Finite-dimensional Irreducible Representations Research Articles

Ghost Distributions on Supersymmetric Spaces I: Koszul Induced Superspaces, Branching, and the Full Ghost Centre

AbstractGiven a Lie superalgebra $$\mathfrak {g}$$ g , Gorelik defined the anticentre $$\mathcal {A}$$ A of its enveloping algebra, which consists of certain elements that square to the center. We seek to generalize and enrich the anticentre to the context of supersymmetric pairs $$(\mathfrak {g},\mathfrak {k})$$ ( g , k ) , or more generally supersymmetric spaces G/K. We define certain invariant distributions on G/K, which we call ghost distributions, and which in some sense are induced from invariant distributions on $$G_0/K_0$$ G 0 / K 0 . Ghost distributions, and in particular their Harish-Chandra polynomials, give information about branching from G to a symmetric subgroup $$K'$$ K ′ which is related (and sometimes conjugate) to K. We discuss the case of $$G\times G/G$$ G × G / G for an arbitrary quasireductive supergroup G, where our results prove the existence of a polynomial which determines projectivity of irreducible G-modules. Finally, a generalization of Gorelik’s ghost centre is defined which we call the full ghost centre, $$\mathcal {Z}_{full}$$ Z full . For type I basic Lie superalgebras $$\mathfrak {g}$$ g we fully describe $$\mathcal {Z}_{full}$$ Z full , and prove that if $$\mathfrak {g}$$ g contains an internal grading operator, $$\mathcal {Z}_{full}$$ Z full consists exactly of those elements in $$\mathcal {U}\mathfrak {g}$$ U g acting by $$\mathbb {Z}$$ Z -graded constants on every finite-dimensional irreducible representation.

Commutative avatars of representations of semisimple Lie groups

Here we announce the construction and properties of a big commutative subalgebra of the Kirillov algebra attached to a finite dimensional irreducible representation of a complex semisimple Lie group. They are commutative finite flat algebras over the cohomology of the classifying space of the group. They are isomorphic with the equivariant intersection cohomology of affine Schubert varieties, endowing the latter with a new ring structure. Study of the finer aspects of the structure of the big algebras will also furnish the stalks of the intersection cohomology with ring structure, thus ringifying Lusztig's q-weight multiplicity polynomials i.e., certain affine Kazhdan-Lusztig polynomials.

So(3) ⊂ su(3) revisited

This paper reproduces the result of Elliot, namely that the irreducible finite dimensional representation of the Lie algebra su(3) of highest weight (m, n) is decomposed according to the embedding so(3) ⊂ su(3). First, a realisation (a representation in terms of vector fields) of the Lie algebra su(3) is constructed on a space of polynomials of three variables. The special polynomial basis of the representation space is given. In this basis, we find the highest weight vectors of the representation of the Lie subalgebra so(3) and in this way the representation space is decomposed to the direct sum of invariant subspaces. The process is illustrated by the example of the decomposition of the representation of highest weight (2, 2). As an additional result, the generating function of the decomposition is given.

Representations of the super-Yangian of type B(n,m)

We are concerned with finite-dimensional irreducible representations of the Yangians associated with the orthosymplectic Lie superalgebras osp2n+1|2m. Every such representation is highest weight and we use embedding theorems and odd reflections of Yangian type to derive necessary conditions for an irreducible highest weight representation to be finite-dimensional. We conjecture that these conditions are also sufficient. We prove the conjecture in the case n=1 and arbitrary m⩾1.

Cartan–Helgason theorem for quaternionic symmetric and twistor spaces

Let (g,k) be a complex quaternionic symmetric pair with k having an ideal sl(2,ℂ), k=sl(2,ℂ)+mc. Consider the representation Sm(ℂ2)=ℂm+1 of k via the projection k→sl(2,ℂ) onto the ideal sl(2,ℂ). We study the finite dimensional irreducible representations V(λ) of g which contain Sm(ℂ2) under k⊆g. We give a characterization of all such representations V(λ) and find the corresponding multiplicity, the dimension of Hom(V(λ)|k,Sm(ℂ2)). We consider also the branching problem of V(λ) under l=u(1)ℂ+mc⊂k and find the multiplicities. Geometrically the Lie subalgebra l⊂k defines a twistor space over the compact symmetric space of the compact real form Gu of Gℂ, Lie(Gℂ)=g, and our results give the decomposition for the L2-spaces of sections of certain vector bundles over the symmetric space and line bundles over the twistor space. This generalizes Cartan–Helgason’s theorem for symmetric spaces (g,k) and Schlichtkrull’s theorem for Hermitian symmetric spaces where one-dimensional representations of k are considered.

