Regularization of Mickelsson generators for nonexceptional quantum groups

Let g′ ? g be a pair of Lie algebras of either symplectic or orthogonal infinitesimal endomorphisms of the complex vector spaces C ^N?2 ? C ^N and U _q (g′) ? U _q (g) be a pair of quantum groups with a triangular decomposition U _q (g) = U _q (g_-)U _q (g₊)U _q (h). Let Z _q (g, g′) be the corresponding step algebra. We assume that its generators are rational trigonometric functions h ? → U _q (g_±). We describe their regularization such that the resulting generators do not vanish for any choice of the weight.     

Lie Algebras with Nilpotent Length Greater than that of each of their Subalgebras

The main purpose of this paper is to study the finite-dimensional solvable Lie algebras described in its title, which we call minimal non- \({\mathcal N}\). To facilitate this we investigate solvable Lie algebras of nilpotent length k, and of nilpotent length ≤k, and extreme Lie algebras, which have the property that their nilpotent length is equal to the number of conjugacy classes of maximal subalgebras. We characterise the minimal non-\({\mathcal N}\) Lie algebras in which every nilpotent subalgebra is abelian, and those of solvability index ≤3.     

On the centers of quantum groups of <Emphasis Type="Italic">A</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>-type

Let g be the finite dimensional simple Lie algebra of type An, and let U? = U _q (g,Λ) and U = U _q (g,Q) be the quantum groups defined over the weight lattice and over the root lattice, respectively. In this paper, we find two algebraically independent central elements in U? for all n ≥ 2 and give an explicit formula of the Casimir elements for the quantum group U? = U _q (g,Λ), which corresponds to the Casimir element of the enveloping algebra U(g). Moreover, for n = 2 we give explicitly generators of the center subalgebras of the quantum groups U? = U _q (g,Λ) and U = U _q (g,Q).     

Group classification of differential-difference equations: Low-dimensional Lie algebras

Differential-difference equations of the form u? _n = F _n (t, u_n?1, u _n , u_n+1, u?_n?1, u? _n , u?_n+1) are classified according to their intrinsic Lie point symmetries, equivalence group and some low-dimensional Lie algebras including the Abelian symmetry algebras, nilpotent nonAbelian symmetry algebras, solvable symmetry algebras with nonAbelian nilradicals, solvable symmetry algebras with Abelian nilradicals and nonsolvable symmetry algebras. Here F _n is a nonlinear function of its arguments and the dot over u denotes differentiation with respect to t.     

SCAP-subalgebras of Lie algebras

A subalgebra H of a finite dimensional Lie algebra L is said to be a SCAP-subalgebra if there is a chief series 0 = L₀ ? L₁ ?... ? L_t = L of L such that for every i = 1, 2,..., t, we have H + L_i = H + L_i-1 or H ∩ L_i = H ∩ L_i-1. This is analogous to the concept of SCAP-subgroup, which has been studied by a number of authors. In this article, we investigate the connection between the structure of a Lie algebra and its SCAP-subalgebras and give some sufficient conditions for a Lie algebra to be solvable or supersolvable.     

A Characterization of Stem Algebras in Terms of Central Derivations

Let L be a Lie algebra, and Der _z (L) denote the set of all central derivations of L, that is, the set of all derivations of L mapping L into the center. In this paper, by using the notion of isoclinism, we study the center of Der _z (L) for nilpotent Lie algebras with nilindex 2. We also give a characterization of stem Lie algebras by their central derivations. In fact we show that for non-abelian nilpotent Lie algebras of finite dimension and any nilpotent Lie algebra with nilindex 2 (not finite dimensional in general), Der _z (L) is abelian if and only if L is a stem Lie algebra.     

On nonabelian composition factors of a finite prime spectrum minimal group

Assume that G is a primitive permutation group on a finite set X, x ∈ X, y ∈ X \ {x}, and G _x,y \(\underline \triangleleft \) G _x . P. Cameron raised the question about the validity of the equality G _x,y = 1 in this case. The author proved earlier that, if soc(G) is not a direct power of an exceptional group of Lie type, then G _x,y = 1. In the present paper, we prove that, if soc(G) is a direct power of an exceptional group of Lie type distinct from E ₆(q), ² E ₆(q), E ₇(q), and E ₈(q), then G _x,y = 1.     

Generalized Wigner-Inönü contractions and Maxwell (super)algebras

We consider a class of generalized Wigner-Inönü contractions for the semidirect product of two particularly related semisimple Lie (super)algebras. A special class of such contractions provides the D = 4 Maxwell algebra and the recently introduced simple D = 4 Maxwell superalgebra. Further we present two types of D = 4 N-extended Maxwell superalgebras, the nonstandard one for any N with ½N(N?1) central charges and the standard one, for even N = 2k, with k(2k ? 1) internal symmetry generators.     

