Quantum enveloping algebras with von Neumann regular Cartan-like generators and the Pierce decomposition
Steven Duplij, Sergey Sinel'shchikov
备注：
18 pages, minor corrections after the referee report, retyped in journal format, svjour style, to appear in Communications in Mathematical Physics (2008), dedicated to the memory of L. L. Vaksman (1951-2007), see http://webusers.physics.umn.edu/~duplij/vaksman
Quantum bialgebras derivable from Uq(sl2) which contain idempotents and von Neumann regular Cartan-like generators are introduced and investigated. Various types of antipodes (invertible and von Neumann regular) on these bialgebras are constructed, which leads to a Hopf algebra structure and a von Neumann-Hopf algebra structure, respectively. For them, explicit forms of some particular R-matrices (also, invertible and von Neumann regular) are presented, and the latter respects the Pierce decomposition.
A perturbative quantization procedure for Lie bialgebras is introduced and used to classify all three dimensional complex quantum algebras compatible with a given coproduct. The role of elements of the quantum universal enveloping algebra that, analogously to generators in Lie algebras, have a distinguished type of coproduct is discussed, and the relevance of a symmetrical basis in the universal enveloping algebra stressed. New quantizations of three dimensional solvable algebras, relevant for possible physical applications for their simplicity, are obtained and all already known related resul…
Let $\mathfrak{g}$ be an affine Kac-Moody algebra with symmetric Cartan datum, $\mathfrak{n^{+}}$ be the maximal nilpotent subalgebra of $\mathfrak{g}$. By the Hall algebra approach, we construct integral bases of the $\mathbb{Z}$-form of the enveloping algebra $U(\mathfrak{n^{+}})$. In particular, the representation theory of tame quivers is essentially used in this paper.
We replace the group of group-like elements of the quantized enveloping algebra $U_q({\frak{g}})$ of a finite dimensional semisimple Lie algebra ${\frak g}$ by some regular monoid and get the weak Hopf algebra ${\frak{w}}_q^{\sf d}({\frak g})$. It is a new subclass of weak Hopf algebras but not Hopf algebras. Then we devote to constructing a basis of ${\frak{w}}_q^{\sf d}({\frak g})$ and determine the group of weak Hopf algebra automorphisms of ${\frak{w}}_q^{\sf d}({\frak g})$ when $q$ is not a root of unity.
For quantized universal enveloping algebras we construct weight modules by inducing representations of the centralizer of the Cartan subalgebra in the quantized universal enveloping algebra. The induced modules arising from finite-dimensional weight modules the centralizer algebra are studied. In particular, we study the induction of one-dimensional modules, and this is related to the study of commutative subalgebras of the centralizer algebra. For the special case of $U_q(\mathfrak{sl}(2,\mathbb{C}))$ we show that we get the admissible unitary representations corresponding to the non-compact …
A generalization of the differential geometry of forms and vector fields to the case of quantum Lie algebras is given. In an abstract formulation that incorporates many existing examples of differential geometry on quantum groups, we combine an exterior derivative, inner derivations, Lie derivatives, forms and functions all into one big algebra. In particular we find a generalized Cartan identity that holds on the whole quantum universal enveloping algebra of the left-invariant vector fields and implicit commutation relations for a left-invariant basis of 1-forms.
A rigid framework for the Cartan calculus of Lie derivatives, inner derivations, functions, and forms is proposed. The construction employs a semi-direct product of two graded Hopf algebras, the respective super-extensions of the deformed universal enveloping algebra and the algebra of functions on a quantum group. Relations in the Cartan calculus follow as consistency conditions. The approach is not a priori based on the Leibniz rule for the exterior derivative and might hence also be of interest in the recent work on its deformations. The Cartan identity for the Lie derivatives is proven. (T…
We study the L-infinity-formality problem for the Hochschild complex of the universal enveloping algebra of some examples of Lie algebras such as Cartan-3-regular quadratic Lie algebras (for example semisimple Lie algebras and in more detail so(3)), and free Lie algebras generated by a vector space of dimension at least 2. We show that for these examples formality in Kontsevich's sense does NOT hold, although some of them allow unconditioned deformability. We compute the L-infinity-structure on the cohomology given by homotopy transfer in certain cases.
