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The loop Virasoro conformal superalgebra associated with the loop super-Virasoro algebra is constructed in the present paper. The conformal superderivation algebra of is completely determined, which is shown to consist of inner superderivations. And nontrivial free and free -graded -modules of rank two are classified. We also give a classification of irreducible free -modules of rank two and all irreducible submodules of each free -graded -module of rank two.

In this paper, we determine the universal central extension, derivation algebra and automorphism group of the twisted N = 1 Schrodinger–Neveu–Schwarz algebra. Furthermore, we generalize these results to the generalized twisted N = 1 Schrodinger–Neveu–Schwarz algebra in the final section.

For a field F and a row-finite directed graph Γ, let L(Γ) be the associated Leavitt path algebra. We find necessary and sufficient conditions for the Lie algebra [L(Γ), L(Γ)] to be simple.

In this article, we identify several harmonic oscillator invariants and calculate the Leibniz homology of the massless harmonic oscillator algebra in terms of these invariants.

In this paper we introduce -deformation of Weyl algebra and study the ladders in this algebra, which give the factorization of certain q- and h-difference operators of second order.

We continue the study of the vertex operator algebra L(k, 0) associated to a type affine Lie algebra at admissible one-third integer levels, , initiated in Axtell and Lee [3]. We show there are only finitely many irreducible L(k, 0)-modules from the category 𝒪. The proof relies on knowledge of an explicit formula for the singular vectors. After obtaining this formula, we are able to prove there are only finitely many irreducible A(L(k, 0))-modules from the category 𝒪. The main result then follows from the bijective correspondence in A(V)-theory.

We establish a connection between planar rook algebras and tensor representations of the natural two-dimensional representation of the general linear Lie superalgebra 𝔤𝔩(1 | 1). In particular, we show that the centralizer algebra is the planar rook algebra for all k ≥ 1, and we exhibit an explicit decomposition of into irreducible 𝔤𝔩(1 | 1)-modules. We obtain similar results for the quantum enveloping algebra and its natural two-dimensional module .

We introduce a new technique called T*-extension, of constructing a metric 3-Lie algebra out of an arbitrary 3-Lie algebra, and explore all possible metrics and corresponding signatures on this resulting metric 3-Lie algebra.

We prove that if 𝒱 is a residually finite variety of nonassociative algebras over a finite field, and the enveloping algebra of each finite member of 𝒱 is finitely generated as a module over its center, then 𝒱 is generated by a single finite algebra.

Let be an affine Lie algebra of the type . Suppose we are given a ℤ-gradation of the corresponding simple finite-dimensional Lie algebra 𝔤 = 𝔤−1 ⊕ 𝔤0 ⊕ 𝔤1; then we also have the induced ℤ-gradation of the affine Lie algebra Let L(Λ) be a standard module of level 1. Feigin–Stoyanovsky's type subspace W(Λ) is the -submodule of L(Λ) generated by the highest-weight vector v Λ, We find a combinatorial basis of W(Λ) given in terms of difference and initial conditions.

We provide necessary details to several arguments that appeared in our previous paper “Drinfeld Realization of Twisted Quantum Affine Algebras,” Commun. Algebra 35 (2007) 3683–3698.

We construct the entire generalized Kac–Moody Lie algebra as a quotient of the positive part of another generalized Kac–Moody Lie algebra. The positive part of a generalized Kac–Moody Lie algebra can be constructed from representations of quivers using Ringel's Hall algebra construction. Thus we give a direct realization of the entire generalized Kac–Moody Lie algebra. This idea arises from the affine Lie algebra construction and evaluation maps. In [16], we give a quantum version of this construction after analyzing Nakajima's quiver variety construction of integral highest weight representations of the quantized enveloping algebras in terms of the irreducible components of quiver varieties.

We compute the Lie algebra of a higher degree hyperbolic alternating form, which generalizes the symplectic algebra to higher degrees. We use this to show that hyperbolic forms descend under scalar extension. An immediate consequence is that a version of the weak Hasse–Minkowski Theorem is valid for alternating forms of higher degree.

We investigate connections between the exponents of a complex simple Lie algebra and the coefficients of its highest root. We also give a new description of the exponents completely in terms of the largest coefficient of the highest root.

Let Der skew (ℂ q ) be the skew derivation algebra of the quantum torus with q ≠ 1 a pth primitive root of unity. In this article, we investigate the automorphism group of the derived Lie subalgebra of the infinite dimensional Lie algebra . The result of the article generalizes those obtained in Xue et al. (2006) and Zheng and Tan (2007) with q = 1 or q being generic.

