Hecke Symmetry Research Articles

Hecke Symmetries, Associated with Artin–Schelter Regular Algebras of Type $$\boldsymbol{E}$$ and $$\boldsymbol{H}$$

Hecke Symmetries, Associated with Artin–Schelter Regular Algebras of Type $$\boldsymbol{E}$$ and $$\boldsymbol{H}$$

Noncommutative geometry on central extension of U(u(2))

In our previous publications, we have introduced analogs of partial derivatives on the reflection equation algebras, associated with Hecke symmetries. As a consequence, we get quantum partial derivatives on the enveloping algebras U(gl(N)). In the current paper, we consider the particular case N = 2 in detail and discuss the problem of a prolongation of these derivatives onto some central extension of the compact form U(u(2)) of the algebra U(gl(2)). Possible applications of this noncommutative geometry are discussed.

Hecke symmetries associated with the polynomial algebra in 3 commuting indeterminates

It is shown in the paper that each Hecke symmetry R with the R-symmetric algebra freely generated by 3 commuting elements is determined by a bivector and a symmetric bilinear form on a 3-dimensional vector space. A general formula for such Hecke symmetries is given and the equivalence classes are described.

On the h-adic Quantum Vertex Algebras Associated with Hecke Symmetries

We study the quantum vertex algebraic framework for the Yangians of RTT-type and the braided Yangians associated with Hecke symmetries, introduced by Gurevich and Saponov. First, we construct several families of modules for the aforementioned Yangian-like algebras which, in the RTT-type case, lead to a certain h-adic quantum vertex algebra $${\mathcal {V}}_c (R)$$ via the Etingof–Kazhdan construction, while, in the braided case, they produce ( $$\phi $$ -coordinated) $${\mathcal {V}}_c (R)$$ -modules. Next, we show that the coefficients of suitably defined quantum determinant can be used to obtain central elements of $${\mathcal {V}}_c (R)$$ , as well as the invariants of such ( $$\phi $$ -coordinated) $${\mathcal {V}}_c (R)$$ -modules. Finally, we investigate a certain algebra which is closely connected with the representation theory of $${\mathcal {V}}_c (R)$$ .

Matrix Capelli identities related to reflection equation algebra

By using the notion of quantum double we introduce analogs of partial derivatives on a reflection equation algebra, associated with a Hecke symmetry of GLN type. We construct the matrix L=MD, where M is the generating matrix of the reflection equation algebra and D is the matrix composed of the quantum partial derivatives and prove that the matrices M, D and L satisfy a matrix identity, called the matrix Capelli one. Upon applying quantum trace, it becomes a scalar relation, which is a far-reaching generalization of the classical Capelli identity. Also, we get a generalization of the some higher Capelli identities proved by A. Okounkov in [6].

On the graded algebras associated with Hecke symmetries, II. The Hilbert series

Hecke symmetries give rise to a family of graded algebras which represent quantum groups and spaces of noncommutative geometry. The present paper continues the work aiming to understand general properties of these algebras without a restriction on the parameter q of Hecke relation used in earlier results. However, if q is a root of 1, we need a restriction on the indecomposable modules for the Hecke algebras of type A that can occur as direct summands of representations in the tensor powers of the initial vector space V. In this setting, we generalize known results on rationality of Hilbert series. The combinatorial nature of this problem stems from a relationship between the Grothendieck ring of the category of comodules for the Faddeev–Reshetikhin–Takhtajan bialgebra A(R) associated with a Hecke symmetry R and the ring of symmetric functions. We then improve two results on monoidal equivalences of corepresentation categories and on Gorensteinness of graded algebras from a previous article.

Quantum doubles of Fock type and bosonization

We introduce analogs of creation and annihilation operators, related to involutive and Hecke symmetries R, and perform bosonic and fermionic realization of the modified Reflection Equation algebras in terms of the so-called Quantum Doubles of Fock type. Also, we introduce Quantum Doubles of Fock type, associated with Birman-Murakami-Wenzl symmetries coming from orthogonal or simplectic Quantum Groups and exhibit the algebras obtained by means of the corresponding bosonization (fermionization). Besides, we apply this scheme to current braidings arising from Hecke symmetries R via the Baxterization procedure.