We construct a crystal basis for the negative half of the quantum group U associated to the standard super Cartan datum of gl(m|1), which is compatible with known crystals on Kac modules and simple modules. We show that these crystals admit globalization functions which produce compatible canonical bases. We then define a braid group action on a family of quantum groups including U, and use this action to show that our canonical basis agrees with those constructed from PBW bases. Finally, we explicitly compute some small-rank examples.
This volume contains a mildly expanded version of lectures and talks at seminars and conferences, as well as review papers on subjects listed in the title of the volume. A great deal of these texts have already been published or sent to press. However, the only way to provide a good exposition of the field we are interested in is to collect all of those papers together. Bounded symmetric domains form a favorite subject of research in function theory, non-commutative harmonic analysis, and representation theory of real reductive Lie groups. The authors introduce the notions of q-analogues of bo…
We construct explicit Drinfel'd twists for the generalized Cartan type $H$ Lie algebras in characteristic $0$ and obtain the corresponding quantizations and their integral forms. Via making modular reductions including modulo $p$ reduction and modulo $p$-restrictedness reduction, and base changes, we derive certain modular quantizations of the restricted universal enveloping algebra $\mathbf u(\mathbf{H}(2n;\underline{1}))$ in characteristic $p$. They are new non-pointed Hopf algebras of truncated $p$-polynomial noncommutative and noncocommutative deformation of prime-power dimension $p^{p^{2n…
There is renewed interest in the coideal subalgebras used to form quantum symmetric pairs because of recent discoveries showing that they play a fundamental role in the representation theory of quantized enveloping algebras. However, there is still no general theory of finite-dimensional modules for these coideals. In this paper, we establish an important step in this direction: we show that every quantum symmetric pair coideal subalgebra admits a quantum Cartan subalgebra which is a polynomial ring that specializes to its classical counterpart. The construction builds on Kostant and Sugiura's…
We construct explicit Drinfel'd twists of Jordanian type for the generalized Cartan type K Lie algebras in characteristic 0 and obtain the corresponding quantizations, especially their integral forms. By making modular reductions including modulo p and modulo p-restrictedness reduction, and base changes, we derive certain modular quantizations of the restricted universal enveloping algebra $\mathbf u(\mathbf{K}(2n{+}1;\underline{1}))$ for the restricted simple Lie algebra of Cartan type K in characteristic p. They are new pointed Hopf algebras of noncommutative and noncocommutative and with di…
In this paper we use refiltering methods to prove that certain types of multi-filtered algebras are Auslander-regular and Cohen-Macaulay. This is applied to obtain that the quantized enveloping algebra associated to a Cartan matrix is Auslander-regular and Cohen-Macaulay.
The generalized Cartan type $\mathbf{S}$ Lie algebras in char 0 with the Lie bialgebra structures involved are quantized, where the Drinfel'd twist we used is proved to be a variation of the Jordanian twist. As the passage from char 0 to char p, their quantization integral forms are given. By the modular reduction and base changes, we obtain certain quantizations of the restricted universal enveloping algebra $\mathbf u(\mathbf{S}(n;\underline{1}))$ (for the Cartan type simple modular restricted Lie algebra $\mathbf{S}(n;\underline{1})$ of $\mathbf{S}$ type). They are new Hopf algebras of trun…
Aspects of the algebraic structure and representation theory of the quantum affine superalgebras with symmetrizable Cartan matrices are studied. The irreducible integrable highest weight representations are classified, and shown to be deformations of their classical counterparts. It is also shown that Jimbo type quantum affine superalgebras can be obtained by deforming universal enveloping algebras of ordinary (i.e., non-graded) affine algebras supplemented by certain parity operators.
We study a class of quantized enveloping algebras, called twisted Yangians, associated with the symmetric pairs of types B, C, D in Cartan's classification. These algebras can be regarded as coideal subalgebras of the extended Yangian for orthogonal or symplectic Lie algebras. They can also be presented as quotients of a reflection algebra by additional symmetry relations. We prove an analogue of the Poincare-Birkoff-Witt Theorem, determine their centres and study also extended reflection algebras.