For any additive subgroup G of an arbitrary field 𝔽 of characteristic zero, there corresponds a generalized Heisenberg–Virasoro algebra ℒ[G]. Given a total order of G compatible with its group structure, and any h, hI, c, cI, cLI ∈ 𝔽, a Verma module over ℒ[G] is defined. In this note, the irreducibility of Verma modules is completely determined.

In this article, we study the representations of the vertex operator algebra (VOA) for m, n ≡ 1 or 2 mod 4. As our main result, we determine all irreducible modules for U and its fusion rules.

We determine the irreducible weight modules with weight multiplicities at most 1 over the derivation algebra of the localization of the quantum plane at q = −1.

In this article, we study the central extensions and derivations of the Lie algebra of skew derivations for the quantum torus. The results of the article generalize those obtained in Jiang and Meng (1998a b) and Kirkman et al. (1994).

We construct a quantum semigroup and an algebra of forms appropriate for the generalised homological algebra of N-complexes (Kapranov, arXiv:q-alg/9611005). This is an analogue to the picture for usual homological algebra, where one has the quantum general linear group (Kassel, 1995) and the differential forms constructed by Wess and Zumino (1990). The two pictures are different and we explain why the dichotomy arises.

We define generalized Virasoro operators acting on a Fock space V(Γ). These generalize the standard construction of Virasoro operators. By using the Jacobi Identity we compute the commutators of these operators. These operators result in an abelian extension of the toroidal Lie algebra. We explicitly describe the abelian extension.

In this paper, we define the strong semi-simple n-Lie algebra, and prove: (1) A finite dimensional n-Lie algebra A over an algebraically closed field F of characteristic 0 is strong semi-simple if and only if A can be decomposed into the direct sum of its simple ideals. (2) Each derivation of a strong semi-simple n-Lie algebra is inner. (3) The Killing form on a strong semi-simple n-Lie algebra is nondegenerate, and other properties.

The purpose of this work is to construct completely reducible vertex representations for the toroidal Lie algebra of type F 4.

This paper describes Oscillator representations of Virasoro algebra over , which is a part of an extension of a simply laced lattice by a hyperbolic lattice of rank . These results are generalizati...

All highest weight vectors of weight one and two in the vertex operator algebra associated with a lattice with respect to a certain choice of mutually orthogonal conformal vectors are determined.

For the q-analogue of boson associated to a hereditary algebra over a finite field k we generalize the result of M. Kashiwara to show that the category of integrable -modules is semisimple, and the positive part of the quantized generalized Kac-Moody algebra associated to represents the unique isoclass of simple objects. Hence we obtain another proof of the result due to B. Sevenhant-M. Van den Bergh that is isomorphic to the twisted Hall algebra .

We consider the class of Noetherian ring, appeared as a result of quantization of algebraic groups and their representations within framework of mathematical physics. One set up the problem of desc...

Let D be the Lie algebra of regular differential operators on , and be the central extension of D. Let W 1+∞,minus;N be the vertex algebra associated to the irreducible vacuum -module with the central charge c = −N. We show that W 1+∞,−N is a subalgebra of the Heisenberg vertex algebra M(1) with 2N generators, and construct 2N-dimensional family of irreducible W 1+∞,−N -modules. Considering these modules as -modules, we identify the corresponding highest weights.

Let Z be the ring of integers. In this paper we classify Z × Z-graded Lie algebras A = i, j∈Z A i, j over a characteristic 0 field F with dim A i, j ≤ 1 for each i and j, satisfying the following three properties: I. A 0 = i∈Z A i, 0 ≃ Vir, the centerless Virasoro algebra; II. A 0,−1, A 0,0, A 0,1 span a Heisenberg algebra; III. A is generated by the centerless Virasoro algebra and the Heisenberg algebra.

Let Γ be a finite connected valued Auslander-Reiten quiver, K an infinite set of some finite fields up to isomorphism, and ℋ(Γ)1 the degenerate Ringel-Hall algebra of Γ on K. We prove that the structural coefficient of ℋ(Γ)1 is rational numbers which is independent of K, and ℋ(Γ)1 ⊗ Z Q is the universal enveloping algebra of the Lie algebra ℒ(Γ) ⊗ Z Q, where ℒ(Γ) is the Lie subalgebra of ℋ(Γ)1 generated by the vertex set of Γ.

We construct explicitly a level two representation of the affine Lie algebra using interactive bosonic fields.

