Representations Of Lie Algebras Research Articles

Some compatible Poisson structures and integrable bi-Hamiltonian systems on four dimensional and nilpotent six dimensional symplectic real Lie groups

We provide an alternative method for obtaining of compatible Poisson structures on Lie groups by means of the adjoint representations of Lie algebras. In this way we calculate some compatible Poisson structures on four dimensional and nilpotent six dimensional symplectic real Lie groups. Then using Magri-Morosi’s theorem we obtain new bi-Hamiltonian systems with four dimensional and nilpotent six dimensional symplectic real Lie groups as phase spaces.

Virasoro and KdV

We investigate the structure of representations of the (positive half of the) Virasoro algebra and situations in which they decompose as a tensor product of Lie algebra representations. As an illustration, we apply these results to the differential operators defined by the Virasoro conjecture and obtain some factorization properties of the solutions as well as a link to the multicomponent KP hierarchy.

Minimal representations for 6-dimensional nilpotent Lie algebra

Given a Lie algebra [Formula: see text], let [Formula: see text] and [Formula: see text] be the minimal dimension of a faithful representation and nilrepresentation of [Formula: see text], respectively. In this paper, we give [Formula: see text] and [Formula: see text] for each nilpotent Lie algebra [Formula: see text] of dimension [Formula: see text] over a field [Formula: see text] of characteristic zero. We also give a minimal faithful representation and nilrepresentation in each case.

Non-local meta-conformal invariance in diffusion-limited erosion

The non-stationary relaxation and physical ageing in the diffusion-limited erosion process (dle) is studied through the exact solution of its Langevin equation, in d spatial dimensions. The dynamical exponent z = 1, the growth exponent and the ageing exponents and are found. In d = 1 spatial dimension, a new representation of the meta-conformal Lie algebra, isomorphic to , acts as a dynamical symmetry of the noise-averaged dle Langevin equation. Its infinitesimal generators are non-local in space. The exact form of the full time-space dependence of the two-time response function of dle is reproduced for d = 1 from this symmetry. The relationship to the terrace-step-kink model of vicinal surfaces is discussed.

Invariant Tensors and the Cyclic Sieving Phenomenon

We construct a large class of examples of the cyclic sieving phenomenon by exploiting the representation theory of semi-simple Lie algebras. Let $M$ be a finite dimensional representation of a semi-simple Lie algebra and let $B$ be the associated Kashiwara crystal. For $r\ge 0$, the triple $(X,c,P)$ which exhibits the cyclic sieving phenomenon is constructed as follows: the set $X$ is the set of isolated vertices in the crystal $\otimes^rB$; the map $c\colon X\rightarrow X$ is a generalisation of promotion acting on standard tableaux of rectangular shape and the polynomial $P$ is the fake degree of the Frobenius character of a representation of $\mathfrak{S}_r$ related to the natural action of $\mathfrak{S}_r$ on the subspace of invariant tensors in $\otimes^rM$. Taking $M$ to be the defining representation of $\mathrm{SL}(n)$ gives the cyclic sieving phenomenon for rectangular tableaux.

Geometric Theory of Heat from Souriau Lie Groups Thermodynamics and Koszul Hessian Geometry: Applications in Information Geometry for Exponential Families

We introduce the symplectic structure of information geometry based on Souriau’s Lie group thermodynamics model, with a covariant definition of Gibbs equilibrium via invariances through co-adjoint action of a group on its moment space, defining physical observables like energy, heat, and moment as pure geometrical objects. Using geometric Planck temperature of Souriau model and symplectic cocycle notion, the Fisher metric is identified as a Souriau geometric heat capacity. The Souriau model is based on affine representation of Lie group and Lie algebra that we compare with Koszul works on G/K homogeneous space and bijective correspondence between the set of G-invariant flat connections on G/K and the set of affine representations of the Lie algebra of G. In the framework of Lie group thermodynamics, an Euler-Poincaré equation is elaborated with respect to thermodynamic variables, and a new variational principal for thermodynamics is built through an invariant Poincaré-Cartan-Souriau integral. The Souriau-Fisher metric is linked to KKS (Kostant–Kirillov–Souriau) 2-form that associates a canonical homogeneous symplectic manifold to the co-adjoint orbits. We apply this model in the framework of information geometry for the action of an affine group for exponential families, and provide some illustrations of use cases for multivariate gaussian densities. Information geometry is presented in the context of the seminal work of Fréchet and his Clairaut-Legendre equation. The Souriau model of statistical physics is validated as compatible with the Balian gauge model of thermodynamics. We recall the precursor work of Casalis on affine group invariance for natural exponential families.