Lusztig has constructed a Frobenius morphism for quantum groups at an $\ell$-th root of unity, which gives an integral lift of the Frobenius map on universal enveloping algebras in positive characteristic. Using the Hall algebra we give a simple construction of this map for the positive part of the quantum group attached to an arbitrary Cartan datum in the nondivisible case.
We give proofs of the PBW and duality theorems for the quantum Kac-Moody algebras and quantum current algebras, relying on Lie bialgebra duality. We also show that the classical limit of the quantum current algebras associated with an untwisted affine Cartan matrix is the enveloping algebra of a quotient of the corresponding toroidal algebra; this quotient is trivial in all cases except the $A_1^{(1)}$ case.
Let U be the quantised enveloping algebra associated to a Cartan matrix of finite type. Let W be the tensor product of a finite list of highest weight representations of U. Then the centraliser algebra of W has a basis called the dual canonical basis which gives an integral form. We show that this integral form is cellular by using results due to Lusztig.
The role of the quantum universal enveloping algebras of symmetries in constructing non-commutative geometry of the space-time including vector bundles, measure, equations of motion and their solutions is discussed. In the framework of the twist theory the Klein-Gordon-Fock and Dirac equations on the quantum Minkowski space are studied from this point of view for the simplest quantum deformation of the Lorentz algebra induced by its Cartan subalgebra twist.
Classical symmetric pairs consist of a symmetrizable Kac-Moody algebra $\mathfrak{g}$, together with its subalgebra of fixed points under an involutive automorphism of the second kind. Quantum group analogs of this construction, known as quantum symmetric pairs, replace the fixed point Lie subalgebras by one-sided coideal subalgebras of the quantized enveloping algebra $U_q(\mathfrak{g})$. We provide a complete presentation by generators and relations for these quantum symmetric pair coideal subalgebras. These relations are of inhomogeneous $q$-Serre type and are valid without restrictions on …
Let ${U}_q(sl_2)$ be the quantized enveloping algebra associated to the simple Lie algebra $sl_2$. In this paper, we study the quantum double $D_q$ of the Borel subalgebra ${U}_q((sl_2)^{\leq 0})$ of ${U}_q(sl_2)$. We construct an analogue of Kostant--Lusztig ${Z}[v,v^{-1}]$-form for $D_q$ and show that it is a Hopf subalgebra. We prove that, over an algebraically closed field, every simple $D_q$-module is the pullback of a simple ${U}_q(sl_2)$-module through certain surjection from $D_q$ onto ${U}_q(sl_2)$, and the category of finite dimensional weight $D_q$-modules is equivalent to a direct …
This is a review of our previous works (some of them joint with B. Feigin and N. Reshetikhin) on the Gaudin model and opers. We define a commutative subalgebra in the tensor power of the universal enveloping algebra of a simple Lie algebra g. This algebra includes the hamiltonians of the Gaudin model, hence we call it the Gaudin algebra. It is constructed as a quotient of the center of the completed enveloping algebra of the affine Kac-Moody algebra g^ at the critical level. We identify the spectrum of the Gaudin algebra with the space of opers associated to the Langlands dual Lie algebra of g…
We start with the observation that the quantum group SL_q(2), described in terms of its algebra of functions has a quantum subgroup, which is just a usual Cartan group. Based on this observation we develop a general method of constructing quantum groups with similar property. We also describe this method in the language of quantized universal enveloping algebras, which is another common method of studying quantum groups. We carry our method in detail for root systems of type SL(2); as a byproduct we find a new series of quantum groups - metaplectic groups of SL(2)-type. Representations of thes…
We describe quantum groups given by multiparametric deformations of enveloping algebras of Kac-Moody algebras as a family of pointed Hopf algebras introduced by Andruskiewitsch and Schneider associated to a generalized Cartan matrix. We show that under some assumptions, these Hopf algebras depend only on one parameter on each connected component of the Dynkin diagram, up to a cocycle deformation. In particular, we obtain in this way a known result of Hu, Pei and Rosso.
