Classical Lie Algebras Research Articles

Quiver theories and Hilbert series of classical Slodowy intersections

We build on previous studies of the Higgs and Coulomb branches of SUSY quiver theories having 8 supercharges, including 3dN=4, and Classical gauge groups. The vacuum moduli spaces of many such theories can be parameterised by pairs of nilpotent orbits of Classical Lie algebras; they are transverse to one orbit and intersect the closure of the second. We refer to these transverse spaces as “Slodowy intersections”. They embrace reduced single instanton moduli spaces, nilpotent orbits, Kraft-Procesi transitions and Slodowy slices, as well as other types. We show how quiver subtractions, between multi-flavoured unitary or ortho-symplectic quivers, can be used to find a complete set of Higgs branch constructions for the Slodowy intersections of any Classical group. We discuss the relationships between the Higgs and Coulomb branches of these quivers and Tσρ theories in the context of 3d mirror symmetry, including problematic aspects of Coulomb branch constructions from ortho-symplectic quivers. We review Coulomb and Higgs branch constructions for a subset of Slodowy intersections from multi-flavoured Dynkin diagram quivers. We tabulate Hilbert series and Highest Weight Generating functions for Slodowy intersections of Classical algebras up to rank 4. The results are confirmed by direct calculation of Hilbert series from a localisation formula for normal Slodowy intersections that is related to the Hall Littlewood polynomials.

Quantum-classical duality for Gaudin magnets with boundary

We establish a remarkable relationship between the quantum Gaudin models with boundary and the classical many-body integrable systems of Calogero-Moser type associated with the root systems of classical Lie algebras (B, C and D). We show that under identification of spectra of the Gaudin Hamiltonians HjG with particles velocities q˙j of the classical model all integrals of motion of the latter take zero values. This is the generalization of the quantum-classical duality observed earlier for Gaudin models with periodic boundary conditions and Calogero-Moser models associated with the root system of the type A.

Hom-Lie structures on 3-dimensional skew symmetric algebras

We describe the dimension of the space of possible linear endomorphisms that turn skew-symmetric three-dimensional algebras into Hom-Lie algebras. We find a correspondence between the rank of a matrix containing the structure constants of the bilinear product and the dimension of the space of Hom-Lie structures. Examples from classical complex Lie algebras are given to demonstrate this correspondence.

Gelfand–Tsetlin Degeneration of Shift of Argument Subalgebras in Types B, C and D

The universal enveloping algebra of any semisimple Lie algebra $$\mathfrak {g}$$ contains a family of maximal commutative subalgebras, called shift of argument subalgebras, parametrized by regular Cartan elements of $$\mathfrak {g}$$. For $$\mathfrak {g}=\mathfrak {gl}_n$$ the Gelfand–Tsetlin commutative subalgebra in $$U(\mathfrak {g})$$ arises as some limit of subalgebras from this family. We study the analogous limit of shift of argument subalgebras for classical Lie algebras ($$\mathfrak {g}=\mathfrak {sp}_{2n}$$ or $$\mathfrak {so}_{n}$$). The limit subalgebra is described explicitly in terms of Bethe subalgebras in twisted Yangians $$Y^-(2)$$ and $$Y^+(2)$$, respectively. We index the eigenbasis of such limit subalgebra in any irreducible finite-dimensional representation of $$\mathfrak {g}$$ by Gelfand–Tsetlin patterns of the corresponding type, and conjecture that this indexing is, in appropriate sense, natural. According to Halacheva et al. (Crystals and monodromy of Bethe vectors. arXiv:1708.05105, 2017) such eigenbasis has a natural $$\mathfrak {g}$$-crystal structure. We conjecture that this crystal structure coincides with that on Gelfand–Tsetlin patterns defined by Littelmann in Cones, crystals, and patterns (Transform Groups 3(2):145–179, 1998).

Projective modules over classical Lie algebras of infinite rank in the parabolic category

We study the truncation functors and show the existence of projective cover with a finite Verma flag of each irreducible module in parabolic BGG category O over infinite rank Lie algebra of types a,b,c,d. Moreover, O is a Koszul category. As a consequence, the corresponding parabolic BGG category O‾ over infinite rank Lie superalgebra of types a,b,c,d through the super duality is also a Koszul category.

The Kostrikin Radical and Similar Radicals of Lie Algebras

The existing notion of the Kostrikin radical as a radical in the Kurosh–Amitsur sense on classes of Mal’tsev algebras over rings with 1/6 is not completely justified. More precisely, to the fullest extent it is true for classes of Lie algebras over fields of characteristic zero and, as shown in the given paper, classes of algebraic Lie algebras of degree not greater than n over rings with 1/n! at all n ≥ 1. Similar conclusions are obtained in the paper also for the Jordan, regular, and extremal radicals constructed analogously to the Kostrikin radical.