On the graded algebras associated with Hecke symmetries

A Hecke symmetry R on a finite dimensional vector space V gives rise to two graded factor algebras \mathbb S (V, R) and \Lambda (V, R) of the tensor algebra of V which are regarded as quantum analogs of the symmetric and the exterior algebras. Another graded algebra associated with R is the Faddeev–Reshetikhin–Takhtajan bialgebra A(R) which coacts on \mathbb S (V, R) and \Lambda (V, R) . There are also more general graded algebras defined with respect to pairs of Hecke symmetries and interpreted in terms of quantum hom-spaces. Their nice behaviour has been known under the assumption that the parameter q of the Hecke relation is such that 1 + q + \cdots + q^{n-1} \neq 0 for all n > 0 . The present paper makes an attempt to investigate several questions without this condition on q . Particularly we are interested in Koszulness and Gorensteinness of those graded algebras. For q a root of 1 positive results require a restriction on the indecomposable modules for the Hecke algebras of type A that can occur as direct summands of representations in the tensor powers of V .

Aspects of Hecke Symmetry:Anomalies, Curves, and Chazy Equations

We study various relations governing quasi-automorphic forms associated to discrete subgroups of ${\rm SL}(2,\mathbb{R}) $ called Hecke groups. We show that the Eisenstein series associated to a Hecke group ${\rm H}(m)$ satisfy a set of $m$ coupled linear differential equations, which are natural analogues of the well-known Ramanujan identities for quasi-modular forms of ${\rm SL}(2,\mathbb{Z})$. Each Hecke group is then associated to a (hyper-)elliptic curve, whose coefficients are determined by an anomaly equation. For the $m=3$ and $4$ cases, the Ramanujan identities admit a natural geometric interpretation as a Gauss-Manin connection on the parameter space of the elliptic curve. The Ramanujan identities also allow us to associate a nonlinear differential equation of order $ m $ to each Hecke group. These equations are higher-order analogues of the Chazy equation, and we show that they are solved by the quasi-automorphic Eisenstein series $E_2^{(m)}$ associated to ${\rm H}(m) $ and its orbit under the Hecke group. We conclude by demonstrating that these nonlinear equations possess the Painlev\'e property.

Bethe Subalgebras in Braided Yangians and Gaudin-Type Models

In \cite{GS1} the notion of braided Yangians of Reflection Equation type was introduced. Each of these algebras is associated with an involutive or Hecke symmetry $R$. Besides, the quantum analogs of certain symmetric polynomials (elementary symmetric ones, power sums) were suggested. In the present paper we show that these quantum symmetric polynomials commute with each other and consequently generate a commutative Bethe subalgebra. As an application, we get some Gaudin-type models and the corresponding Bethe subalgebras.

Braided Yangians

Yangian-like algebras associated with current R-matrices different from the Yang ones are introduced. These algebras are of two types. The so-called braided Yangians are close to the Reflection Equation algebras, arising from involutive or Hecke symmetries. The Yangians of RTT type are close to the corresponding RTT algebras. Some properties of these two classes of the Yangian-like algebras are studied. Thus, evaluation morphisms for them are constructed, their bi-algebra structures are described, and quantum analogs of certain symmetric polynomials, in particular, the quantum determinant, are introduced. It is proved that in any braided Yangian this determinant is always central, whereas, in general, this is not true for the Yangians of RTT type. Analogs of the Cayley–Hamilton–Newton identities in the braided Yangians are exhibited. A bosonic realization of the braided Yangians is performed.

Drinfeld–Sokolov reduction in quantum algebras: canonical form of generating matrices

We define the second canonical forms for the generating matrices of the Reflection Equation algebras and the braided Yangians, associated with all even skew-invertible involutive and Hecke symmetries. By using the Cayley–Hamilton identities for these matrices, we show that they are similar to their canonical forms in the sense of Chervov and Talalaev (J Math Sci (NY) 158:904–911, 2008).

