In this paper we develop the theory of a necklace ring and the theory of a logarithmic function (associated with graded Lie algebras and Lie superalgebras). Related to a necklace ring, we show that its construction is functorial. On a logarithmic function we generalize the results in Bryant (J. Algebra. 253 (2002); no.1, 167-188) to the cases of graded Lie (super)algebras with a group action by...

Please let me know if you find any mistakes! Friday, October 3, 2014. Grading will be based on an exam on Friday, December 12. Groups of general interest often have some additional natural structure: Topological groups, Lie groups. From Lie groups, we get Lie algebras via a standard procedure in differential topology. There’s also an association in the reverse direction, from Lie algebras to Li...

By exploring the description of chiral blocks in terms of co-invariants, a proof of the Verlinde formula for WZW models is obtained which is entirely based on the representation theory of affine Lie algebras. In contrast to other proofs of the Verlinde formula, this approach works for all untwisted affine Lie algebras. As a by-product we obtain a homological interpretation of the Verlinde multi...

The general aim of this paper is to describe a particular case of a classical scheme which involves a whole class of spaces, and geometries associated to a family of Lie groups. At all different levels of this scheme, either the spaces, the Lie groups or their Lie algebras are related among themselves by contractions, yet their properties can be dealt with in a completely unified way. The famil...

Causal and non-causal observability are discussed in this paper for nonlinear timedelay systems. By extending the Lie derivative for time-delay systems in the algebraic framework introduced by Xia et al. (2002), we present a canonical form and give sufficient condition in order to deal with causal and non-causal observations of state and unknown inputs of time-delay systems.

In this paper, we study the simple modules for the restricted Lie superalgebra gl(m|n). A condition for the simplicity of the induced modules is given, and an analogue of Kac-Weisfeiler theorem is proved. Mathematics Subject Classification (2000): 17B50; 17B10.

Let F be the underlying base field of characteristic p > 3 and denote by W and S the even parts of the finite-dimensional generalized Witt Lie superalgebra W and the special Lie superalgebra S, respectively. We first give the generator sets of the Lie algebras W and S. Using certain properties of the canonical tori of W and S, we then determine the derivation algebra of W and the derivation spa...

We prove that the normalizer of any diffuse amenable subalgebra of a free group factor L(Fr) generates an amenable von Neumann subalgebra. Moreover, any II1 factor of the form Q ⊗̄L(Fr), with Q an arbitrary subfactor of a tensor product of free group factors, has no Cartan subalgebras. We also prove that if a free ergodic measure preserving action of a free group Fr, 2 ≤ r ≤ ∞, on a probability ...

Let G denote an infinite-dimensional Heisenberg-like group, which is a class of infinite-dimensional step 2 stratified Lie groups. We consider holomorphic functions on G that are square integrable with respect to a heat kernel measure which is formally subelliptic, in the sense that all appropriate finite dimensional projections are smooth measures. We prove a unitary equivalence between a subc...

The branching problem for a couple of non-compatible Lie algebras and their parabolic subalgebras applied to generalized Verma modules was recently discussed in [15]. In the present article, we employ the recently developed F-method, [10], [11] to the couple of non-compatible Lie algebras Lie G2 i ↪→ so(7), and generalized conformal so(7)-Verma modules of scalar type. As a result, we classify t...