124,464 research outputs found
Invariants of automorphic lie algebras
No full textAutomorphic Lie Algebras arise in the context of reduction groups introduced in the late 1970s [35] in the field of integrable systems. They are subalgebras of Lie algebras over a ring of rational functions, denied by invariance under the action of a finite group, the reduction group. Since their introduction in 2005 [29, 31], mathematicians aimed to classify Automorphic Lie Algebras. Past work shows remarkable uniformity between the Lie algebras associated to different reduction groups. That is, many Automorphic Lie Algebras with nonisomorphic reduction groups are isomorphic [4, 30]. In this thesis we set out to find the origin of these observations by searching for properties that are independent of the reduction group, called invariants of Automorphic Lie Algebras.
The uniformity of Automorphic Lie Algebras with nonisomorphic reduction groups starts at the Riemann sphere containing the spectral parameter, restricting the finite groups to the polyhedral groups. Through the use of classical invariant theory and the properties of this class of groups it is shown that Automorphic Lie Algebras are freely generated modules over the polynomial ring in one variable. Moreover, the number of generators equals the dimension of the base Lie algebra, yielding an invariant. This allows the definition of the determinant of invariant vectors which will turn out to be another invariant. A surprisingly simple formula is given expressing this determinant as a monomial in ground forms.
All invariants are used to set up a structure theory for Automorphic Lie Algebras. This naturally leads to a cohomology theory for root systems. A first exploration of this structure theory narrows down the search for Automorphic Lie Algebras signicantly. Various particular cases are fully determined by their invariants, including most of the previously studied Automorphic Lie Algebras, thereby providing an explanation for their uniformity.In addition, the structure theory advances the classification project. For example, it clarifies the effect of a change in pole orbit resulting in various new Cartan-Weyl normal form generators for Automorphic Lie Algebras. From a more general perspective, the success of the structure theory and root system cohomology in absence of a field promises interesting theoretical developments for Lie algebras over a graded ring
Two Generator Subalgebras Of Lie Algebras.
No full textIn [14] Thompson showed that a finite group G is solvable if and only if every twogenerated subgroup is solvable (Corollary 2, p. 388). Recently, Grunevald et al. [10] have shown that the analogue holds for finite-dimensional Lie algebras over infinite fields of characteristic greater than 5. It is a natural question to ask to what extent the two-generated subalgebras determine the structure of the algebra. It is to this question that this paper is addressed. Here, we consider the classes of strongly-solvable and of supersolvable Lie algebras, and the property of triangulability
Lie 2-algebra models
No full textIn this paper, we begin the study of zero-dimensional field theories with fields taking values in a semistrict Lie 2-algebra. These theories contain the IKKT matrix model and various M-brane related models as special cases. They feature solutions that can be interpreted as quantized 2-plectic manifolds. In particular, we find solutions corresponding to quantizations of 3, S 3 and a five-dimensional Hpp-wave. Moreover, by expanding a certain class of Lie 2-algebra models around the solution corresponding to quantized 3, we obtain higher BF-theory on this quantized space.</p
Inductive constructions for Lie bialgebras and Hopf algebras
No full textIn recent years, two generalisations of the theory of Lie algebras have become prominent, namely the "semi-classical" theory of Lie bialgebras and the "quantum" theory of Hopf algebras, including the quantized enveloping algebras. I develop an inductive approach to the study of these objects. An important tool is a construction called double-bosonisation defined by Majid for both Lie bialgebras and Hopf algebras, inspired by the triangular decomposition of a Lie algebra into positive and negative roots and a Cartan subalgebra. We describe two specific applications. The first uses double-bosonisation to add positive and negative roots and considers the relationship between two algebras when there is an inclusion of the associated Dynkin diagrams. In this setting, which we call Lie induction, double-bosonisation realises the addition of nodes to Dynkin diagrams. We use our methods to obtain necessary conditions for such an induction to be simple, using representation theory, providing a different perspective