Faithful completely reducible representations of modular Lie algebras

ABSTRACTLet L be a Lie algebra of dimension n over a field F of characteristic p>0. I prove the existence of a faithful completely reducible L-module of dimension less than or equal to .

Minimal faithful upper-triangular matrix representations for solvable Lie algebras

The existence of matrix representations for any given finite-dimensional complex Lie algebra is a classic result on Lie Theory. In particular, such representations can be obtained by means of an isomorphic matrix Lie algebra consisting of upper-triangular square matrices. Unfortunately, there is no general information about the minimal order for the matrices involved in such representations. In this way, our main goal is to revisit, debug and implement an algorithm which provides the minimal order for matrix representations of any finite-dimensional solvable Lie algebra when inserting its law, as well as returning a matrix representative of such an algebra by using the minimal order previously computed. In order to show the applicability of this procedure, we have computed minimal representatives not only for each solvable Lie algebra with dimension less than 6, but also for some solvable Lie algebras of arbitrary dimension.

Combinatorial bases of basic modules for affine Lie algebras Cn(1)

Lepowsky and Wilson initiated the approach to combinatorial Rogers-Ramanujan type identities via vertex operator constructions of standard (i.e., integrable highest weight) representations of affine Kac-Moody Lie algebras. Meurman and Primc developed further this approach for sl(2,C) ˜ by using vertex operator algebras and Verma modules. In this paper, we use the same method to construct combinatorial bases of basic modules for affine Lie algebras of type Cn(1) and, as a consequence, we obtain a series of Rogers-Ramanujan type identities. A major new insight is a combinatorial parametrization of leading terms of defining relations for level one standard modules for affine Lie algebra of type Cn(1).

Representations of Lie algebras by non-skewselfadjoint operators in Hilbert space

We study non-skewselfadjoint representations of a finite dimensional real Lie algebra g. To this end we embed a non-skewselfadjoint representation of g into a more complicated structure, that we call a g-operator vessel and that is associated to an overdetermined linear conservative input/state/output system on the corresponding simply connected Lie group G. We develop the frequency domain theory of the system in terms of representations of G, and introduce the joint characteristic function of a g-operator vessel which is the analogue of the classical notion of the characteristic function of a single non-selfadjoint operator. As the first non-commutative example, we apply the theory to the Lie algebra of the ax+b group, the group of affine transformations of the line.

Kraśkiewicz–Pragacz modules and Ringel duality

In [5,6] Kraśkiewicz and Pragacz introduced representations of the upper-triangular Lie algebra b whose characters are Schubert polynomials. In [12] the author studied the properties of Kraśkiewicz–Pragacz modules using the theory of highest weight categories. From the results there, in particular we obtain a certain highest weight category whose standard modules are KP modules. In this paper we show that this highest weight category is self Ringel-dual. This leads to an interesting symmetry relation on Ext groups between KP modules. We also show that the tensor product operation on b-modules is compatible with Ringel duality functor.

A new functor from D 5-Mod to E 6-Mod

We find a new representation of the simple Lie algebra of type E 6 on the polynomial algebra in 16 variables, which gives a fractional representation of the corresponding Lie group on 16-dimensional space. Using this representation and Shen’s idea of mixed product, we construct a new functor from D 5-Mod to E 6-Mod. A condition for the functor to map a finite-dimensional irreducible D 5-module to an infinite-dimensional irreducible E 6-module is obtained. Our results yield explicit constructions of certain infinite-dimensional irreducible weight E6-modules with finite-dimensional weight subspaces. In our approach, the idea of Kostant’s characteristic identities plays a key role.