Vertex operator representations for toroidal Lie algebras of simply-laced type were studied in [MRY] and [EM]. In this paper we give vertex operator constructions for the toroidal Lie algebra of type Bl (l ≥ 3). The constructions are related to the folding of Dynkin diagram of and a two-cocycle necessary for the vertex operator constructions.

(1999). On radicals of a left-symmetric algebra. Communications in Algebra: Vol. 27, No. 7, pp. 3161-3175.

A quantization of a non-standard rational solution of CYBE for sit is given explicitly. We obtain the quantization with the help of a twisting of the usual Yangian Y (sl(2)). This quantum object (deformed Yangian Y-eta,Y-xi(sl(2))) is a two-parametric deformation of the universal enveloping algebra U(sl(2)[u]) of the polynomial current algebra sl(2)[u]. We consider the pseudotriangular structure on Y-eta,Y-xi(sl(2)), the quantum double DYeta,xi(sl(2)), its the universal R-matrix and also the RTT-realization of Y-eta,Y-xi(sl(2)).

Let A [1] be a 1-fold affinization of a generalized Cartan matrix A of finite or affine type so that A [1] has two identical columns and two identical rows. We consider a presentation of a toroidal Lie algebra as a homomorphic image of the generalized intersection matrix Lie algebra of A [1] and thus obtain an explicit description of the generators of the kernel from the defining relations

In this paper, the derivation algebra and the universal central extension of the q-analog of the Virasoro-like algebra are given.

(1996). The 2 × 2 quantum matrix weyl algebra. Communications in Algebra: Vol. 24, No. 4, pp. 1409-1434.

After Drinfe'd and Jimbo construction [3,7] of quantized universal envelop¬ing algebra associated to each complex simple Lie algebra, larger classes of quasitriangular Hopf algebras as been founded [1,12] and studied [8]. We prove that the multiparameter quantum group defined by Reshetikhin and De Concini-Kac-Procesi are indeed equivalent. Besides we write this algebra as the quantum double of a Borel-type sub-Hopf-algebra

(1994). A q-analog for the virasoro algebra. Communications in Algebra: Vol. 22, No. 10, pp. 3755-3774.

We give an explicit description of the Kac-Peterson-Lepowsky constr tion of the basic representation for the affine Lie algebra sˆo2n(C). Using the conjugacy classes of the Weyl group of sˆo2n(C), we describe all equivalent maximal Heisenberg subalgebras (HSA's) of the correspond: affine Lie algebra. We associate to these HSA's multicomponent charged and neutral free fermionic fields. The boson-fermion correspondence these fields provides us with fermionic vertex operators, whose 'norr, ordered products' give the (twisted) vertex operators of the Kac-Peters( Lepowsky construction.

(1991). Linear Algebra and Quantum Chemistry. The American Mathematical Monthly: Vol. 98, No. 8, pp. 710-718.

[No abstract available]

(1991). Two dual classes of bialgebras related to the concepts of “quantum group” and “quantum lie algebra”. Communications in Algebra: Vol. 19, No. 12, pp. 3295-3345.

An application of the theory of Noetherian rings is used to produce a new family of irreducible representations for the Virasoro algebra.

A Lie algebra L(Σ) can be associated with each nonlinear filtering problem, and the realizability or, better, the representability of L(σ) or quotients of L(σ) by means of vector fields on a finite dimensional manifold is related to the existence of finite dimensional recursive filters. In this paper, the structure and representability properties of L(σ) are analyzed for several interesting and/or well known classes of problems. It is shown that, for certain nonlinear filtering problems, L(σ) is given by the Weyl algebra It is proved that neither W n nor any quotient of W n can be realized with C ∞ or analytic vector fields on a finite dimensional manifold, thus suggesting that for these problems, no statistic of the conditional density can be computed with a finite dimensional recursive filter. For another class of problems (including bilinear systems with linear observations), it is shown that L(Σ) is a certain type of filtered Lie algebra. The algebras of this class are of a type which suggest that “su...

We decompose the six tensor products involving the three unitarizable ir-reducible representations of the Virasoro algebra with central charge into irreducible components with central charge 1, and determine the symmetric and antisymmetric parts of the tensor squares.

The gap equation for the chiral-limiting quark mass is shown to be the basis of the Goldberger-Treiman relation at the quark level, again a gap equation, for both the massless it and massive U(1) ηo states. The strength β of the isoscalar quark annihilation graphs is also demonstrated to be the origin of isoscalar qq particle mixing. Combining these three results, the perturbative second order U(1) anomaly contribution βp (2)/m2 ηo then generates a third U(1) gap equation. The latter resolves the U(1) problem in QCD because it explains why the η′ mass is significantly larger than the π,κ,η octet masses.