on the classification of simple Lie algebras. We consider the corresponding scheme for quantized enveloping algebras, based on inclusions of the associated root data. We call this quantum Lie induction. We prove that we have a double-bosonisation associated to these inclusions and investigate the structure of the resulting objects, which are Hopf algebras in braided categories, that is, covariant Hopf algebras. The second application generalises one of the most important constructions in this field, namely the Drinfel'd double of a Lie bialgebra, which has dimension twice that of the underlying algebra. Our construction, the triple, has dimension three times that of the input algebra. Our main result is that when the input algebra is factorisable, this is isomorphic to the triple direct sum as an algebra and a twisting as a coalgebra. We also indicate a number of ways in which the triple is related to the double
Lie algebras: infinite generalizations and deformations
No full textThere are many applications of Lie algebras to theoretical physics. This thesis is a study of some new mathematical structures which also are applicable to current physical ideas. The structures studied are Lie algebras of infinite dimension and the deformations of Lie algebras known as quantum algebras. The approach is algebraic, although physical applications are indicated. Chapter 1 The mathematics of finite and infinite dimensional Lie algebras is reviewed, together with an indication of well established uses in physics. The terms and notation used in the rest of the thesis are introduced. Chapter 2 Explicit examples of new infinite dimensional algebras of a type related to the algebras of conformal transformations on arbitrary genus Riemann surfaces are given. The relationship of these algebras to the Virasoro algebra is discussed. Chapter 3 The sine algebra is introduced and its relationship to the Moyal bracket discussed. The finite Lie algebras are given in a trigonometric basis. The many applications of the Moyal algebra are reviewed. Chapter 4 An original proof of the uniqueness of the Moyal algebra is presented. It is shown that the Moyal bracket is the most general Lie bracket of functions of two variables, and thus that the underlying associative star product is unique. It follows that all 2-index Lie algebras correspond to the Moyal algebra in some basis. Chapter 5 Quantum deformations of Lie algebras, or quantum algebras, are introduced. The many deformations of su(2) are described and the associativity conditions are discussed. Some new higher dimensional and infinite dimensional quantum algebras are given. Chapter 6 Quantum groups are discussed as groups of transformations of the quantum plane. Higher dimensional quantum groups and quantum supergroups are also described
The sheets of a classical lie algebra
No full textWe consider the adjoint action of a connected complex semisimple group G on its Lie
algebra g. A sheet of g is a maximal irreducible subset of g consisting of G-orbits of
a fixed dimension. The Lie algebra g is the finite union of its (not necessarily disjoint)
sheets. It is known how sheets are classified, and how they intersect (see [2] for the whole
story).
Let S be a sheet of g. A fundamental result says that S contains a unique nilpotent
orbit. Let {e, h, f} be a standard triple in g such that e is contained in S. Let gf be the
centralizer of f in g and define X � gf by e +X = S \ (e + gf ). Katsylo then constructs
in [9] a geometric quotient : S ! (e+X)/A where A denotes the centralizer of the triple
in G.
On the other hand, Borho and Kraft consider the categorical quotient �
S
: S ! S//G
and the normalization map of S//G. They construct a homeomorphism from the normalization
of S//G to the orbit space S/G, which is equipped with the quotient topology.
Suppose S were smooth (or normal). The restriction of �S
to S then factors through
the normalization of S//G and the induced map is a geometric quotient by a standard
criterion of geometric invariant theory ([15], Proposition 0.2). We note that the induced
map may be a geometric quotient without S being smooth (or normal).
The purpose of this work, however, is to investigate the smoothness of sheets. The
main result is:
Theorem. The sheets of classical Lie algebras are smooth.
If g is sln, this is a result of Kraft and Luna ([13]), and of Peterson ([17]) (see also [1] for
a detailed proof). For the other classical Lie algebras a few partial results were obtained
by Broer ([4]) and Panyushev ([16]). They both heavily use some additional symmetry.
On the other hand, one of the sheets of G2 is not normal (see [19]), the remaining ones
being smooth. For most of the sheets of exceptional Lie algebras it is not known whether
they are smooth or not.
