−1 using the modular representation theory of the affine Hecke algebras Hn of type A and their level two cyclotomic quotients with Hecke parameter being a primitive 2ℓth root of unity. We construct “hat” versions of i-induction and i-restriction functors on the category RepI(Hn) of finite dimensional integral modules over Hn, which induce Kashiwara operators on a suitable subgroup of the Grothendieck groups of RepI(Hn). For any simple module M∈RepI(Hn), we prove that the simple submodules of resHn−2HnM which belong to Bˆ(∞) (Definition 5.1) occur with multiplicity two. The main results generalize the earlier work of Grojnowski and Vazirani on the relations between the crystal of slˆℓ and the affine Hecke algebras of type A at a primitive ℓth root of unity.

SPECIAL VINBERG CONES

The paper is devoted to the generalization of the Vinberg theory of homogeneous convex cones. Such a cone is described as the set of “positive definite matrices” in the Vinberg commutative algebra ℋn of Hermitian T-matrices. These algebras are a generalization of Euclidean Jordan algebras and consist of n × n matrices A = (aij), where aii ∈ ℝ, the entry aij for i < j belongs to some Euclidean vector space (Vij ; \U0001d524) and {a}_{ji}={a}_{ij}^{ast }=mathfrak{g}left({a}_{ij},cdot right)in {V}_{ij}^{ast } belongs to the dual space {V}_{ij}^{ast }. The multiplication of T-Hermitian matrices is defined by a system of “isometric” bilinear maps Vij × Vjk → Vij ; i < j < k, such that |aij ⋅ ajk| = |aij| ⋅ |aik|, alm ∈ Vlm. For n = 2, the Hermitian T-algebra ℋn= ℋ2 (V) is determined by a Euclidean vector space V and is isomorphic to a Euclidean Jordan algebra called the spin factor algebra and the associated homogeneous convex cone is the Lorentz cone of timelike future directed vectors in the Minkowski vector space ℝ1,1⊕ V . A special Vinberg Hermitian T-algebra is a rank 3 matrix algebra ℋ3(V; S) associated to a Clifford Cl(V )-module S together with an “admissible” Euclidean metric \U0001d524S.We generalize the construction of rank 2 Vinberg algebras ℋ2(V ) and special Vinberg algebras ℋ3(V; S) to the pseudo-Euclidean case, when V is a pseudo-Euclidean vector space and S = S0 ⊕ S1 is a ℤ2-graded Clifford Cl(V )-module with an admissible pseudo-Euclidean metric. The associated cone \U0001d4b1 is a homogeneous, but not convex cone in ℋm; m = 2; 3. We calculate the characteristic function of Koszul-Vinberg for this cone and write down the associated cubic polynomial. We extend Baez’ quantum-mechanical interpretation of the Vinberg cone \U0001d4b12 ⊂ ℋ2(V ) to the special rank 3 case.

Sur une construction algébrique d'isomorphismes génériques d'algèbres de Hecke de type A

We construct explicit homomorphisms between the generic Hecke algebra Hn(q) of type A and the degenerate affine Hecke algebra DH(Sn) by means of a generic element Rqν(X,S) which satisfies the quadratic Hecke relation. For the specialization of DH(Sn) by means of the Jucys-Murphy's elements associated with the symmetric group Sn, we obtain explicit isomorphisms, after extension of the scalars to C(q), between Hn(q) and the group algebra of the symmetric group defined over C(q). Moreover, this isomorphism is compatible with the Hoefsmit's model of irreducible representations of Hn(q) and Sn.

THE CATEGORY OF WEIGHT MODULES FOR SYMPLECTIC OSCILLATOR LIE ALGEBRAS

The rank n symplectic oscillator Lie algebra 𝔤n is the semidirect product of the symplectic Lie algebra 𝔰𝔭2n and the Heisenberg algebra Hn. In this paper, we first study weight modules with finite-dimensional weight spaces over 𝔤n. When the central charge \( \dot{z} \) ≠ 0, it is shown that there is an equivalence between the full subcategory 𝒪𝔤n \( \left[\dot{z}\right] \) of the BGG category 𝒪𝔤n for 𝔤n and the BGG category 𝒪𝔰𝔭2n for 𝔰𝔭2n. Then using the technique of localization and the structure of generalized highest weight modules, we give the classification of simple weight modules over 𝔤n with finite-dimensional weight spaces. As a byproduct we also determine all simple 𝔤n-modules (not necessarily weight modules) that have a simple Hn-submodule.

Stability of the centers of the symplectic group rings [formula omitted

We investigate the structure constants of the center Hn of the group algebra Z[Spn(q)] over a finite field. The reflection length on the group GL2n(q) induces a filtration on the algebras Hn. We prove that the structure constants of the associated graded algebra Sn are independent of n. As tool in the proof we consider the embedding Spm↪Spn(q) and determine the behavior of the centralizers and intersection of centralizers under the embedding Spm↪Spn(q). We determine explicit formulas for rate of the growth of the centralizer of an element U∈Spm(q) in terms of dimension of the fixed space of U.

Generalized Heisenberg algebra applied to realizations of the orthogonal, Lorentz, and Poincaré algebras and their dual extensions

We introduce the generalized Heisenberg algebra Hn and construct realizations of the orthogonal and Lorentz algebras by a formal power series in a semicompletion of Hn. The obtained realizations are given in terms of the generating function for the Bernoulli numbers. We also introduce an extension of the orthogonal and Lorentz algebras by quantum angles and study realizations of the extended algebras in Hn. Furthermore, we show that by extending the generalized Heisenberg algebra Hn, one can also obtain realizations of the Poincaré algebra and its extension by quantum angles.