The R-matrix presentation for the rational form of a quantized enveloping algebra

Let Uq(g) denote the rational form of the quantized enveloping algebra associated to a complex simple Lie algebra g. Let λ be a nonzero dominant integral weight of g, and let V be the corresponding type 1 finite-dimensional irreducible representation of Uq(g). Starting from this data, the R-matrix formalism for quantum groups outputs a Hopf algebra URλ(g) defined in terms of a pair of generating matrices satisfying well-known quadratic matrix relations. In this paper, we prove that this Hopf algebra admits a Chevalley–Serre type presentation which can be recovered from that of Uq(g) by adding a single invertible quantum Cartan element. We simultaneously establish that URλ(g) can be realized as a Hopf subalgebra of the tensor product of the space of Laurent polynomials in a single variable with the quantized enveloping algebra associated to the lattice generated by the weights of V. The proofs of these results are based on a detailed analysis of the homogeneous components of the matrix equations and generating matrices defining URλ(g), with respect to a natural grading by the root lattice of g compatible with the weight space decomposition of End(V).

Twisted conjugacy in residually finite groups of finite Prüfer rank

Abstract Suppose 𝐺 is a residually finite group of finite upper rank admitting an automorphism 𝜑 with finite Reidemeister number R ⁢ ( φ ) R(\varphi) (the number of 𝜑-twisted conjugacy classes). We prove that such a 𝐺 is soluble-by-finite (in other words, any residually finite group of finite upper rank that is not soluble-by-finite has the R ∞ R_{\infty} property). This reduction is the first step in the proof of the second main theorem of the paper: suppose 𝐺 is a residually finite group of finite Prüfer rank and 𝜑 is its automorphism. Then R ⁢ ( φ ) R(\varphi) (if it is finite) is equal to the number of equivalence classes of finite-dimensional irreducible unitary representations of 𝐺, which are fixed points of the dual map φ ̂ : [ ρ ] ↦ [ ρ ∘ φ ] \hat{\varphi}\colon[\rho]\mapsto[\rho\circ\varphi] (i.e. we prove the TBFT𝑓, the finite version of the conjecture about the twisted Burnside–Frobenius theorem, for such groups).

Finite dimensional irreducible representations of Lie superalgebra D (2, 1; α)

This paper focuses on the finite dimensional irreducible representations of Lie superalgebra D(2, 1; α). We explicitly construct the finite dimensional representations of the superalgebra D(2, 1; α) by using the shift operator and differential operator representations. Unlike ordinary Lie algebra representation, there are typical and atypical representations for most superalgebras. Therefore, its typical and atypical representation conditions are also given. Our results are expected to be useful for the construction of primary fields of the corresponding current superalgebra of D(2, 1; α).

Models of representations for classical series of Lie algebras

By a model of representations of a Lie algebra we mean a representation which is a direct sum of all irreducible finite-dimensional representations taken with multiplicity $1$. An explicit construction of a model of representations for all classical series of simple Lie algebras is given. This construction is generic for all classical series of Lie algebras. The space of the model is constructed as the space of polynomial solutions of a system of partial differential equations, where the equations are constructed form relations between minors of matrices taken from the corresponding Lie group. This system admits a simplification very close to the GKZ system, which is satisfied by $A$-hypergeometric functions.

Matrix elements of SO(3) in sl3 representations as bispectral multivariate functions

We compute the matrix elements of SO(3) in any finite-dimensional irreducible representation of sl3. They are expressed in terms of a double sum of products of Krawtchouk and Racah polynomials which generalize the Griffiths–Krawtchouk polynomials. Their recurrence and difference relations are obtained as byproducts of our construction. The proof is based on the decomposition of a general three-dimensional rotation in terms of elementary planar rotations and a transition between two embeddings of sl2 in sl3. The former is related to monovariate Krawtchouk polynomials and the latter, to monovariate Racah polynomials. The appearance of Racah polynomials in this context is algebraically explained by showing that the two sl2 Casimir elements related to the two embeddings of sl2 in sl3 obey the Racah algebra relations. We also show that these two elements generate the centralizer in U(sl3) of the Cartan subalgebra and its complete algebraic description is given.

