Lie Algebras Of Type Research Articles

An isomorphism between unitals and between related classical groups

Abstract An isomorphism between two hermitian unitals is provided, and used to treat isomorphisms of classical groups that are related to the isomorphism between certain simple real Lie algebras of types A and D (and rank 3).

On the full Kostant–Toda hierarchy and its ℓ-banded reductions for the Lie algebras of type A,B and G

Abstract This paper is concerned with the solutions of the full Kostant–Toda (f-KT) hierarchy in the Hessenberg form and their reductions to the ℓ -banded Kostant–Toda ( ℓ -KT) hierarchies for 1 ⩽ ℓ ⩽ n , where the case with ℓ = 1 is the classical tridiagonal Toda hierarchy and the case with ℓ = n is the f-KT hierarchy. Based on the Hessenberg form of the f-KT hierarchy, we study the f-KT hierarchy and the corresponding ℓ -KT hierarchies on simple Lie algebras of type A , B and G as the root space reductions with proper choices of Chevalley systems. Explicit formulas of the polynomial solutions for the τ-functions are also given in terms of the Schur functions and Schur’s Q-functions.

Quantization of length in spaces with position-dependent noncommutativity

In this paper, we present a novel approach to quantizing the length in noncommutative spaces with positional-dependent noncommutativity. The method involves constructing ladder operators that change the length not only along a plane but also along the third direction due to a noncommutative parameter that is a combination of canonical/Weyl–Moyal-type and Lie algebraic-type. The primary quantization of length in canonical-type noncommutative space takes place only on a plane, while in the present case, it happens in all three directions. We establish an operator algebra that allows for the raising or lowering of eigenvalues of the operator corresponding to the square of the length. We also attempt to determine how the obtained ladder operators act on different states and work out the eigenvalues of the square of the length operator in terms of eigenvalues corresponding to the ladder operators. We conclude by discussing the results obtained.

Schur-Weyl duality for toroidal algebras of type A

We state and prove an analog of the Schur-Weyl duality for a quotient of the classical 2-toroidal Lie algebra of type A. We then provide a method to extend this duality to the m-toroidal case, m > 2.

Local properties of Schrödinger-Virasoro type Lie algebras and a full diamond thin Lie algebra

Let S V ( z ) be a class of Schrödinger-Virasoro type Lie algebras, where z ∈ Z . In this paper, we show that S V ( z ) is 2-local complete (namely, every 2-local derivation is a derivation) for almost all integers z’s. We also introduce the notion of full diamond thin Lie algebras, and construct an example F from S V ( − 1 ) . We completely determine the derivations of F by combinatoric techniques, and show that F is not 2-local complete.

2-Local derivations of Heisenberg–Virasoro type Lie algebras

Let [Formula: see text] with [Formula: see text] be the Heisenberg–Virasoro type Lie algebras, and [Formula: see text] be a thin Lie algebra with only one diamond. In this paper, we study [Formula: see text]-local derivations on [Formula: see text] and [Formula: see text]. For [Formula: see text], we show that every [Formula: see text]-local derivation is a derivation for almost all parameter pairs [Formula: see text], generalizing some previous results, while for [Formula: see text], we show that it admits infinite many [Formula: see text]-local derivations which are not derivations. Along the way, we also prove that the space of outer derivations of [Formula: see text] is infinite-dimensional.

Computational Construction of Weierstrass Sections for Semi-Invariant Polynomial Functions on Lie Algebras

In this paper, we present a computational approach to construct Weierstrass sections for semi-invariant polynomial functions on Lie algebras, extending the foundational work of Bourbaki and Popov. We focus on simple Lie algebras of type B, C, or D, and their associated parabolic subalgebras, particularly those with Levi factors composed of successive blocks of size two. Our method extends the notion of Weierstrass sections introduced by Popov, enabling us to explicitly construct these sections and establish their polynomiality. Furthermore, we demonstrate how these sections facilitate the linearization of semi-invariant generators. Central to our approach is the construction of an adapted pair, akin to a principal sl_2-triple in the non-reductive case. We provide computational algorithms and implementations for constructing these Weierstrass sections, offering a novel avenue for research in Lie algebra theory and algebraic geometry.

Symplectic Form yang Berkaitan Dengan Satu-form Suatu Aljabar Lie Berdimensi Rendah

In this paper, we study symplectic form on low dimensional real Lie algebra. A symplectic form is very important in classifying of Lie algebra types. Based on their dimension and certain conditions, there are two types of Lie algebras. A lie algebra with odd dimension endowed with one-form such that is called a contact Lie algebra, while a Lie algebra whose dimension is even and it is endowed with zero index is called a Frobenius Lie algebra. The research aimed to give explicit formula of a symplectic form of low dimensional contact Lie algebras and Frobenius Lie algebras. We established that a one-form associated to simplectic form determine a type of a Lie algebra whether a contact or a Frobenius Lie algebras.To clearer the main results, we give some examples of one-form and symplectic form of Frobenius and contact Lie algebras.

On the combinatorial rank of quantum groups of type F4

In this paper, we calculate the combinatorial rank of the positive part [Formula: see text] of the multiparameter version of the small Lusztig quantum group, where [Formula: see text] is a simple Lie algebra of type [Formula: see text]. For this purpose, we also present the basis and relations of the quantum algebra, calculate the coproduct of its PBW-generators and determine all skew-primitive elements of this quantum group. In conclusion, we obtain that the combinatorial rank of [Formula: see text] is 4.

Block Basis for Modular Coinvariants of Pseudo-Reflection Groups

We investigate the block basis for modular coinvariants of finite pseudo-reflection groups. We are particularly interested in the case of a subgroup [Formula: see text] of the parabolic subgroups of [Formula: see text] which generalizes the Weyl groups of restricted Cartan type Lie algebra.

