The orbits of Weyl groups W(A(n)) of simple A(n) type Lie algebras are reduced to the union of orbits of the Weyl groups of maximal reductive subalgebras of A(n). Matrices transforming points of the orbits of W(An) into points of subalgebra orbits are listed for all cases n<=8 and for the infinite series of algebra-subalgebra pairs A(n) - A(n-k-1) x A(k) x U(1), A(2n) - B(n), A(2n-1) - C(n), A(2n-1) - D(n). Numerous special cases and examples are shown.
The paper contains the Weyl formula for the counting function of the interior transmission problem when the latter is parameter-elliptic. Branching billiard trajectories are constructed, and the second term of the Weyl asymptotics is estimated from above under some conditions on the set of periodic billiard trajectories.
The orbits of Weyl groups W(B(n)), W(C(n)) and W(D(n)) of the simple Lie algebras B(n), C(n) and D(n) are reduced to the union of the orbits of Weyl groups of the maximal reductive subalgebras of B(n), C(n) and D(n). Matrices transforming points of W(B(n)), W(C(n)) and W(D(n)) orbits into points of subalgebra orbits are listed for all cases n<=8 and for the infinite series of algebra-subalgebra pairs B(n) - B(n-1) x U(1), B(n) - D(n), B(n) - B(n-k) x D(k), B(n) - A(1), C(n) - C(n-k) x C(k), C(n) - A(n-1) x U(1), D(n) - A(n-1) x U(1), D(n) - D(n-1) x U(1), D(n) -B(n-1), D(n) - B(n-k-1) x B(k), …
Weyl denominator identity for the affinization of a basic Lie superalgebra with a non-zero Killing form was formulated by Kac and Wakimoto and was proven by them in defect one case. In this paper we prove this identity.
We prove the Weyl denominator identity for the affine Lie superalgebra gl(2|2)^ conjectured by V. Kac and M. Wakimoto. As it was pointed out in their paper, this gives another proof of Jacobi identity for the number of presentations of a given integer as a sum of eight squares.
Recurrent relations for branching coefficients in affine Lie algebras integrable highest weight modules are studied. The decomposition algorithm based on the injection fan technique is developed for the case of an arbitrary reductive subalgebra. In particular we consider the situation where the Weyl denominator becomes singular with respect to the subalgebra. We demonstrate that for any reductive subalgebra it is possible to define the injection fan and the analogue of the Weyl numerator - the tools that describe explicitly the recurrent properties of branching coefficients. Possible applicati…
The weights of finite-dimensional representations of simple Lie algebras are naturally associated with Weyl polytopes. Representation characters decompose into multiplicity-free sums over the weights in Weyl polytopes. The Brion formula for these Weyl polytope sums is remarkably similar to the Weyl character formula. Moreover, the same Lie characters are also expressible as Demazure character formulas. This motivates a search for new expressions for Weyl polytope sums, and we prove such a formula involving Demazure operators. It applies to the Weyl polytope sums of the simple Lie algebras $A_n…
The purpose of this paper is to answer the question whether it is possible to realize simultaneously the relations $N_{α,β}=-N_{-α,-β}$, $N_{α,β}=N_{β,-α-β}=N_{-α-β,α}$ and $N_{α,β}N_{-α,-β}=-\frac{1}{2}q(p+1)\langleα,H_α\rangle$ by the structure constants of the Lie algebra ${\mathfrak g}_2$. We show that if the structure constants obey the first relation, the three last ones are violated, and vice versa. Contrary to the second case, the first one uses the Cartan matrix elements to derive the structure constants in the form of $\langleβ,H_α\rangle$. The commutation relations corresponding to …
This paper generalizes Weyl realization to a class of Lie superalgebras $\mathfrak{g}=\mathfrak{g}_0\oplus \mathfrak{g}_1$ satisfying $[\mathfrak{g}_1,\mathfrak{g}_1]=\{0\}$. First, we give a novel proof of the Weyl realization of a Lie algebra $\mathfrak{g}_0$ by deriving a functional equation for the function that defines the realization. We show that this equation has a unique solution given by the generating function for the Bernoulli numbers. This method is then generalized to Lie superalgebras of the above type.
We investigate dual realizations of non--commutative spaces of Lie algebra type in terms of formal power series in the Weyl algebra. To each realization of a Lie algebra $\g$ we associate a star--product on the symmetric algebra $S(\g)$ and an ordering on the enveloping algebra $U(\g)$. Dual realizations of $\g$ are defined in terms of left--right duality of the star--products on $S(\g)$. It is shown that the dual realizations are related to an extension problem for $\g$ by shift operators whose action on $U(\g)$ describes left and right shift of the generators of $U(\g)$ in a given monomial. …
We develop and describe continuous and discrete transforms of class functions on a compact semisimple, but not simple, Lie group $G$ as their expansions into series of special functions that are invariant under the action of the even subgroup of the Weyl group of $G$. We distinguish two cases of even Weyl groups -- one is the direct product of even Weyl groups of simple components of $G$, the second is the full even Weyl group of $G$. The problem is rather simple in two dimensions. It is much richer in dimensions greater than two -- we describe in detail $E-$transforms of semisimple Lie groups…
Using a left multiplication defined on a right quaternionic Hilbert space, linear self-adjoint momentum operators on a right quaternionic Hilbert space are defined in complete analogy with their complex counterpart. With the aid of the so-obtained position and momentum operators, we study the Heisenberg uncertainty principle on the whole set of quaternions and on a quaternionic slice, namely on a copy of the complex plane inside the quaternions. For the quaternionic harmonic oscillator, the uncertainty relation is shown to saturate on a neighborhood of the origin in the case we consider the wh…
As Lie algebras of compact connected Lie groups, semisimple Lie algebras have wide applications in the description of continuous symmetries of physical systems. Mathematically, semisimple Lie algebra admits a Cartan-Weyl basis of generators which consists of a Cartan subalgebra of mutually commuting generators H_I and a number of step generators E^αthat are characterized by a root space of non-degenerate one-forms α. This simple decomposition in terms of the root space allows for a complete classification of semisimple Lie algebras. In this paper, we introduce the analogous concept of a Cartan…
When it is based on Kac-Peterson form of Affine Weyl Groups, Weyl-Kac character formula could be formulated in terms of Theta functions and a sum over finite Weyl groups. We, instead, give a reformulation in terms of Schur functions which are determined by the so-called Permutation Weights and there are only a finite number of permutation weights at each and every order of weight depths . Affine signatures are expressed in terms of an index which by definition is based on a decomposition of horizontal weights in terms of some Fundamental Weights.