Tangent Lie Algebra of a Diffeomorphism Group and Application to Holonomy Theory

In this paper we introduce the notion of tangent space TG of a (not necessary smooth) subgroup G of the diffeomorphism group Diff(M) of a compact manifold M. We prove that TG is a Lie subalgebra of the Lie algebra of smooth vector fields on M. The construction can be generalized to subgroups of any (finite or infinite dimensional) Lie groups. The tangent Lie algebra TG introduced this way is a generalization of the classical Lie algebra in the smooth cases. As a working example, we discuss in detail the tangent structure of the holonomy group and fibered holonomy group of Finsler manifolds.

Duality Identities for Moduli Functions of Generalized Melvin Solutions Related to Classical Lie Algebras of Rank 4

We consider generalized Melvin-like solutions associated with nonexceptional Lie algebras of rank 4 (namely, A4, B4, C4, and D4) corresponding to certain internal symmetries of the solutions. The system under consideration is a static cylindrically symmetric gravitational configuration in D dimensions in presence of four Abelian 2-forms and four scalar fields. The solution is governed by four moduli functions Hs(z) (s=1,…,4) of squared radial coordinate z=ρ2 obeying four differential equations of the Toda chain type. These functions turn out to be polynomials of powers (n1,n2,n3,n4)=(4,6,6,4),(8,14,18,10),(7,12,15,16),(6,10,6,6) for Lie algebras A4, B4, C4, and D4, respectively. The asymptotic behaviour for the polynomials at large distances is governed by some integer-valued 4×4 matrix ν connected in a certain way with the inverse Cartan matrix of the Lie algebra and (in A4 case) the matrix representing a generator of the Z2-group of symmetry of the Dynkin diagram. The symmetry properties and duality identities for polynomials are obtained, as well as asymptotic relations for solutions at large distances. We also calculate 2-form flux integrals over 2-dimensional discs and corresponding Wilson loop factors over their boundaries.

On self-similar Lie algebras and virtual endomorphisms

We introduce the notion of virtual endomorphisms of Lie algebras and use it as an approach for constructing self-similarity of Lie algebras. This is done in particular for a class of metabelian Lie algebras having homological type $$FP_n$$ , which are Lie algebra analogues of lamplighter groups. We establish several criteria when the existence of virtual endomorphism implies a self-similar Lie structure. Furthermore, we prove that the classical Lie algebra $$sl_n(k)$$ , where char(k) does not divide n affords non-trivial faithful self-similarity.

A Lax type operator for quantum finite W-algebras

For a reductive Lie algebra $$\mathfrak {g}$$ , its nilpotent element f and its faithful finite dimensional representation, we construct a Lax operator L(z) with coefficients in the quantum finite W-algebra $$W(\mathfrak {g},f)$$ . We show that for the classical linear Lie algebras $$\mathfrak {gl}_N$$ , $$\mathfrak {sl}_N$$ , $$\mathfrak {so}_N$$ and $$\mathfrak {sp}_N$$ , the operator L(z) satisfies a generalized Yangian identity. The operator L(z) is a quantum finite analogue of the operator of generalized Adler type which we recently introduced in the classical affine setup. As in the latter case, L(z) is obtained as a generalized quasideterminant.

Commuting variety of Witt algebra

Let $$\mathfrak{g} = W_1 $$ be the Witt algebra over an algebraically closed field k of characteristic p > 3; and let be the commuting variety of g. In contrast with the case of classical Lie algebras of P. Levy [J. Algebra, 2002, 250: 473–484], we show that the variety is reducible, and not equidimensional. Irreducible components of and their dimensions are precisely given. As a consequence, the variety is not normal.

The R-Matrix Presentation for the Yangian of a Simple Lie Algebra

Starting from a finite-dimensional representation of the Yangian $Y(\mathfrak{g})$ for a simple Lie algebra $\mathfrak{g}$ in Drinfeld's original presentation, we construct a Hopf algebra $X_\mathcal{I}(\mathfrak{g})$, called the extended Yangian, whose defining relations are encoded in a ternary matrix relation built from a specific $R$-matrix $R(u)$. We prove that there is a surjective Hopf algebra morphism $X_\mathcal{I}(\mathfrak{g})\twoheadrightarrow Y(\mathfrak{g})$ whose kernel is generated as an ideal by the coefficients of a central matrix $\mathcal{Z}(u)$. When the underlying representation is irreducible, we show that this matrix becomes a grouplike central series, thereby making available a proof of a well-known theorem stated by Drinfeld in the 1980's. We then study in detail the algebraic structure of the extended Yangian, and prove several generalizations of results which are known to hold for Yangians associated to classical Lie algebras in their $R$-matrix presentations.