This work is organized as follows:
In the first chapter, we recall the notions of decomposition class and of induced orbit,
as well as their relevance to the theory of sheets. Let l be a Levi subalgebra of g and x 2 l
a nilpotent element. The G-conjugates of elements y = z + x such that the centralizer
of z is equal to l form a decomposition class of g (“similar Jordan decomposition”). The
fact that every sheet contains a dense decomposition class leads to the classification of
sheets by G-conjugacy classes of pairs (l,Ol) consisting of a Levi subalgebra of g and a so
called rigid orbit Ol in the derived algebra of l. A rigid orbit is a (nilpotent) orbit which
itself is a sheet. The unique nilpotent orbit in the sheet corresponding to a pair (l,Ol) as
above is obtained by inducing Ol from l to g: Let p be any parabolic subalgebra of g with
Levi part l, and pu its unipotent radical. The induced orbit Indg
l Ol is then defined as the
unique orbit of maximal dimension in G(Ol + pu).
In the second chapter, we explain Katsylo’s results on sheets in detail. Let S be the
sheet corresponding to a pair (l,Ol) and let {e, h, f} be a standard triple in g such that
e is contained in S. If the triple is suitably chosen the sheet S may be described as
G(e + k) where k denotes the center of l. We use the canonical isomorphism attached to
the triple (2.1), and obtain a morphism ": e + k ! e + gf such that e + z and "(e + z)
are G-conjugate for every z 2 k. It turns out that "(e + k) is an irreducible component of
e + X, the intersection of S and e + gf . Moreover, the centralizer of the triple in G acts
transitively on the set of irreducible components of e + X, and its connected component
acts trivially on e + X. Essentially by sl2 theory, the two varieties S and e + X are
smoothly equivalent. This is the approach we use to investigate smoothness of sheets. At
the end of the chapter, we apply these ideas to the regular sheet of g and to admissible
sheets of g. The regular sheet is the (very well known) open, dense subset consisting of
the regular elements of g. It corresponds to the pair (h, 0) where h is a Cartan subalgebra
of g. By Kostant, e + gf is contained in the regular sheet and every regular element is
G-conjugate to a unique element of e + gf . Hence " maps e + h onto e + gf ; it is the
quotient by the Weyl group of G. The admissible sheets, in this context, are those coming
nearest to the regular sheet.
In the remaining chapters, we deal with sheets in classical Lie algebras (in fact, our
setting is slightly more general (3.1)). We prove that " maps e+k onto e+X; it turns out
to be the quotient by some reflection group acting on k. Therefore e+X is isomorphic to
affine space and so S is smooth.
We first take a look at the linear group, that is, G is equal to GL(V ) for some complex
vector space V . In this case, the sheets of g are in one-to-one correspondence to the
partitions of dim V (3.3). In order to make this explicit, we associate a partition to every
y 2 g as follows: We decompose V as a C[y]-module into a direct sum of cyclic submodules
by successively cutting off cyclic submodules of maximal dimension. The dimensions of
these direct summands define a partition of dim V . The sheets of g are then the sets S(l)
consisting of elements y 2 g with fixed partition l. The crucial observation is the fact
that there is a decomposition of V into direct summands Vi which respects the setting of
the second chapter in the following sense (Chapter 5): Let S be a sheet of g described as
G(e + k) and let ": e + k ! e + gf be the corresponding map. For every y 2 e + k, the
C[y]-module V decomposes into a direct sum of the same cyclic submodules Vi. We find
elements ei and subspaces ki of gi = gl(Vi) such that Gi(ei +ki) is the regular sheet of gi,
and such that e =
P
i ei and k � �iki. Let "i : ei + ki ! ei + gfi
i be the corresponding
maps. Then " is the restriction of
P
i "i to k. But we already know that "i is the quotient
by the Weyl group of Gi. Finally, a straightforward calculation using basic invariants
(power sums) shows that " is the quotient by the normalizer of k in the Weyl group of G,
which in this case acts as reflection group on k. Since the centralizer of the triple {e, h, f} in G is connected, the image of " is equal to e + X.