Braided algebras and their applications to Noncommutative Geometry

We introduce the notion of a braided algebra and study some examples of these. In particular, R-symmetric and R-skew-symmetric algebras of a linear space V equipped with a skew-invertible Hecke symmetry R are braided algebras. We prove the “mountain property” for the numerators and denominators of their Poincaré–Hilbert series (which are always rational functions).Also, we further develop a differential calculus on modified Reflection Equation algebras. Thus, we exhibit a new form of the Leibniz rule for partial derivatives on such algebras related to involutive symmetries R. In particular, we present this rule for the algebra U(gl(m)). The case of the algebra U(gl(2)) and its compact form U(u(2)) (which can be treated as a deformation of the Minkowski space algebra) is considered in detail. On the algebra U(u(2)) we introduce the notion of the quantum radius, which is a deformation of the usual radius, and compute the action of rotationally invariant operators and in particular of the Laplace operator. This enables us to define analogs of the Laplace–Beltrami operators corresponding to certain Schwarzschild-type metrics and to compute their actions on the algebra U(u(2)) and its central extension. Some “physical” consequences of our considerations are presented.

Braided Weyl algebras and differential calculus on [formula omitted

On any reflection equation algebra corresponding to a skew-invertible Hecke symmetry (i.e., a special type solution of the Quantum Yang–Baxter equation) we define analogs of the partial derivatives. Together with elements of the initial reflection equation algebra they generate a “braided analog” of the Weyl algebra. When q→1, the braided Weyl algebra corresponding to the Quantum Group Uq(sl(2)) goes to the Weyl algebra defined on the algebra Sym(u(2)) or U(u(2)) depending on the way of passing to the limit. Thus, we define partial derivatives on the algebra U(u(2)), find their “eigenfunctions”, and introduce an analog of the Laplace operator on this algebra. Also, we define the “radial part” of this operator, express it in terms of “quantum eigenvalues”, and sketch an analog of the de Rham complex on the algebra U(u(2)). Eventual applications of our approach are discussed.

Generic super-orbits in [formula omitted] and their braided counterparts

We introduce some braided varieties–braided orbits–by considering quotients of the so-called Reflection Equation Algebras associated with Hecke symmetries (i.e. special type solutions of the quantum Yang–Baxter equation). Such a braided variety is called regular if there exists a projective module on it, which is a counterpart of the cotangent bundle on a generic orbit O ∈ g l ( m ) ∗ in the framework of the Serre approach. We give a criterium of regularity of a braided orbit in terms of roots of the Cayley–Hamilton identity valid for the generating matrix of the Reflection Equation Algebra in question. By specializing our general construction we get super-orbits in g l ( m | n ) ∗ and a criterium of their regularity.

Representation theory of (modified) reflection equation algebra of $GL(m|n)$ type

Let R : V 2 → V 2 be a Hecke type solution of the quantum Yang-Baxter equation (a Hecke symmetry). Then, the Hilbert-Poincre series of the associated R-exterior algebra of the space V is a ratio of two polynomials of degree m (numerator) and n (denominator). Assuming R to be skew-invertible, we define a rigid quasitensor category SW(V(m|n)) of vector spaces, generated by the space V and its dual V � , and compute certain numerical char- acteristics of its objects. Besides, we introduce a braided bialgebra structure in the modified Reflection Equation Algebra, associated with R, and equip objects of the category SW(V(m|n)) with an action of this algebra. In the case related to the quantum group Uq(sl(m)), we con- sider the Poisson counterpart of the modified Reflection Equation Algebra and compute the semiclassical term of the pairing, defined via the categorical (or quantum) trace.

