Sabino Di Trani
In the paper we complete a case by case proof of Reeder's Conjecture started in our previous work, proving the conjecture for simple Lie algebras of type $D$ and for the exceptional cases.
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In the paper we propose a proof of Reeder's Conjecture on the graded multiplicities of small representations in the exterior algebra $Λ$g for the simple Lie algebras of type B and C.
Let $\mathfrak g$ be a simple Lie algebra with Cartan subalgebra $\mathfrak h$ and Weyl group $W$. We build up a graded map $(\mathcal H\otimes \bigwedge\mathfrak h\otimes \mathfrak h)^W\to (\bigwedge \mathfrak g\otimes \mathfrak g)^\mathfrak g$ of $(\bigwedge \mathfrak g)^\mathfrak g\cong S(\mathfrak h)^W$-modules, where $\mathcal H$ is the space of $W$-harmonics. In this way we prove an enhanced form of a conjecture of Reeder for the adjoint representation. New version with different title. Various improvements. New section 7.
This note presents a procedure to determine the reduction of the irreducible and the induced characters of the symmetric group in terms of the irreducible and induced characters of the hyperoctahedral group Key Words: Symmetric Group, Hyperoctahedral group, Representations, Characters, Re- duction.
Let k be a field of characteristic not two or three, let $\mathfrak{g}$ be a finite-dimensional colour Lie algebra and let V be a finite-dimensional representation of $\mathfrak{g}$. In this article we give various ways of constructing a colour Lie algebra $\tilde{\mathfrak{g}}$ whose bracket in some sense extends both the bracket of $\mathfrak{g}$ and the action of $\mathfrak{g}$ on V. Colour Lie algebras, originally introduced by R. Ree ([Ree60]), generalise both Lie algebras and Lie superalgebras, and in those cases our results imply many known results ([Kos99], [Kos01], [CK15], [SS15]). Fo…
Linear codes with large minimal distances are important error correcting codes in information theory.Orthogonal codes have more applications in the other fields of mathematics. In this paper, we study the binary and ternary orthogonal codes generated by the weight matrices on finite-dimensional modules of simple Lie algebras. The Weyl groups of the Lie algebras act on these codes isometrically. It turns out that certain weight matrices of $sl(n,\mbb{C})$ and $o(2n,\mbb{C})$ generate doubly-even binary orthogonal codes and ternary orthogonal codes with large minimal distances. Moreover, we prov…
For the orthogonal Lie algebra O(2n+1), in addition to the conventional set of orthogonal polynomials, another set is produced with the help of the Lie superalgebra OSP(1|2n). Difficulties related with expression of Dyson's constant for the Lie superalgebras are discussed.
Symmetric matrices, orthogonal Lie algebras, and Lie-Yamaguti algebras
本文利用对称矩阵上的 Jordan 与三元系结构构造了一族以正交李代数 so(n,K) 为导子代数的不可约 Lie-Yamaguti 代数，并用计算机代数研究了 n<5 时的低次多项式恒等式。
Symplectic (respectively orthogonal) triple systems provide constructions of Lie algebras (resp. superalgebras). However, in characteristic 3, it is shown that this role can be interchanged and that Lie superalgebras (resp. algebras) can be built out of symplectic triple systems (resp. orthogonal triple systems) with a different construction. As a consequence, new simple finite dimensional Lie superalgebras, as well as new models of some nonclassical simple Lie algebras, are obtained over fields of characteristic 3.
It was studied the growth of Temperley-Lieb type algebras with orthogonal and commutative relations associated with 2-colored edges graphs. Depending on the structure of the graphs they can be finite-dimensional or to have linear or exponential growth.
In this note, we compute the homology with trivial coefficients of Lie algebras of generalized Jacobi matrices of type $B, C$ and $D$ over an associative unital $k$-algebra with $k$ being a field of characteristic $0$.
We give a uniform interpretation of the classical continuous Chebyshev's and Hahn's orthogonal polynomials of discrete variable in terms of Feigin's Lie algebra gl(N), where N is any complex number. One can similarly interpret Chebyshev's and Hahn's q-polynomials and introduce orthogonal polynomials corresponding to Lie superlagebras. We also describe the real forms of gl(N), quasi-finite modules over gl(N), and conditions for unitarity of the quasi-finite modules. Analogs of tensors over gl(N) are also introduced.