Ten types of discrete Fourier transforms of Weyl orbit functions are developed. Generalizing one-dimensional cosine, sine and exponential, each type of the Weyl orbit function represents an exponential symmetrized with respect to a subgroup of the Weyl group. Fundamental domains of even affine and dual even affine Weyl groups, governing the argument and label symmetries of the even orbit functions, are determined. The discrete orthogonality relations are formulated on finite sets of points from the refinements of the dual weight lattices. Explicit counting formulas for the number of points of …
One of the most important questions in the Bagger-Lambert-Gustavsson (BLG) theory of multiple M2-branes is the choice of the Lie 3-algebra. The Lie 3-algebra should be chosen such that the corresponding BLG model is unitary and admits fuzzy 3-sphere as a solution. In this paper we propose another new condition: the Lie 3-algebras of use must be connected to the semisimple Lie algebras describing the gauge symmetry of D-branes via a certain reduction condition. We show that this reduction condition leads to a natural generalization of the Cartan-Weyl 3-algebras introduced in arXiv:1004.1397. Si…
4-dimensional $F_{4} $ polytopes and their dual polytopes have been constructed as the orbits of the Coxeter-Weyl group $W(F_{4})$ where the group elements and the vertices of the polytopes are represented by quaternions. Branchings of an arbitrary \textbf{$W(F_{4})$} orbit under the Coxeter groups $W(B_{4} $ and $W(B_{3}) \times W(A_{1})$ have been presented. The role of group theoretical technique and the use of quaternions have been emphasized
In 1994, Kac and Wakimoto suggested a generalization of Bernstein and Leites character formula for basic Lie superalgebras, and the natural question was raised: to which simple highest weight modules does it apply? In this paper, we prove a similar formula for a large class of finite-dimensional simple modules over the Lie superalgebra gl(m|n), which we call piecewise disconnected modules, or PDC. The class of PDC modules naturally includes totally connected modules and totally disconnected modules, the two families for which similiar character formulas were proven by Su and Zhang as special c…
4-dimensional H4 polytopes and their dual polytopes have been constructed as the orbits of the Coxeter-Weyl group W(H4) where the group elements and the vertices of the polytopes are represented by quaternions. Projection of an arbitrary W(H4) orbit into three dimensions is made preserving the icosahedral subgroup W(H3) and the tetrahedral subgroup W(A3), the latter follows a branching under the Coxeter group W(A4) . The dual polytopes of the semi-regular and quasi-regular H4 polytopes have been constructed.
We present LieART 2.0 which contains substantial extensions to the Mathematica application LieART (Lie Algebras and Representation Theory) for computations frequently encountered in Lie algebras and representation theory, such as tensor product decomposition and subalgebra branching of irreducible representations. The basic procedure is unchanged: it computes root systems of Lie algebras, weight systems and several other properties of irreducible representations, but new features and procedures have been included to allow the extensions to be seamless. The new version of LieART continues to be…
Einstein-Weyl structures on a three-dimensional manifold $M$ is given by a system $E$ of PDEs on sections of a bundle over $M$. This system is invariant under the Lie pseudogroup $G$ of local diffeomorphisms on $M$. Two Einstein-Weyl structures are locally equivalent if there exists a local diffeomorphism taking one to the other. Our goal is to describe the quotient equation $E/G$ whose solutions correspond to nonequivalent Einstein-Weyl structures. The approach uses symmetries of the Manakov-Santini integrable system and the action of the corresponding Lie pseudogroup.
Poincare Polynomial of a Kac-Moody Lie algebra can be obtained by classifying the Weyl orbit $W(ρ)$ of its Weyl vector $ρ$. A remarkable fact for Affine Lie algebras is that the number of elements of $W(ρ)$ is finite at each and every depth level though totally it has infinite number of elements. This allows us to look at $W(ρ)$ as a manifold graded by depths of its elements and hence a new kind of Poincare Polynomial is defined. We give these polynomials for all Affine Kac-Moody Lie algebras, non-twisted or twisted. The remarkable fact is however that, on the contrary to the ones which are cl…
It is well-known that the commutant algebra of the $U_q(\mathfrak{sl}_2)$-action on the $n$-fold tensor product of its fundamental module is isomorphic to the Temperley-Lieb algebra TL$_n(ν)$ with fugacity parameter $ν= -q - q^{-1}$ (at least in the generic case, i.e., when $q$ is not a root of unity, or $n$ is small enough). Furthermore, the simple $U_q(\mathfrak{sl}_2)$-modules appearing in the direct-sum decomposition of the $n$-fold tensor product module are in one-to-one correspondence with those of the Temperley-Lieb algebra. This double-commutant property is referred to as quantum Schur…