Leibniz Algebras Associated with Representations of the Diamond Lie Algebra

In this paper we describe some Leibniz algebras whose corresponding Lie algebra is four-dimensional Diamond Lie algebra \(\mathfrak {D}\) and the ideal generated by the squares of elements (further denoted by I) is a right \(\mathfrak {D}\)-module. Using description (Casati et al J. Math. Phys. 51: 033515 (2010)) of representations of algebra \(\mathfrak {D}\) in \(\mathfrak {sl}(3,{\mathbb {C}})\) and \(\mathfrak {sp}(4,{\mathbb {F}})\) where \({\mathbb {F}}={\mathbb {R}}\) or \({\mathbb {C}}\) we obtain the classification of above mentioned Leibniz algebras. Moreover, Fock representation of Heisenberg Lie algebra was extended to the case of the algebra \(\mathfrak {D}.\) Classification of Leibniz algebras with corresponding Lie algebra \(\mathfrak {D}\) and with the ideal I as a Fock right \(\mathfrak {D}\)-module is presented. The linear integrable deformations in terms of the second cohomology groups of obtained finite-dimensional Leibniz algebras are described. Two computer programs in Mathematica 10 which help to calculate for a given Leibniz algebra the general form of elements of spaces B L 2 and Z L 2 are constructed, as well.

Standard parabolic subsets of highest weight modules

In this paper we study certain fundamental and distinguished subsets of weights of an arbitrary highest weight module over a complex semisimple Lie algebra. These sets ${\rm wt}_J \mathbb{V}^\lambda$ are defined for each highest weight module $\mathbb{V}^\lambda$ and each subset $J$ of simple roots; we term them "standard parabolic subsets of weights". It is shown that for any highest weight module, the sets of simple roots whose corresponding standard parabolic subsets of weights are equal form intervals in the poset of subsets of the set of simple roots under containment. Moreover, we provide closed-form expressions for the maximum and minimum elements of the aforementioned intervals for all highest weight modules $\mathbb{V}^\lambda$ over semisimple Lie algebras $\mathfrak{g}$. Surprisingly, these formulas only require the Dynkin diagram of $\mathfrak{g}$ and the integrability data of $\mathbb{V}^\lambda$. As a consequence, we extend classical work by Satake, Borel-Tits, Vinberg, and Casselman, as well as recent variants by Cellini-Marietti to all highest weight modules. We further compute the dimension, stabilizer, and vertex set of standard parabolic faces of highest weight modules, and show that they are completely determined by the aforementioned closed-form expressions. We also compute the $f$-polynomial and a minimal half-space representation of the convex hull of the set of weights. These results were recently shown for the adjoint representation of a simple Lie algebra, but analogues remain unknown for any other finite- or infinite-dimensional highest weight module. Our analysis is uniform and type-free, across all semisimple Lie algebras and for arbitrary highest weight modules.

SU(2)/SL(2) knot invariants and Kontsevich–Soibelman monodromies

We review the Reshetikhin–Turaev approach for constructing noncompact knot invariants involving Rmatrices associated with infinite-dimensional representations, primarily those constructed from the Faddeev quantum dilogarithm. The corresponding formulas can be obtained from modular transformations of conformal blocks as their Kontsevich–Soibelman monodromies and are presented in the form of transcendental integrals, where the main issue is working with the integration contours. We discuss possibilities for extracting more explicit and convenient expressions that can be compared with the ordinary (compact) knot polynomials coming from finite-dimensional representations of simple Lie algebras, with their limits and properties. In particular, the quantum A-polynomials and difference equations for colored Jones polynomials are the same as in the compact case, but the equations in the noncompact case are homogeneous and have a nontrivial right-hand side for ordinary Jones polynomials.

A Steinberg type decomposition theorem for higher level Demazure modules

We study Demazure modules which occur in a level ℓ irreducible integrable representation of an affine Lie algebra. We also assume that they are stable under the action of the standard maximal parabolic subalgebra of the affine Lie algebra. We prove that such a module is isomorphic to the fusion product of ‘‘prime’’ Demazure modules, where the prime factors are indexed by dominant integral weights which are either a multiple of ℓ or take value less than ℓ on all simple coroots. Our proof depends on a technical result which we prove in all the classical cases and G2. Calculations with mathematica show that this result is correct for small values of the level. Using our result, we show that there exist generalizations of Q-systems to pairs of weights where one of the weights is not necessarily rectangular and is of a different level. Our results also allow us to compare the multiplicities of an irreducible representation occurring in the tensor product of certain pairs of irreducible representations, i.e., we establish a version of Schur positivity for such pairs of irreducible modules for a simple Lie algebra.