A weight basis for each finite-dimensional irreducible representation of the orthogonal Lie algebra o(2n) is constructed. The basis vectors are parametrized by the D-type Gelfand--Tsetlin patterns. Explicit formulas for the matrix elements of generators of o(2n) in this basis are given. The construction is based on the representation theory of the Yangians and extends our previous results for the symplectic Lie algebras.
On the action of the cactus group on the set of Gelfand-Tsetlin patterns for orthogonal Lie algebras
本文利用Howe对偶性定义仙人掌群在正交李代数Gelfand-Tsetlin模式集合上的自然作用，通过建立主基与Gelfand-Tsetlin基的双射实现。
Geometric properties of special orthogonal representations associated to exceptional Lie superalgebras
本文基于特征不为2或3的域上的八元数代数，证明了导子代数与特殊正交李代数的基础表示和旋量表示为特殊正交表示，揭示了其与二元三次式相关的几何性质及与Fano平面的联系，并给出了例外李超代数的构造。
An orthogonal realization of representations of the Temperley-Lieb algebra
本文通过在 $\mathbf{U}(\mathfrak{sl}_2)$ 的张量模 $V^{\otimes n}$ 中构造两两正交的极大向量，为 Temperley–Lieb 代数 $\text{TL}_n$ 的单模建立正交基，并阐明其与标准胞腔基的关系。
The conformal transformations with respect to the metric defining the orthogonal Lie algebra o(n) give rise to a one-parameter (c) family of inhomogeneous first-order differential operator representations of the orthogonal Lie algebra o(n+2). Letting these operators act on the space of exponential-polynomial functions that depend on a parametric vector a, we prove that the space forms an irreducible o(n+2)-module for any constant c if the vector a is not on a certain hypersurface. By partially swapping differential operators and multiplication operators, we obtain more general differential ope…
Polynomials in this paper are defined starting from a compact semisimple Lie group. A known classification of maximal, semisimple subgroups of simple Lie groups is used to select the cases to be considered here. A general method is presented and all the cases of rank not greater then 3 are explicitly studied. We derive the polynomials of simple Lie groups B_3 and C_3 as they are not available elsewhere. The results point to far reaching Lie theoretical connections to the theory of multivariable orthogonal polynomials.
Let $\mathcal{N}_{\mathfrak{g}^*}$ be the variety of nilpotent elements in the dual of the Lie algebra of a reductive algebraic group over an algebraically closed field. In \cite{Lu2} Lusztig proposes a definition of a partition of $\mathcal{N}_{\mathfrak{g}^*}$ into smooth locally closed subvarieties (which are indexed by the unipotent classes in the corresponding group over complex numbers) and gives explicit results in types $A$, $C$ and $D$. We discuss type $B$ in this note.
The aim of our paper is to construct pseudo $H$-type algebras from the covering free nilpotent two-step Lie algebra as the quotient algebra by an ideal. We propose an explicit algorithm of construction of such an ideal by making use of a non-degenerate scalar product. Moreover, as a bypass result, we recover the existence of a rational structure on pseudo $H$-type algebras, which implies the existence of lattices on the corresponding pseudo $H$-type Lie groups. Our approach substantially uses combinatorics and reveals the interplay of pseudo $H$-type algebras with combinatorial and orthogonal …
We generalize Baranov and Shlaka's results about bar-minimal Jordan-Lie and regular inner ideals of finite dimensional associative algebras. Let A be a finite dimensional 1-perfect associative algebras A over an algebraically closed field \mathbb{F} of characteristic p\ge0 and let A' be a subalgebra of A. We prove that for any bar-minimal Jordan-Lie inner ideal B' a A', there is a bar-minimal Jordan-Lie inner ideal of A that contains B' and if B' is regular, then is a regular inner ideal of A that contains B'. We also prove that for any strict orthogonal pair (e',f') in A', there is a strict o…
Let $G$ be a semisimple Lie group and let $\g =\n_- +\hh +\n$ be a triangular decomposition of $\g= \hbox{Lie}\,G$. Let $\b =\hh +\n$ and let $H,N,B$ be Lie subgroups of $G$ corresponding respectively to $\hh,\n$ and $\b$. We may identify $\n_-$ with the dual space to $\n$. The coadjoint action of $N$ on $\n_-$ extends to an action of $B$ on $\n_-$. There exists a unique nonempty Zariski open orbit $X$ of $B$ on $\n_-$. Any $N$-orbit in $X$ is a maximal coadjoint orbit of $N$ in $\n_-$. The cascade of orthogonal roots defines a cross-section $\r_-^{\times}$ of the set of such orbits leading to…
In this lecutre note, we consider infinite dimensional Lie algebras of generalized Jacobi matrices $\mathfrak{g}J(k)$ and $\mathfrak{gl}_\infty(k)$, which are important in soliton theory, and their orthogonal and symplectic subalgebras. In particular, we construct the homology ring of the Lie algebra $\mathfrak{g}J(k)$ and of the orthogonal and symplectic subalgebras.