Computing abelian subalgebras for linear algebras of upper-triangular matrices from an algorithmic perspective

Abstract In this paper, the maximal abelian dimension is algorithmically and computationally studied for the Lie algebra hn, of n×n upper-triangular matrices. More concretely, we define an algorithm to compute abelian subalgebras of hn besides programming its implementation with the symbolic computation package MAPLE. The algorithm returns a maximal abelian subalgebra of hn and, hence, its maximal abelian dimension. The order n of the matrices hn is the unique input needed to obtain these subalgebras. Finally, a computational study of the algorithm is presented and we explain and comment some suggestions and comments related to how it works.

Leibniz algebras of Heisenberg type

We introduce and provide a classification theorem for the class of Heisenberg–Fock Leibniz algebras. This type of algebras is formed by those Leibniz algebras L whose corresponding Lie algebras are isomorphic to Heisenberg algebras Hn and whose Hn-modules I, where I denotes the ideal generated by the squares of elements of L, are isomorphic to Fock modules. We also consider the three-dimensional Heisenberg algebra H3 and study three classes of Leibniz algebras with H3 as corresponding Lie algebra, by taking certain generalizations of the Fock module. Moreover, we describe the class of Leibniz algebras with Hn as corresponding Lie algebra and such that the action I×Hn→I gives rise to a minimal faithful representation of Hn. The classification of this family of Leibniz algebras for the case of n=3 is given.

Graded decomposition numbers of Ariki–Koike algebras for blocks of small weight

We present some blocks of Ariki–Koike algebras Hn,r where the decomposition matrices are independent of the characteristic of the underlying field. We complete the description of the graded decomposition numbers for blocks of Ariki–Koike algebras of weight at most two, which consists in analysing the indecomposable core blocks at level r=3, and give a closed formula for the decomposition numbers in this case.

Geometric construction of Hopf cyclic characteristic classes

In earlier joint work with A. Connes on transverse index theory on foliations, cyclic cohomology adapted to Hopf algebras has emerged as a decisive tool in deciphering the total index class of the hypoelliptic signature operator. We have found a Hopf algebra Hn, playing the role of a ‘quantum structure group’ for the ‘space of leaves’ of a codimension n foliation, whose Hopf cyclic cohomology is canonically isomorphic to the Gelfand–Fuks cohomology of the Lie algebra of formal vector fields. However, with a few low-dimensional exceptions, no explicit construction was known for its Hopf cyclic classes. This paper provides an effective method for constructing the Hopf cyclic cohomology classes of Hn and of Hn relative to On, in the spirit of the Chern–Weil theory, which completely elucidates their relationship with the characteristic classes of foliations.

Decomposable Specht modules for the Iwahori–Hecke algebra [formula omitted

Let Sλ denote the Specht module defined by Dipper and James for the Iwahori–Hecke algebra Hn of the symmetric group Sn. When e=2 we determine the decomposability of all Specht modules corresponding to hook partitions (a,1b). We do so by utilising the Brundan–Kleshchev isomorphism between H and a Khovanov–Lauda–Rouquier algebra and working with the relevant KLR algebra, using the set-up of Kleshchev–Mathas–Ram. When n is even, we easily arrive at the conclusion that Sλ is indecomposable. When n is odd, we find an endomorphism of Sλ and use it to obtain a generalised eigenspace decomposition of Sλ.

A classification of SU(d)-type C*-tensor categories

Kazhdan and Wenzl classified all rigid tensor categories with fusion ring isomorphic to the fusion ring of the group SU(d). In this paper we consider the C*-analogue of this problem. Given a rigid C*-tensor category 𝒞 with fusion ring isomorphic to the fusion ring of the group SU(d), we can extract a constant q from 𝒞 such that there exists a *-representation of the Hecke algebra Hn(q) into 𝒞. The categorical trace on 𝒞 induces a Markov trace on Hn(q). Using this Markov trace and a representation of Hn(q) in [Formula: see text] we show that 𝒞 is equivalent to a twist of the category [Formula: see text]. Furthermore a sufficient condition on a C*-tensor category 𝒞 is given for existence of an embedding of a twist of [Formula: see text] in 𝒞.

POSITIVE PERMUTATION BRAIDS AND PERMUTATION INVERSIONS WITH SOME APPLICATIONS

In this paper we constructed an isomorphic group of binary matrices to a finite symmetric group. Our method is based on the inversion of permutations. Using this embedding we find an algorithm for writing down a standard braid word representation for each positive permutation braid. Also an algorithm for writing basis of Hecke algebra Hn+1from such basis of Hnis given.

An algorithm for semistandardising homomorphisms

Suppose μ is a partition of n and λ a composition of n, and let Sμ, Mλ denote the Specht module and permutation module defined by Dipper and James for the Iwahori–Hecke algebra Hn of the symmetric group Wn. We give an explicit fast algorithm for expressing a tableau homomorphism ϕˆA:Sμ→Mλ as a linear combination of semistandard homomorphisms. Along the way we provide a utility result related to removing rows from tableaux.

Computational calculus of the law of a family of solvable Lie algebras

This paper studies the law of the Lie algebras hn associated with a particular type of Lie groups: the Lie groups Hn formed by all the n × n upper-triangular matrices without zeros in their main diagonal. Indeed, these laws are obtained by means of a computational algorithm which we have constructed and particularly implemented by using the symbolic computation package MAPLE. Besides, the complexity of this algorithm is studied by considering the number of computations carried out with this implementation.