Characterizing projective general unitary groups PGU<Subscript>3</Subscript>(<Emphasis Type="Italic">q</Emphasis><Superscript>2</Superscript>) by their complex group algebras

Let G be a finite group. Let X ₁(G) be the first column of the ordinary character table of G. We will show that if X ₁(G) = X1(PGU₃(q ²)), then G ? PGU₃(q ²). As a consequence, we show that the projective general unitary groups PGU₃(q ²) are uniquely determined by the structure of their complex group algebras.     

Conformally Invariant Elliptic Liouville Equation and Its Symmetry-Preserving Discretization

The symmetry algebra of the real elliptic Liouville equation is an infinite-dimensional loop algebra with the simple Lie algebra o(3, 1) as its maximal finite-dimensional subalgebra. The entire algebra generates the conformal group of the Euclidean plane E₂. This infinite-dimensional algebra distinguishes the elliptic Liouville equation from the hyperbolic one with its symmetry algebra that is the direct sum of two Virasoro algebras. Following a previously developed discretization procedure, we present a difference scheme that is invariant under the group O(3, 1) and has the elliptic Liouville equation in polar coordinates as its continuous limit. The lattice is a solution of an equation invariant under O(3, 1) and is itself invariant under a subgroup of O(3, 1), namely, the O(2) rotations of the Euclidean plane.     

On lie algebras of affine vector fields of ideal realizations of holomorphic linear connections

We study the properties of real realizations of holomorphic linear connections over associative commutative algebras \(\mathbb{A}\) _m with unity. The following statements are proved.If a holomorphic linear connection ? on M _n over \(\mathbb{A}\) _m (m ≥ 2) is torsion-free and R ≠ 0, then the dimension over ? of the Lie algebra of all affine vector fields of the space (M _mn ^? , ?^?) is no greater than (mn)² ? 2mn + 5, where m = dim_? \(\mathbb{A}\), \(n = dim_\mathbb{A} \) M _n , and ?^? is the real realization of the connection ?.Let ?^? =¹ ? ײ ? be the real realization of a holomorphic linear connection ? over the algebra of double numbers. If the Weyl tensor W = 0 and the components of the curvature tensor ¹ R ≠ 0, ² R ≠ 0, then the Lie algebra of infinitesimal affine transformations of the space (M _2n ^? , ?^?) is isomorphic to the direct sum of the Lie algebras of infinitesimal affine transformations of the spaces ( ^a M _n , ^a ?) (a = 1, 2).     

Family algebras and generalized exponents for polyvector representations of simple Lie algebras of type <Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

We give an explicit formula for the exterior powers ∧ ^k π ₁ of the defining representation π ₁ of the simple Lie algebra ?ο(2n + 1, ?). We use the technique of family algebras. All representations in question are children of the spinor representation σ of g2ο(2n + 1, ?). We also give a survey of main results on family algebras.     

A class of simple Lie algebras attached to unit forms

Let n ≥ 3. The complex Lie algebra, which is attached to a unit form q(x ₁, x ₂,..., x _n) = \({\sum\nolimits_{i = 1}^n {x_i^2 + \sum\nolimits_{1 \leqslant i \leqslant j \leqslant n} {\left( { - 1} \right)} } ^{j - i}}{x_i}{x_j}\) and defined by generators and generalized Serre relations, is proved to be a finite-dimensional simple Lie algebra of type A _n , and realized by the Ringel-Hall Lie algebra of a Nakayama algebra of radical square zero. As its application of the realization, we give the roots and a Chevalley basis of the simple Lie algebra.     

Span of a DL-algebra

Unless otherswise specified, all objects are defined over a field k of characteristic 0. Let K be a field. The unessentialness of an extension of the algebra Der K by means of a splittable semisimple Lie algebra is established. Let D _K be the category of differential Lie algebras (DL-algebras) (g;K). In this paper for an extension L/K the functor η:D _K → D _L , defining the tensor product L ? _K of vector spaces and the homomorphism of Lie algebras, is constructed. If the extension L/K is algebraic, then η is unique. The results will be required for strengthening the progress on Gelfand–Kirillov problem and weakened conjecture [1, 2].     