Algebraic approach and exact solutions of superintegrable systems in 2D Darboux spaces

Abstract Superintegrable systems in two-dimensional (2D) Darboux spaces were classified and it was found that there exist 12 distinct classes of superintegrable systems with quadratic integrals of motion (and quadratic symmetry algebras generated by the integrals) in the Darboux spaces. In this paper, we obtain exact solutions via purely algebraic means for the energies of all the 12 existing classes of superintegrable systems in four different 2D Darboux spaces. This is achieved by constructing the deformed oscillator realization and finite-dimensional irreducible representation of the underlying quadratic symmetry algebra generated by quadratic integrals respectively for each of the 12 superintegrable systems. We also introduce generic cubic and quintic algebras, generated respectively by linear and quadratic integrals and linear and cubic integrals, and obtain their Casimir operators and deformed oscillator realizations. As examples of applications, we present three classes of new superintegrable systems with cubic symmetry algebras in 2D Darboux spaces.

Minimal boundaries for operator algebras

We study boundaries for unital operator algebras. These are sets of irreducible ∗ * -representations that completely capture the spatial norm attainment for a given subalgebra. Classically, the Choquet boundary is the minimal boundary of a function algebra and it coincides with the collection of peak points. We investigate the question of minimality for the non-commutative counterpart of the Choquet boundary and show that minimality is equivalent to what we call the Bishop property. Not every operator algebra has the Bishop property, but we exhibit classes of examples that do. Throughout our analysis, we exploit various non-commutative notions of peak points for an operator algebra. When specialized to the setting of C ∗ \mathrm {C}^* -algebras, our techniques allow us to provide a new proof of a recent characterization of those C ∗ \mathrm {C}^* -algebras admitting only finite-dimensional irreducible representations.

Affinization of q-oscillator representations of $$U_q(\mathfrak {gl}_n)$$

We introduce a category $\widehat{\mathcal{O}}_{\rm osc}$ of $q$-oscillator representations of the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_n)$. We show that $\widehat{\mathcal{O}}_{\rm osc}$ has a family of irreducible representations, which naturally corresponds to finite-dimensional irreducible representations of quantum affine algebra of untwisted affine type $A$. It is done by constructing a category of $q$-oscillator representations of the quantum affine superalgebra of type $A$, which interpolates these two family of irreducible representations. The category $\widehat{\mathcal{O}}_{\rm osc}$ can be viewed as a quantum affine analogue of the semisimple tensor category generated by unitarizable highest weight representations of $\mathfrak{gl}_{u+v}$ ($n=u+v$) appearing in the $(\mathfrak{gl}_{u+v},\mathfrak{gl}_\ell)$-duality on a bosonic Fock space.

Explicit constructions of some infinite families of finite-dimensional irreducible representations of the type $${\textsf {E}}_{6}$$ and $${\textsf {E}}_{7}$$ simple Lie algebras

Explicit constructions of some infinite families of finite-dimensional irreducible representations of the type $${\textsf {E}}_{6}$$ and $${\textsf {E}}_{7}$$ simple Lie algebras

Ghost center and representations of the diagonal reduction algebra of osp(1|2)

Reduction algebras are known by many names in the literature, including step algebras, Mickelsson algebras, Zhelobenko algebras, and transvector algebras, to name a few. These algebras, realized by raising and lowering operators, allow for the calculation of Clebsch-Gordan coefficients, branching rules, and intertwining operators; and have connections to extremal equations and dynamical R-matrices in integrable face models. In this paper we continue the study of the diagonal reduction superalgebra A of the orthosymplectic Lie superalgebra osp(1|2). We construct a Harish-Chandra homomorphism, Verma modules, and study the Shapovalov form on each Verma module. Using these results, we prove that the ghost center (center plus anti-center) of A is generated by two central elements and one anti-central element (analogous to the Scasimir due to Leśniewski for osp(1|2)). As another application, we classify all finite-dimensional irreducible representations of A. Lastly, we calculate an infinite-dimensional tensor product decomposition explicitly.