Superelliptic Affine Lie algebras and orthogonal polynomials
本文通过超椭圆李代数的泛中心扩张构造两类正交多项式，证明其生成函数满足可用超椭圆积分显式求解的四阶线性偏微分方程，从而给出不完全可积的四阶线性PDE实例。
Classical invariants for representations of one Lie group can often be related to invariants of some other Lie group. Physics suggests that the right objects to consider for these questions are certain refinements of classical invariants known as conformal blocks. Conformal blocks appear in algebraic geometry as spaces of global sections of line bundles on the moduli stack of parabolic bundles on a smooth curve. Rank-level duality connects a conformal block associated to one Lie algebra to a conformal block for a different Lie algebra. In this paper we prove a rank-level duality for type so(2r…
Rankin-Cohen brackets for orthogonal Lie algebras and bilinear conformally invariant differential operators
本文利用代数傅里叶变换的李理论方法，分类了正交李代数情形下标量广义Verma模对角分支规则中的奇异向量，并应用于分类共形球面上的双线性共形不变微分算子。
We show that the centraliser of the maximal compact subgroup of the real orthogonal or symplectic groups acting on tensors of their standard representation are isomorphic to cyclotomic Brauer algebras. We also show that for the symplectic group this cyclotomic Brauer algebra splits into summands of walled Brauer algebras.
We prove the equivalence of two presentations of the Yangian $Y(\mathfrak{g})$ of a simple Lie algebra $\mathfrak{g}$ and we also show the equivalence with a third presentation when $\mathfrak{g}$ is either an orthogonal or a symplectic Lie algebra. As an application, we obtain an explicit correspondence between two versions of the classification theorem of finite-dimensional irreducible modules for orthogonal and symplectic Yangians.
Recursive algebraic construction of two infinite families of polynomials in $n$ variables is proposed as a uniform method applicable to every semisimple Lie group of rank $n$. Its result recognizes Chebyshev polynomials of the first and second kind as the special case of the simple group of type $A_1$. The obtained not Laurent-type polynomials are proved to be equivalent to the partial cases of the Macdonald symmetric polynomials. Basic relation between the polynomials and their properties follow from the corresponding properties of the orbit functions, namely the orthogonality and discretizat…
We investigate a certain class of solvable metric Lie algebras. For this purpose a theory of twofold extensions associated to an orthogonal representation of an abelian Lie algebra is developed. Among other things, we obtain a classification scheme for indecomposable metric Lie algebras with maximal isotropic centre and the classification of metric Lie algebras of index 2.
We define a class of algebras which are distinguished by a PBW property and an orthogonality condition, and which we call Hopf-Hecke algebras, since they generalize the Drinfeld Hecke algebras defined by Drinfeld. In the course of studying the orthogonality condition and in analogy to the orthogonal group we show the existence of a pin cover for cocommutative Hopf algebras over $\mathbb{C}$ with an orthogonal module or, more generally, pointed cocommutative Hopf algebras over a field of characteristic $0$ with an orthogonal module. Following the suggestion of Dan Barbasch and Siddhartha Sahi, …
We prove a structure theorem for Lie n-algebras possessing an invariant inner product. We define the notion of a double extension of a metric Lie n-algebra by another Lie n-algebra and prove that all metric Lie n-algebras are obtained from the simple and one-dimensional ones by iterating the operations of orthogonal direct sum and double extension.