Maximal subalgebras of Cartan type in the exceptional Lie algebras

In this paper, we initiate the study of the maximal subalgebras of exceptional simple classical Lie algebras \(\mathfrak {g}\) over algebraically closed fields k of positive characteristic p, such that the prime characteristic is good for \(\mathfrak {g}\). We deal with what is surely the most unnatural case; that is, where the maximal subalgebra in question is a simple subalgebra of non-classical type. We show that only the first Witt algebra can occur as a subalgebra of \(\mathfrak {g}\) and give an explicit classification of when it is maximal in \(\mathfrak {g}\).     

A Nilpotent Ideal in the Lie Rings with Automorphism of Prime Order

We improve the conclusion in Khukhro's theorem stating that a Lie ring (algebra) L admitting an automorphism of prime order p with finitely many m fixed points (with finite-dimensional fixed-point subalgebra of dimension m) has a subring (subalgebra) H of nilpotency class bounded by a function of p such that the index of the additive subgroup |L: H| (the codimension of H) is bounded by a function of m and p. We prove that there exists an ideal, rather than merely a subring (subalgebra), of nilpotency class bounded in terms of p and of index (codimension) bounded in terms of m and p. The proof is based on the method of generalized, or graded, centralizers which was originally suggested in [E. I. Khukhro, Math. USSR Sbornik 71 (1992) 51–63]. An important precursor is a joint theorem of the author and E. I. Khukhro on almost solubility of Lie rings (algebras) with almost regular automorphisms of finite order.     

On the nilpotent residuals of all subalgebras of Lie algebras

Let \(\mathcal{N}\) denote the class of nilpotent Lie algebras. For any finite-dimensional Lie algebra L over an arbitrary field \(\mathbb{F}\), there exists a smallest ideal I of L such that L/I ∈ \(\mathcal{N}\). This uniquely determined ideal of L is called the nilpotent residual of L and is denoted by L\(\mathcal{N}\). In this paper, we define the subalgebra S(L) = ∩_H≤LI_L(H\(\mathcal{N}\)). Set S₀(L) = 0. Define S_i+1(L)/S_i(L) = S(L/S_i(L)) for i > 1. By S_∞(L) denote the terminal term of the ascending series. It is proved that L = S_∞(L) if and only if L\(\mathcal{N}\) is nilpotent. In addition, we investigate the basic properties of a Lie algebra L with S(L) = L.     

Hodge Star as Braided Fourier Transform

We study super-braided Hopf algebras Λ primitively generated by finite-dimensional right crossed (or Drinfeld-Radford-Yetter) modules Λ¹ over a Hopf algebra A which are quotients of the augmentation ideal A ⁺ as a crossed module by right multiplication and the adjoint coaction. Here super-bosonisation Open image in new window provides a bicovariant differential graded algebra on A. We introduce Λ _{m a x} providing the maximal prolongation, while the canonical braided-exterior algebra Λ _min = B _?(Λ¹) provides the Woronowicz exterior calculus. In this context we introduce a Hodge star operator ? by super-braided Fourier transform on B _?(Λ¹) and left and right interior products by braided partial derivatives. Our new approach to the Hodge star (a) differs from previous approaches in that it is canonically determined by the differential calculus and (b) differs on key examples, having order 3 in middle degree on k[S ₃] with its 3D calculus and obeying the q-Hecke relation ?² = 1 + (q ? q ^?1)? in middle degree on k _q [S L ₂] with its 4D calculus. Our work also provides a Hodge map on quantum plane calculi and a new starting point for calculi on coquasitriangular Hopf algebras A whereby any subcoalgebra \(\mathcal {L}\subseteq A\) defines a sub-braided Lie algebra and \({\Lambda }^{1}\subseteq \mathcal {L}^{*}\) provides the required data A ⁺ → Λ¹.     

Quantum quaternion spheres

For the quantum symplectic group SP _q (2n), we describe the C ^?-algebra of continuous functions on the quotient space S P _q (2n)/S P _q (2n?2) as an universal C ^?-algebra given by a finite set of generators and relations. The proof involves a careful analysis of the relations, and use of the branching rules for representations of the symplectic group due to Zhelobenko. We then exhibit a set of generators of the K-groups of this C ^?-algebra in terms of generators of the C ^?-algebra.     

Characterization of 2-Local Derivations and Local Lie Derivations on Some Algebras

We prove that each 2-local derivation from the algebra M_n(A ) (n > 2) into its bimodule M_n(M) is a derivation, where A is a unital Banach algebra and M is a unital A -bimodule such that each Jordan derivation from A into M is an inner derivation, and that each 2-local derivation on a C*-algebra with a faithful traceable representation is a derivation. We also characterize local and 2-local Lie derivations on some algebras such as von Neumann algebras, nest algebras, the Jiang–Su algebra, and UHF algebras.