Gruson–Serganova character formulas and the Duflo–Serganova cohomology functor

Abstract We establish an explicit formula for the character of an irreducible finite-dimensional representation of g ⁢ l ⁢ ( m | n ) \mathfrak{gl}(m|n) . The formula is a finite sum with integer coefficients in terms of a basis E μ \mathcal{E}_{\mu} (Euler characters) of the character ring. We prove a simple formula for the behavior of the “superversion” of E μ \mathcal{E}_{\mu} in the g ⁢ l ⁢ ( m | n ) \mathfrak{gl}(m|n) and o ⁢ s ⁢ p ⁢ ( m | 2 ⁢ n ) \mathfrak{osp}(m|2n) -case under the map ds \mathrm{ds} on the supercharacter ring induced by the Duflo–Serganova cohomology functor DS \mathrm{DS} . As an application, we get combinatorial formulas for superdimensions, dimensions and g 0 \mathfrak{g}_{0} -decompositions for g ⁢ l ⁢ ( m | n ) \mathfrak{gl}(m|n) and o ⁢ s ⁢ p ⁢ ( m | 2 ⁢ n ) \mathfrak{osp}(m|2n) .

Transfer Matrices of Rational Spin Chains via Novel BGG-Type Resolutions

We obtain BGG-type formulas for transfer matrices of irreducible finite-dimensional representations of the classical Lie algebras $\mathfrak{g}$, whose highest weight is a multiple of a fundamental one and which can be lifted to the representations over the Yangian $Y(\mathfrak{g})$. These transfer matrices are expressed in terms of transfer matrices of certain infinite-dimensional highest weight representations (such as parabolic Verma modules and their generalizations) in the auxiliary space. We further factorise the corresponding infinite-dimensional transfer matrices into the products of two Baxter $Q$-operators, arising from our previous study (arXiv:2001.04929, arXiv:2104.14518) of the degenerate Lax matrices. Our approach is crucially based on the new BGG-type resolutions of the finite-dimensional $\mathfrak{g}$-modules, which naturally arise geometrically as the restricted duals of the Cousin complexes of relative local cohomology groups of ample line bundles on the partial flag variety $G/P$ stratified by $B_{-}$-orbits.

Coloured $$\mathfrak {sl}_r$$ invariants of torus knots and characters of $${\mathcal {W}}_r$$ algebras

Let $$p<p'$$ be a pair of coprime positive integers. In this note, generalizing Morton’s work in the case of $$\mathfrak {sl}_2$$ , we give a formula for the $$\mathfrak {sl}_r$$ Jones invariants of torus knots $$T (p,p')$$ coloured with the finite-dimensional irreducible representations $$L_r(n\Lambda _1)$$ . When $$r \le p$$ , we show that appropriate limits of the shifted (non-normalized, framing dependent) invariants calculated along $$L_r(nr\Lambda _1)$$ are essentially the characters of certain minimal model principal $${\mathcal {W}}$$ algebras of type $${{\,\textrm{A}\,}}$$ , namely, $${\mathcal {W}}_r(p,p')$$ , up to some factors independent of p and $$p'$$ but depending on r. In particular, these limits are essentially modular. We expect these limits to be the 0-tails of corresponding sequences of invariants. At the end, we formulate a conjecture on limits for $$p<r$$ .

A functional realization of the Gelfand-Tsetlin base

A realization of a finite dimensional irreducible representation of the Lie algebra $\mathfrak{gl}_n$ in the space of functions on the group $\mathrm{GL}_n$ is considered. It is proved that functions corresponding to Gelfand-Tsetlin diagrams are linear combinations of some new functions of hypergeometric type which are closely related to $A$-hypergeometric functions. These new functions are solution of a system of partial differential equations which follows from the Gelfand-Kapranov-Zelevinsky by an "antisymmetrization". The coefficients in the constructed linear combination are hypergeometric constants, that is, they are values of some hypergeometric functions when instead of all arguments ones are substituted.

A functional realization of the Gelfand-Tsetlin base

A realization of a finite dimensional irreducible representation of the Lie algebra $\mathfrak{gl}_n$ in the space of functions on the group $\mathrm{GL}_n$ is considered. It is proved that functions corresponding to Gelfand-Tsetlin diagrams are linear combinations of some new functions of hypergeometric type which are closely related to $A$-hypergeometric functions. These new functions are solution of a system of partial differential equations which follows from the Gelfand-Kapranov-Zelevinsky by an "antisymmetrization". The coefficients in the constructed linear combination are hypergeometric constants, that is, they are values of some hypergeometric functions when instead of all arguments ones are substituted. Bibliography: 16 titles.