Leibniz algebras associated with some finite-dimensional representation of Diamond Lie algebra

In this paper we classify Leibniz algebras whose associated Lie algebra is four-dimensional Diamond Lie algebra \U0001d56f and the ideal generated by squares of elements is represented by one of the finite-dimensional indecomposable D-modules Un1, Un2 or Wn1 or Wn2.

Weight posets associated with gradings of simple Lie algebras, Weyl groups, and arrangements of hyperplanes

The set of weights of a finite-dimensional representation of a reductive Lie algebra has a natural poset structure ("weight poset"). Studying certain combinatorial problems related to antichains in weight posets, we realised that the best setting is provided by the representations associated with $\mathbb Z$-gradings of simple Lie algebras (arXiv: math.CO 1411.7683). If $\mathfrak g$ is a simple Lie algebra, then a $\mathbb Z$-grading of $\mathfrak g$ induces a $\mathbb Z$-grading of the corresponding root system $\Delta$. In this article, we elaborate on a general theory of lower ideals (or antichains) in the corresponding weight posets $\Delta(1)$. In particular, we provide a bijection between the lower ideals in $\Delta(1)$ and certain elements of the Weyl group of $\mathfrak g$. An inspiring observation is that, to a great extent, the theory of lower ideals in $\Delta(1)$ is similar to the theory of upper (= ad-nilpotent) ideals in the whole poset of positive roots $\Delta^+$.

On the derivation representation of the fundamental Lie algebra of mixed elliptic motives

Richard Hain and Makoto Matsumoto constructed a category of universal mixed elliptic motives, and described the fundamental Lie algebra of this category: it is a semi-direct product of the fundamental Lie algebra \({{\mathrm{Lie}}}\pi _1(\mathsf {MTM})\) of the category of mixed Tate motives over \(\mathbb {Z}\) with a filtered and graded Lie algebra \(\mathfrak {u}\). This Lie algebra, and in particular \(\mathfrak {u}\), admits a representation as derivations of the free Lie algebra on two generators. In this paper we study the image \(\mathscr {E}\) of this representation of \(\mathfrak {u}\), starting from some results by Aaron Pollack, who determined all the relations in a certain filtered quotient of \(\mathscr {E}\), and gave several examples of relations in low weights in \(\mathscr {E}\) that are connected to period polynomials of cusp forms on \({{\mathrm{SL}}}_2(\mathbb {Z})\). Pollack’s examples lead to a conjecture on the existence of such relations in all depths and all weights, that we state in this article and prove in depth 3 in all weights. The proof follows quite naturally from Ecalle’s theory of moulds, to which we give a brief introduction. We prove two useful general theorems on moulds in the appendices.

Haar expectations of ratios of random characteristic polynomials

We compute Haar ensemble averages of ratios of random characteristic polynomials for the classical Lie groups K = O(N), SO(N), and USp(N). To that end, we start from the Clifford-Weyl algebera in its canonical realization on the complex of holomorphic differential forms for a C-vector space V. From it we construct the Fock representation of an orthosymplectic Lie superalgebra osp associated to V. Particular attention is paid to defining Howe's oscillator semigroup and the representation that partially exponentiates the Lie algebra representation of sp in osp. In the process, by pushing the semigroup representation to its boundary and arguing by continuity, we provide a construction of the Shale-Weil-Segal representation of the metaplectic group. To deal with a product of n ratios of characteristic polynomials, we let V = C^n \otimes C^N where C^N is equipped with its standard K-representation, and focus on the subspace of K-equivariant forms. By Howe duality, this is a highest-weight irreducible representation of the centralizer g of Lie(K) in osp. We identify the K-Haar expectation of n ratios with the character of this g-representation, which we show to be uniquely determined by analyticity, Weyl group invariance, certain weight constraints and a system of differential equations coming from the Laplace-Casimir invariants of g. We find an explicit solution to the problem posed by all these conditions. In this way we prove that the said Haar expectations are expressed by a Weyl-type character formula for all integers N \ge 1. This completes earlier work by Conrey, Farmer, and Zirnbauer for the case of U(N).