李超代数与Frobenius代数

利用限制李超代数的新定义,给出了李超代数的p-包络的一些相关结果,并将李代数中表示理论的一些结果推广到李超代数上,进而研究了限制李超代数与Frobenius代数的关系．     

On Certain Semiprime Associative Superalgebras

In this note we emphasise the relationship between the structure of an associative superalgebra with superinvolution and the structure of the Lie substructure of skewsymmetric elements. More explicitly, we show that if A is a semiprime associative superalgebra with superinvolution and K is the Lie superalgebra of skewsymmetric elements satisfying [K ², K ²] = 0, then A is a subdirect product of orders in simple superalgebras each at most 4-dimensional over its center.     

Whittaker Categories and Whittaker Modules for Lie Superalgebras

Following analogous constructions for Lie algebras, we define Whittaker modules and Whittaker categories for finite-dimensional simple Lie superalgebras. Results include a decomposition of Whittaker categories for a Lie superalgebra according to the action of an appropriate sub-superalgebra; and, for basic classical Lie superalgebras of type I, the construction of Whittaker modules from Whittaker modules for the even part.     

Alternative Superalgebras with DCC on Two-Sided Ideals#

We study the structure of alternative superalgebras that satisfy the descending chain condition (DCC) for two-sided ideals. The main results state that the Baer radical in an alternative superalgebra of characteristic ≠ 2, 3 with DCC on two-sided ideals is solvable and every such a semiprime superalgebra (of arbitrary characteristic) is isomorphic to a subdirect sum of an associative superalgebra with this property and a finite direct sum of simple alternative non-associative superalgebras.     

Alternative rings of small order

We construct all the alternative (but not associative) algebras of dimension at most 5 over a perfect field. For any prime p, we show that there are fifteen alternative rings of prime power order pⁿ ,n≤5, which are not associative. None of these rings is nil. Just one has a unity.     

An $${n = \left( {1,1} \right)}$$ Super-Toda Model Based on OSp $$1|{\kern 1pt} 4$$

We show that a Hamiltonian reduction of affine Lie superalgebras having bosonic simple roots (such as OSp ) does produce supersymmetric Toda models, with superconformal symmetry being nonlinearly realized for those fields of the Toda system which are related to the bosonic simple roots of the superalgebra. A fermionic b-c system of conformal spin is a natural ingredient of such models.     

Beltrami—McKeon—Schubert Realizations of osp(1|4) Algebra and Possible Supersymmetric Kinematics

The possible supersymmetric kinematics are presented by the contraction method in different limits from Beltrami—McKeon—Schubert realizations of osp(1|4). The connections among the superalgebras are established. The comparison with the superalgebras contracted from Beltrami—Ivanov—Sorini realization is made.     

Super-WZNW with reductions to supersymmetric and fermionic integrable models

A systematic construction for an action describing a class of supersymmetric integrable models as well as for pure fermionic theories is discussed in terms of the gauged WZNW model associated to half integer graded affine Kac–Moody algebras. Explicit examples of the N=1,2 super-sinh(sine)-Gordon models are discussed in detail. Pure fermionic theories arises for cosets sl(p,1)/sl(p)u(1) when a maximal kernel condition is fulfilled. The integrability condition for such models is discussed and it is shown that the simplest example when p=2 leads to the constrained Bukhvostov–Lipatov, Thirring, scalar massive and pseudo-scalar massless Gross–Neveu models.     

Crystal graphs for general linear Lie superalgebras and quasi-symmetric functions

We give a new representation theoretic interpretation of the ring of quasi-symmetric functions. This is obtained by showing that the super analogue of Gessel's fundamental quasi-symmetric function can be realized as the character of a connected crystal for the Lie superalgebra associated to its non-standard Borel subalgebra with a maximal number of odd isotropic simple roots. We also present an algebraic characterization of these super quasi-symmetric functions.     

Weak Quantum Enveloping Algebras of Borcherds Superalgebras

In present paper we define a new kind of weak quantized enveloping algebra of Borcherds superalgebras . It is a noncommutative and noncocommutative weak graded Hopf algebra. Using localizing with some Ore set, we obtain a different kind of quantized enveloping algebras of Borcherds superalgebras . It has a homomorphic image which is isomorphic to the usual quantum enveloping algebra of . Moreover, is isomorphic to a direct sum of and an other algebra as algebras. The author is sponsored by ZJNSF No. Y607136.