Deformed orthogonal and pseudo-orthogonal Lie algebras are constructed which differ from deformations of Lie algebras in terms of Cartan subalgebra and root vectors and which make it possible to construct representations by operators acting according to Gel'fand--Tsetlin-type formulas. Unitary representations of the q-deformed algebras U_q(so_{n,1}) are found.
We describe a family of polynomials discovered via a particular recursion relation, which have connections to Chebyshev polynomials of the first and the second kind, and the polynomial version of Pell's equation. Many of their properties are listed in Section 3. We show that these families of polynomials in the variable $t$ satisfy certain second order linear differential equations that may be of interest to mathematicians in conformal field theory and number theory. We also prove that these families of polynomials in the setting of Date-Jimbo-Kashiwara-Miwa algebras when multiplied by a suita…
The conformal transformations with respect to the metric defining $o(n,\mbb{C})$ give rise to a nonhomogeneous polynomial representation of $o(n+2,\mbb{C})$. Using Shen's technique of mixed product, we generalize the above representation to a non-homogenous representation of $o(n+2,\mbb{C})$ on the tensor space of any finite-dimensional irreducible $o(n,\mbb{C})$-module with the polynomial space, where a hidden central transformation is involved. Moreover, we find a condition on the constant value taken by the central transformation such that the generalized conformal representation is irreduc…
On the Divisibility of Degrees of Representations of Lie Algebras
本文研究约化李代数不可约表示次数不被给定正整数整除的自然密度，证明在一般线性李代数情形下该密度随维数增大指数衰减至零，并将结果推广到单李代数、李群及自对偶与正交表示。
FFT and orthogonal discrete transform on weight lattices of semi-simple Lie groups
本文提出两种基于代数几何的傅里叶变换快速算法，将Gauss-Jacobi正交变换过程推广至多变量情形，并推导了基于多变量Chebyshev多项式的权格快速变换方案。
Orthogonal Oscillator Representations, Laplace Equations and Intersections of Determinantal Varieties
该研究证明了由正交振子表示经傅里叶变换得到的特殊线性李代数无限维不可约表示的相伴簇是显式行列式簇的交，建立了表示论、偏微分方程与代数几何间的显式关联。
In 1971 Griffiths used a generating function to define polynomials in d variables orthogonal with respect to the multinomial distribution. The polynomials possess a duality between the discrete variables and the degree indices. In 2004 Mizukawa and Tanaka related these polynomials to character algebras and the Gelfand hypergeometric series. Using this approach they clarified the duality and obtained a new proof of the orthogonality. In the present paper, we interpret these polynomials within the context of the Lie algebra sl_{d+1}. Our approach yields yet another proof of the orthogonality. It…
We exhibit explicit orthogonal decompositions of every multidimensional restricted root space of a real semi-simple Lie algebra. We then show a link between this result and a radiality property of smooth functions on G-homogeneous spaces when G is a real semi-simple Lie group.
We construct generators of the center of the universal enveloping algebra of the complex orthogonal Lie algebra realized as the alternative matrices of size $n$. These elements are constructed in accordance with the Iwasawa decomposition of the real rank one indefinite orthogonal Lie algebra. We also discuss the Iwasawa decomposition of the Pfaffian.