A Duality Theorem for Ergodic Actions of Compact Quantum Groups on C *-Algebras

The spectral functor of an ergodic action of a compact quantum group G on a unital C *-algebra is quasitensor, in the sense that the tensor product of two spectral subspaces is isometrically contained in the spectral subspace of the tensor product representation, and the inclusion maps satisfy natural properties. We show that any quasitensor *-functor from Rep(G) to the category of Hilbert spaces is the spectral functor of an ergodic action of G on a unital C *-algebra.As an application, we associate an ergodic G-action on a unital C *-algebra to an inclusion of Rep(G) into an abstract tensor C *-category \({\mathcal{T}}\) .If the inclusion arises from a quantum subgroup K of G, the associated G-system is just the quotient space K\G. If G is a group and \({\mathcal{T}}\) has permutation symmetry, the associated G-system is commutative, and therefore isomorphic to the classical quotient space by a subgroup of G.If a tensor C *-category has a Hecke symmetry making an object ρ of dimension d and μ-determinant 1, then there is an ergodic action of S μ U(d) on a unital C *-algebra having the \((\iota,\rho^r)\) as its spectral subspaces. The special case of \(S_\mu U(2)\) is discussed.

The homological determinant of quantum groups of type 𝐴

LetRRbe a Hecke symmetry depending algebraically on a parameterq∈Cq\in \mathbb {C}. We show that the homology of the Koszul complex associated withRRis one-dimensional whenqqis not a root of unity. A generator of this homology group then induces the homological determinant of the quantum group associated withRR.

Irreducible representations of quantum linear groups of type [formula omitted

A Hecke symmetry is an invertible operator defined on the tensor square of a finite dimensional vector space, which satisfied the (quantized) Yang-Baxter equation and the Hecke equation (x + 1)(x − q) = 0, see [4, 6] for more details. Given such a matrix one can construct a Hopf algebra which is “the function algebra” over an (algebraic) matrix quantum group. One is interested in the representations of this quantum group, that is the comodules over the Hopf algebra. It turns out that the dimension of the vector space on which the Hecke symmetry is defined does not play a significant role but rather a certain “inner” characteristics of the Hecke symmetry, the birank – a pair of non-negative integers. A Hecke symmetry is called even if its birank is of the form (r, 0) and odd if its birank is of the form (0, r). In these cases, the representation category of the corresponding quantum group is like the representation category of the matrix group GL(r), it is semisimple and generally does not depend on the Hecke symmetry but only on the birank. If the Hecke symmetry is neither even nor odd the representation category of the corresponding quantum group seems to be similar to the representation category of the super group GL(r|s). This problem has not been settled yet. The aim of this paper is to classify irreducible representation of the matrix quantum group corresponding to a Hecke symmetry of birank (2, 1). Such a quantum group is usually called matrix quantum group of type (1, 0). In this work we show that irreducible representations of a quantum group of type (1, 0) can be indexed by tuples (m,n, p) of integers with m ≥ n. We exhibit basic decomposition rules of the tensor product of these representations and compute their dimensions. Recall that representations of a quantum group are by definition comodules over the corresponding Hopf algebra (“of functions”). Using the Koszul komplex we construct for each tuple (m,n, p),m ≥ n a comodule and prove that they are simple. The difficulty here is that the comodule category is not semisimple. Our main technique is based on the theory of Hopf algebras with integral (co-Frobenius Hopf algebras). For such Hopf algebras there is a special class of simple comodules that split in any comodule. We show that our Hopf algebra is co-Frobenius and use a criterion for a comodule to be splitting in any comodule (which is equivalent to being injective and projective). (In Kac’s theory of representation of Lie superalgebras irreducible representations that split in any representations are called typical [9].)

Traces in braided categories

With any even Hecke symmetry R (that is a Hecke type solution of the Yang–Baxter equation) we associate a quasitensor category. We formulate a condition on R implying that the constructed category is rigid and its commutativity isomorphisms R U, V are natural in the sense of [20]. We show that this condition leads to rescaling the initial Hecke symmetry. We suggest a new way of introducing traces as properly normalized categorical morphisms End(V)→ K and deduce the corresponding normalization from categorical dimensions.