Weight multiplicity formulas for bivariate representations of classical Lie algebras

A bivariate representation of a complex simple Lie algebra is an irreducible representation having highest weight a combination of the first two fundamental weights. For a complex classical Lie algebra, we establish an expression for the weight multiplicities of bivariate representations.

Resolutions of nilpotent orbit closures via Coulomb branches of 3-dimensional mathcal{N}=4 theories

The Coulomb branches of certain 3-dimensional mathcal{N}=4 quiver gauge theories are closures of nilpotent orbits of classical or exceptional Lie algebras. The monopole formula, as Hilbert series of the associated Coulomb branch chiral ring, has been successful in describing the singular hyper-Kähler structure. By means of the monopole formula with background charges for flavour symmetries, which realises real mass deformations, we study the resolution properties of all (characteristic) height two nilpotent orbits. As a result, the monopole formula correctly reproduces (i) the existence of a symplectic resolution, (ii) the form of the symplectic resolution, and (iii) the Mukai flops in the case of multiple resolutions. Moreover, the (characteristic) height two nilpotent orbit closures are resolved by cotangent bundles of Hermitian symmetric spaces and the unitary Coulomb branch quiver realisations exhaust all the possibilities.

Koszul properties of the moment map of some classical representations

This work concerns the moment map $$\mu $$ associated with the standard representation of a classical Lie algebra. For applications to deformation quantization it is desirable that $$S/(\mu )$$ , the coordinate algebra of the zero fibre of $$\mu $$ , be Koszul. The main result is that this algebra is not Koszul for the standard representation of $$\mathfrak {sl}_{n}$$ , and of $$\mathfrak {sp}_{n}$$ . This is deduced from a computation of the Betti numbers of $$S/(\mu )$$ as an S-module, which are of interest also from the point of view of commutative algebra.

Classical Affine $${\mathcal W}$$-Algebras and the Associated Integrable Hamiltonian Hierarchies for Classical Lie Algebras

We prove that any classical affine W-algebra W(g,f), where g is a classical Lie algebra and f is an arbitrary nilpotent element of g, carries an integrable Hamiltonian hierarchy of Lax type equations. This is based on the theories of generalized Adler type operators and of generalized quasideterminants, which we develop in the paper. Moreover, we show that under certain conditions, the product of two generalized Adler type operators is a Lax type operator. We use this fact to construct a large number of integrable Hamiltonian systems, recovering, as a special case, all KdV type hierarchies constructed by Drinfeld and Sokolov.

Cohomological orientifold Donaldson–Thomas invariants as Chow groups

We establish a geometric interpretation of orientifold Donaldson–Thomas invariants of $$\sigma $$ -symmetric quivers with involution. More precisely, we prove that the cohomological orientifold Donaldson–Thomas invariant is isomorphic to the rational Chow group of the moduli space of $$\sigma $$ -stable self-dual quiver representations. As an application we prove that the Chow Betti numbers of moduli spaces of stable m-tuples in classical Lie algebras can be computed numerically. We also prove a cohomological wall-crossing formula relating semistable Hall modules for different stabilities.

Branes and the Kraft-Procesi transition: classical case

Moduli spaces of a large set of 3d mathcal{N}=4 effective gauge theories are known to be closures of nilpotent orbits. This set of theories has recently acquired a special status, due to Namikawa’s theorem. As a consequence of this theorem, closures of nilpotent orbits are the simplest non-trivial moduli spaces that can be found in three dimensional theories with eight supercharges. In the early 80’s mathematicians Hanspeter Kraft and Claudio Procesi characterized an inclusion relation between nilpotent orbit closures of the same classical Lie algebra. We recently [1] showed a physical realization of their work in terms of the motion of D3-branes on the Type IIB superstring embedding of the effective gauge theories. This analysis is restricted to A-type Lie algebras. The present note expands our previous discussion to the remaining classical cases: orthogonal and symplectic algebras. In order to do so we introduce O3-planes in the superstring description. We also find a brane realization for the mathematical map between two partitions of the same integer number known as collapse. Another result is that basic Kraft-Procesi transitions turn out to be described by the moduli space of orthosymplectic quivers with varying boundary conditions.

Gradings on classical central simple real Lie algebras

For any abelian group G, we classify up to isomorphism all G-gradings on the classical central simple Lie algebras, except those of type D4, over the field of real numbers (or any real closed field).

Generalized Young Walls for Classical Lie Algebras

In this paper, we introduce an new combinatorial model, which we call generalized Young walls for classical Lie algebras, and we give two realizations of the crystal B(∞) over classical Lie algebras using generalized Young walls. Also, we construct natural crystal isomorphisms between generalized Young wall realizations and other realizations, for example, monomial realization, polyhedral realization and tableau realization. Moreover, as applications, we obtain a crystal isomorphism between two different polyhedral realizations of B(∞).