The present paper contains a systematic study of the structure of metric Lie algebras, i.e., finite-dimensional real Lie algebras equipped with a non-degenerate invariant symmetric bilinear form. We show that any metric Lie algebra without simple ideals has the structure of a so called balanced quadratic extension of an auxiliary Lie algebra l by an orthogonal l-module A in a canonical way. Identifying equivalence classes of quadratic extensions of l by A with a certain cohomology set H^2_Q(l,A) we obtain a classification scheme for general metric Lie algebras and a complete classification of …
The finite dimensional simple modular Lie algebras with Cartan matrix cannot be deformed if the characteristic p of the ground field is equal to 0 or greater than 3. If p=3, the orthogonal Lie algebra o(5)is one of the two simple modular Lie algebras with Cartan matrix that have deformations (the Brown algebras br(2; a) are among these 10-dimensional deforms and hence are not counted separately); the 29-dimensional Brown algebra br(3) is the only other simple Lie algebra with Cartan matrix that has deformations. Kostrikin and Kuznetsov described the orbits (isomorphism classes) under the actio…
This book is mainly an exposition of the author's works and his joint works with his former students on explicit representations of finite-dimensional simple Lie algebras, related partial differential equations, linear orthogonal algebraic codes, combinatorics and algebraic varieties. Various oscillator generalizations of the classical representation theorem on harmonic polynomials are presented. New functors from the representation category of a simple Lie algebra to that of another simple Lie algebra are given. Partial differential equations play key roles in solving certain representation p…
Unitary representations of the Temperley-Lieb algebra $TL_N(Q)$ on the tensor space $({\mathbb C^n})^{\otimes N}$ are considered. Two criteria are given for determining when an orthogonal projection matrix $P$ of a rank $r$ gives rise to such a representation. The first of them is the equality of traces of certain matrices and the second is the unitary condition for a certain partitioned matrix. Some estimates are obtained on the lower bound of $Q$ for a given dimension $n$ and rank $r$. It is also shown that if $4r>n^2$, then $Q$ can take only a discrete set of values determined by the value …
We give the number of nilpotent orbits in the Lie algebras of orthogonal groups under the adjoint action of the groups over $\tF_{2^n}$. Let $G$ be an adjoint algebraic group of type $B,C$ or $D$ defined over an algebraically closed field of characteristic 2. We construct the Springer correspondence for the nilpotent variety in the Lie algebra of $G$.
We define a pair of symplectic Dirac operators $(D^+,D^-)$ in an algebraic setting motivated by the analogy with the algebraic orthogonal Dirac operators in representation theory. We work in the settings of $\mathbb Z/2$-graded quadratic Lie algebras $\mathfrak g=\mathfrak k+\mathfrak p$ and of graded affine Hecke algebras $\mathbb H$.
Let $G$ be a real compact Lie group, such that $G=G^0\rtimes C_2$, with $G^0$ simple. Here $G^0$ is the connected component of $G$ containing the identity and $C_2$ is the cyclic group of order $2$. We give a criterion for whether an orthogonal representation $π: G \to \mathrm{O}(V)$ lifts to $\mathrm{Pin}(V)$ in terms of the highest weights of $π$. We also calculate the first and second Stiefel-Whitney classes of the representations of the Orthogonal groups.
In this note we present the Stiefel-Whitney classes (SWCs) for orthogonal representations of several finite groups of Lie type, namely for $G=\text{SL}(2,q),$ $\text{SL}(3,q),$ $\text{Sp}(4,q)$, and $\text{Sp}(6,q)$, with $q$ odd. We also describe the SWCs for $G=\text{SL}(2,q)$ when $q$ is even.
Recently Alberto Elduque listed all simple and graded modulo 2 finite dimensional Lie algebras and superalgebras whose odd component is the spinor representation of the orthogonal Lie algebra equal to the even component, and discovered one exceptional such Lie superalgebra in characteristic 5. For this Lie superalgebra all inequivalent Cartan matrices (in other words, inequivalent systems of simple roots) are listed together with defining relations between analogs of its Chevalley generators.
