On growth of Lie algebras, generalized partitions, and analytic functions

AbstractIn this paper we discuss some recent results on two different types of growth of Lie algebras that lead to some combinatorial problems. First, we study the growth of finitely generated Lie algebras (Sections 1–4). This problem leads to a study of generalized partitions. Recently the author has suggested a series of q-dimensions of algebras Dimq,q∈N which includes, as first terms, dimensions of vector spaces, Gelfand–Kirillov dimensions, and superdimensions. These dimensions enabled us to describe the change of a growth in transition from a Lie algebra to its universal enveloping algebra. In fact, this is a result on some generalized partitions. In this paper we give some results on asymptotics for those generalized partitions. As a main application, we obtain an asymptotical result for the growth of free polynilpotent finitely generated Lie algebras. As a corollary, we specify the asymptotic growth of lower central series ranks for free polynilpotent finitely generated groups. We essentially use Hilbert–Poincaré series and some facts on growth of complex functions which are analytic in the unit circle. By growth of such functions we mean their growth when the variable tends to 1. Also we discuss for all levels q=2,3,… what numbers α>0 can be a q-dimension of some Lie (associative) algebra. Second, we discuss a ‘codimension growth’ for varieties of Lie algebras (Sections 5 and 6). It is useful to consider some exponential generating functions called complexity functions. Those functions are entire functions of a complex variable provided the varieties of Lie algebras are nontrivial. We compute the complexity functions for some varieties. The growth of a complexity function for an arbitrary polynilpotent variety is evaluated. Here we need to study the connection between the growth of a fast increasing entire function and the behavior of its Taylor coefficients. As a result we obtain a result for the asymptotics of the codimension growth of a polynilpotent variety of Lie algebras. Also we obtain an upper bound for a growth of an arbitrary nontrivial variety of Lie algebras

On the Complexity of Lie Algebras

Let U, V, W be finite dimensional vector spaces over a field k and let : U x V → W be a bilinear mapping. The (multiplieative) complexity L(~) of ~ is defined as the least r ∈~ such that there are linear forms Ul,...,u r , Vl,...,v r ∈ (U xV) and elements Wl,...,w r ∈ W satisfying r ~(x,y) = ~ Up(X,y) Vp(X,y) Wp for all (x~y) ∈ U× V .Departamento de Matemátic

Satisfiability in multi-valued circuits

Satisfiability of Boolean circuits is among the most known and important problems in theoretical computer science. This problem is NP-complete in general but becomes polynomial time when restricted either to monotone gates or linear gates. We go outside Boolean realm and consider circuits built of any fixed set of gates on an arbitrary large finite domain. From the complexity point of view this is strictly connected with the problems of solving equations (or systems of equations) over finite algebras. The research reported in this work was motivated by a desire to know for which finite algebras $\mathbf A$ there is a polynomial time algorithm that decides if an equation over $\mathbf A$ has a solution. We are also looking for polynomial time algorithms that decide if two circuits over a finite algebra compute the same function. Although we have not managed to solve these problems in the most general setting we have obtained such a characterization for a very broad class of algebras from congruence modular varieties. This class includes most known and well-studied algebras such as groups, rings, modules (and their generalizations like quasigroups, loops, near-rings, nonassociative rings, Lie algebras), lattices (and their extensions like Boolean algebras, Heyting algebras or other algebras connected with multi-valued logics including MV-algebras). This paper seems to be the first systematic study of the computational complexity of satisfiability of non-Boolean circuits and solving equations over finite algebras. The characterization results provided by the paper is given in terms of nice structural properties of algebras for which the problems are solvable in polynomial time.Comment: 50 page

Generalizations of entanglement based on coherent states and convex sets

Unentangled pure states on a bipartite system are exactly the coherent states with respect to the group of local transformations. What aspects of the study of entanglement are applicable to generalized coherent states? Conversely, what can be learned about entanglement from the well-studied theory of coherent states? With these questions in mind, we characterize unentangled pure states as extremal states when considered as linear functionals on the local Lie algebra. As a result, a relativized notion of purity emerges, showing that there is a close relationship between purity, coherence and (non-)entanglement. To a large extent, these concepts can be defined and studied in the even more general setting of convex cones of states. Based on the idea that entanglement is relative, we suggest considering these notions in the context of partially ordered families of Lie algebras or convex cones, such as those that arise naturally for multipartite systems. The study of entanglement includes notions of local operations and, for information-theoretic purposes, entanglement measures and ways of scaling systems to enable asymptotic developments. We propose ways in which these may be generalized to the Lie-algebraic setting, and to a lesser extent to the convex-cones setting. One of our original motivations for this program is to understand the role of entanglement-like concepts in condensed matter. We discuss how our work provides tools for analyzing the correlations involved in quantum phase transitions and other aspects of condensed-matter systems.Comment: 37 page

Some Problems in the Representation Theory of Simple Modular Lie Algebras

The finite-dimensional restricted simple Lie algebras of characteristic p > 5 are classical or of Cartan type. The classical algebras are analogues of the simple complex Lie algebras and have a well-advanced representation theory with important connections to Kazhdan-Lusztig theory, quantum groups at roots of unity, and the representation theory of algebraic groups. We survey progress that has been made towards developing a representation theory for the restricted simple Cartan-type Lie algebras, discuss comparable results in the classical case, formulate a couple of conjectures, and pose a dozen open problems for further study.Comment: References updated; a few minor changes made in this versio