Towards geometric Satake correspondence for Kac-Moody algebras --- Cherkis bow varieties and affine Lie algebras of type $A$

Towards geometric Satake correspondence for Kac-Moody algebras --- Cherkis bow varieties and affine Lie algebras of type $A$

The Universe Inside Hall Algebras of Coherent Sheaves on Toric Resolutions

Abstract 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 $v=\sqrt {|\Bbbk |}$.

Lie-Poisson gauge theories and κ-Minkowski electrodynamics

We consider gauge theories on Poisson manifolds emerging as semiclassical approximations of noncommutative spacetime with Lie algebra type noncommutativity. We prove an important identity, which allows to obtain simple and manifestly gauge-covariant expressions for the Euler-Lagrange equations of motion, the Bianchi and the Noether identities. We discuss the non-Lagrangian equations of motion, and apply our findings to the κ-Minkowski case. We construct a family of exact solutions of the deformed Maxwell equations in the vacuum. In the classical limit, these solutions recover plane waves with left-handed and right-handed circular polarization, being classical counterparts of photons. The deformed dispersion relation appears to be nontrivial.

Gravity = Yang–Mills

This essay’s title is justified by discussing a class of Yang–Mills-type theories of which standard Yang–Mills theories are special cases but which is broad enough to include gravity as a double field theory. We use the framework of homotopy algebras, where conventional Yang–Mills theory is the tensor product K⊗g of a ‘kinematic’ algebra K with a color Lie algebra g. The larger class of Yang–Mills-type theories are given by the tensor product of K with more general Lie-type algebras, of which K itself is an example, up to anomalies that can be canceled for the tensor product with a second copy K¯. Gravity is then given by K⊗K¯.

On the spectrum of exterior algebra, and generalized exponents of small representations

We present some results about the irreducible representations appearing in the exterior algebra Λg\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda \\mathfrak {g}$$\\end{document}, where g\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {g}$$\\end{document} is a simple Lie algebra over C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {C}}$$\\end{document}. For Lie algebras of type B, C or D we prove that certain irreducible representations, associated to weights characterized in a combinatorial way, appear as irreducible components of Λg\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda \\mathfrak {g}$$\\end{document}. Moreover, we propose an analogue of a conjecture of Kostant, about irreducibles appearing in the exterior algebra of the little adjoint representation. Finally, we give some closed expressions, in type B, C and D, for generalized exponents of small representations that are fundamental representations and we propose a generalization of some results of De Concini, Möseneder Frajria, Procesi and Papi about the module of special covariants of adjoint and little adjoint type.

Multicomponent KP type hierarchies and their reductions, associated to conjugacy classes of Weyl groups of classical Lie algebras

This, to a large extent, expository paper describes the theory of multicomponent hierarchies of evolution equations of XKP type, where X = A, B, C, or D, and AKP = KP and their reductions, associated with the conjugacy classes of the Weyl groups of classical Lie algebras of type X. As usual, the main tool is the multicomponent boson–fermion correspondence, which leads to the corresponding tau-functions, wave functions, dressing operators, and Lax operators.

Trigonometric Lie algebras, affine Kac-Moody Lie algebras, and equivariant quasi modules for vertex algebras

In this paper, we study a family of infinite-dimensional Lie algebras XˆS, where X stands for the type: A,B,C,D, and S is an abelian group, which generalize the A,B,C,D series of trigonometric Lie algebras. Among the main results, we identify XˆS with what are called the covariant algebras of the affine Lie algebra LSˆ with respect to some automorphism groups, where LS is an explicitly defined associative algebra viewed as a Lie algebra. We then show that restricted XˆS-modules of level ℓ naturally correspond to equivariant quasi modules for affine vertex algebras related to LS. Furthermore, for any finite cyclic group S, we completely determine the structures of these four families of Lie algebras, showing that they are essentially affine Kac-Moody Lie algebras of certain types.

Q-Poincaré Inequalities on Carnot Groups with Filiform Type Lie Algebra

In this paper we prove (global) q- Poincaré inequalities for probability measures on nilpotent Lie groups with filiform Lie algebra of any length. The probability measures under consideration have a density with respect to the Haar measure given as a function of a suitable homogeneous norm.

Categorical braid group actions and cactus groups

Let g be a semisimple simply-laced Lie algebra of finite type. Let C be an abelian categorical representation of the quantum group Uq(g) categorifying an integrable representation V. The Artin braid group B of g acts on Db(C) by Rickard complexes, providing a triangulated equivalenceΘw0:Db(Cμ)→Db(Cw0(μ)) where μ is a weight of V, and Θw0 is a positive lift of the longest element of the Weyl group.We prove that this equivalence is t-exact up to shift when V is isotypic, generalising a fundamental result of Chuang and Rouquier in the case g=sl2. For general V, we prove that Θw0 is a perverse equivalence with respect to a Jordan-Hölder filtration of C.Using these results we construct, from the action of B on V, an action of the cactus group on the crystal of V. This recovers the cactus group action on V defined via generalised Schützenberger involutions, and provides a new connection between categorical representation theory and crystal bases. We also use these results to give new proofs of theorems of Berenstein-Zelevinsky, Rhoades, and Stembridge regarding the action of symmetric group on the Kazhdan-Lusztig basis of its Specht modules.

On classical Z2×Z2-graded Lie algebras

We construct classes of Z2×Z2-graded Lie algebras corresponding to the classical Lie algebras in terms of their defining matrices. For the Z2×Z2-graded Lie algebra of type A, the construction coincides with the previously known class. For the Z2×Z2-graded Lie algebra of types B, C, and D, our construction is new and gives rise to interesting defining matrices closely related to the classical ones but undoubtedly different. We also give some examples and possible applications of parastatistics.