A well-known result of Kostant gives a description of the G-module structure for the exterior algebra of Lie algebra $\frak g$. We give a generalization of this result for the isotropy representations of symmetric spaces. If $\frak g={\frak g}_0+{\frak g_1}$ is a Z_2-grading of a simple Lie algebra, we explicitly describe a ${\frak g}_0$-module $Spin_0({\frak g}_1)$ such that the exterior algebra of ${\frak g}_1$ is the tensor square of this module times some power of 2. Although $Spin_0({\frak g}_1)$ is usually reducible, we show that a Casimir element for ${\frak g}_0$ always acts scalarly o…
Full Conformal Oscillator Representations of Orthogonal Lie Algebras and Combinatorial Identities
本文通过连续使用 Zhao 等人构造的函子，给出正交李代数最高权表示的一阶微分算子实现，并用于研究张量分解、Clebsch-Gordan 系数及 KZ 方程，同时得到 Steinberg 模维数的新组合恒等式。
Let $G$ be an adjoint algebraic group of type $B$, $C$, or $D$ over an algebraically closed field of characteristic 2. We construct a Springer correspondence for the Lie algebra of $G$. In particular, for orthogonal Lie algebras in characteristic 2, the structure of component groups of nilpotent centralizers is determined and the number of nilpotent orbits over finite fields is obtained.
Let $G$ be a complex simply-connected semisimple Lie group and let $\g=\hbox{\rm Lie}\,G$. Let $\g = \n_- +\hh + \n$ be a triangular decomposition of $\g$. One readily has that $\hbox{\rm Cent}\,U(\n)$ is isomorphic to the ring $S(\n)^{\n}$ of symmetric invariants. Using the cascade ${\cal B}$ of strongly orthogonal roots, some time ago we proved (see [K]) that $S(\n)^{\n}$ is a polynomial ring $\Bbb C[ξ_1,...,ξ_m]$ where $m$ is the cardinality of ${\cal B}$. The authors in [LW] introduce a very nice representation-theoretic method for the construction of certain elements in $S(\n)^{\n}$. A ke…
Generalizing Feingold-Frenkel's construction we use Weyl bosonic fields to construct toroidal Lie algebras of types $A_n, B_n$, $C_n$ and $D_n$ of level $-1, -2, -1/2$ and -2 respectively. In particular, our construction also gives new bosonic construction for the orthogonal Lie algebras in the cases of affine Lie algebras.
It is shown that any generalized Kac-Moody Lie algebra g that has no mutually orthogonal imaginary simple roots can be written as the vector space direct sum of a Kac-Moody subalgebra and subalgebras isomorphic to free Lie algebras over certain modules for the Kac-Moody subalgebra. Also included is a detailed discussion of Borcherds' construction of the Monster Lie algebra from a vertex algebra and an elementary proof of Borcherds' theorem relating Lie algebras with `an almost positive definite bilinear form' to generalized Kac-Moody algebras. (Preprint version 1996)
Let $G$ be a simply connected algebraic group of type $B,C$ or $D$ over an algebraically closed field of characteristic 2. We construct a Springer correspondence for the dual vector space of the Lie algebra of $G$. In particular, we classify the nilpotent orbits in the duals of symplectic and orthogonal Lie algebras over algebraically closed or finite fields of characteristic 2.
Algebraic interpretation of discrete families of matrix valued orthogonal polynomials
本文通过李代数表示构造矩阵值正交多项式，探讨su(2)、su(1,1)及q-变形so_q(3)代数的矩阵类比，生成Krawtchouk、Meixner和离散Chebyshev多项式。
The Lie algebra generated by $m\ $ $p$-dimensional Grassmannian Dirac operators and $m\ $ $p$-dimensional vector variables is identified as the orthogonal Lie algebra $\mathfrak{so}(2m+1)$. In this paper, we study the space $\mathcal{P}$ of polynomials in these vector variables, corresponding to an irreducible $\mathfrak{so}(2m+1)$ representation. In particular, a basis of $\mathcal{P}$ is constructed, using various Young tableaux techniques. Throughout the paper, we also indicate the relation to the theory of parafermions.
We describe families of polynomials arising in the study of the universal central extensions of Lie algebras introduced by Date, Jimbo, Kashiwara, and Miwa in their work on the Landau-Lifshitz equations. Two of the families of polynomials we show satisfy certain forth order linear differential equations, are orthogonal and are not of classical type.
Reduction theory for stably graded Lie algebras
本文为稳定分次李代数的Vinberg表示定义约化协变，并给出奇分裂特殊正交群的LLL算法类比，用于约化与奇超椭圆曲线2-下降相关的向量系数。
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