The proof for the symplectic groups Sp(V ) and for the orthogonal groups O(V ) follows
along the same lines. We begin with a classification of sheets in combinatorial terms (3.4).
Then we use the combinatorial data to decompose V into a direct sum of subspaces Vi
such that a proceeding similar to the linear case is possible (6.1). To be more precise,
V decomposes as C[y]-module into the direct sum of submodules Vi for every y 2 e + k.
These submodules may not be cyclic; however, they decompose into at most two cyclic
submodules. The next step consists of identifying the maps "i : ei + ki ! ei + gfi
i as
quotients by some reflection group acting on ki. The case of Vi decomposing into two
cyclic submodules of different dimension is the core of this work (6.3). It requires a lot
of ad hoc calculation. The two other cases are readily reduced to the case of the regular
sheet (6.2). At last, a calculation using basic invariants shows that " is the quotient by
some reflection group acting on k (6.4).
Acknowledgments. I am grateful to Hanspeter Kraft for arousing my interest in
this subject, for all his valuable suggestions and support during the course of this work,
and for making it possible to stay at the University of Michigan for a year. I got financial
support from the Max Geldner Stiftung, Basel, during that year abroad. Many thanks go
to Pavel Katsylo and Bram Broer for sharing their ideas, to Stephan Mohrdieck
for his constant interest, and to Jan Draisma for numerous helpful conversations
Outsmarting the liars: toward a cognitive lie detection approach
No full text Five decades of lie detection research have shown that people’s ability to detect deception by observing behavior and listening to speech is limited. The problem is that cues to deception are typically faint and unreliable. The aim for interviewers, therefore, is to ask questions that actively elicit and amplify verbal and nonverbal cues to deceit. We present an innovative lie detection perspective based on cognitive load, demonstrating that it is possible to ask questions that raise cognitive load more in liars than in truth tellers. This cognitive lie detection perspective consists of two approaches. The imposing-cognitive-load approach aims to make the interview setting more difficult for interviewees. We argue that this affects liars more than truth tellers, resulting in more, and more blatant, cues to deceit. The strategic-questioning approach examines different ways of questioning that elicit the most differential responses between truth tellers and liars. </jats:p
The Role of sh-Lie Algebras in Lagrangian Field Theory.
No full textThe purpose of this dissertation is to study strongly homotopy Lie algebras (sh-Lie algebras) and their applications with primary emphasis on applications to field theory. Strongly homotopy Lie algebras are defined on graded vector spaces. They generally consist of an infinite sequence of mappings , which satisfy certain identities. We show that, in the presence of appropriate hypotheses, there always exists a simplified sh-Lie algebra structure with for . This is a special case which has occured in several applications. While it is known that sh-Lie algebras arise in field theory as a homological resolution of a Poisson bracket defined on the space of local functionals, we show how these sh-Lie algebras transform in the event of canonical transformations on the space of local functionals. Additionally, it is shown how a group which acts via canonical transformations transforms the sh-Lie structure and eventually leads to reduction theorems. Two kinds of reduction are obtained corresponding to two different kinds of group action and, in each case it is shown how to obtain an induced sh-Lie algebra on a corresponding reduced graded vector space. Several applications of the theory are considered as well
Maktabat Al Muthanna Baghdad Feb-May 1962
No full textOn the same date, Ali Al-Mansouri issued an official financial statement confirming that the Al-Khanji Foundation owed a total of 11.375.أصدر علي المنصوري بيانًا ماليًا رسميًا بتاريخ 25 نيسان 1962 يُفيد بأن مؤسسة الخانجي مدينة بمبلغ إجمالي قدره 11,375
Lie Symmetry Methods for Multidimensional Linear, Parabolic PDES and Diffusions
No full textIn this paper we introduce methods based upon Lie symmetry analysis for the construction of explicit fundamental solutions of multidimensional parabolic PDEs. We give applications to the problem of finding transition probability densities for multidimensional diffusions and to representation theory.Lie symmetry groups; fundamental solutions; transition densities; representation theory
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