Two "quantum enveloping algebras", here denoted by $U(R)$ and $U^{\sim}(R)$, are associated in [FRTa] and [FRTb] to any Yang-Baxter operator R. The latter is only a bialgebra, in general; the former is a Hopf algebra. In this paper, we study the pointed Hopf algebras $U(R_Q)$, where $R_Q$ is the Yang-Baxter operator associated with the multi-parameter deformation of $GL_n$ supplied in [AST]; cf also [S,Re]. Some earlier results concerning these Hopf algebras $U(R_Q)$ were obtained in [To,CLMT,CM]; a related (but different) Hopf algebra was studied in [DP]. The main new results obtained here co…
Applications of algebras in physics are related to the connection of measurable observables to relevant elements of the algebras, usually the generators. However, in the determination of the generators in Lie algebras there is place for some arbitrary conventions. The situation is much more involved in the context of quantum algebras, where inside the quantum universal enveloping algebra, we have not enough primitive elements that allow for a privileged set of generators and all basic sets are equivalent. In this paper we discuss how the Drinfeld double structure underlying every simple Lie bi…
We report some observations concerning two well-known approaches to construction of quantum groups. Thus, starting from a bialgebra of inhomogeneous type and imposing quadratic, cubic or quartic commutation relations on a subset of its generators we come, in each case, to a q-deformed universal enveloping algebra of a certain simple Lie algebra. An interesting correlation between the order of initial commutation relations and the Cartan matrix of the resulting algebra is observed. Another example demonstrates that the bialgebra structure of sl_q(2) can be completely determined by requiring the…
Shapovalov elements $θ_{β,m}$ of the classical or quantized universal enveloping algebra of a simple Lie algebra $\mathfrak{g}$ are parameterized by a positive root $β$ and a positive integer $m$. They relate the highest vector of a reducible Verma module with highest vectors of its submodules. We obtain a factorization of $θ_{β,m}$ to a product of $θ_{β,1}$ and calculate $θ_{β,1}$ as a residue of a matrix element of the inverse Shapovalov form via a generalized Nigel-Moshinsky algorithm. This way we explicitly express $θ_{β,m}$ of a classical simple Lie algebra through the Cartan-Weyl basis i…
Let $C$ be a symmetrizable generalized Cartan Matrix, and $q$ an indeterminate. ${\fg}(C)$ is the Kac-Moody Lie algebra and $U=U_q({\fg}(C))$ the associated quantum enveloping algebra over $ k={\Bbb Q}(q)$. The quantum function algebra ${\Bbb C}_{q}[G]$ is defined as a suitable $U$-bisubalgebra of the dual space $\hom_{k}(U,k)$ which can be described using matrix elements of integrable $U$-modules. For $\fg$ affine, the highest weight modules of $C_q[G]$ are constructed and, assuming a minimality condition, their (unitarizable) irreducible quotients are shown to be in a 1-1 correspondence with…
In these lectures we develop the projection operator method for quantum groups. Here the term "quantum groups" means q-deformed universal enveloping algebras of contragredient Lie (super)algebras of finite growth. Contains of the lectures can be divided on two parts. Basis fragments of the first part are: combinatorial structure of root systems, the q-analog of the Cartan-Weyl basis, the extremal projector and the universal R-matrix for any contragredient Lie (super)algebra of finite growth. The explicit expressions for the extremal projectors and the universal R-matrices are ordered products …
As a natural generalization of ordinary Lie algebras we introduce the concept of quantum Lie algebras ${\cal L}_q(g)$. We define these in terms of certain adjoint submodules of quantized enveloping algebras $U_q(g)$ endowed with a quantum Lie bracket given by the quantum adjoint action. The structure constants of these algebras depend on the quantum deformation parameter $q$ and they go over into the usual Lie algebras when $q=1$. The notions of q-conjugation and q-linearity are introduced. q-linear analogues of the classical antipode and Cartan involution are defined and a generalised Killing…
For a Hopf algebra A, we define the structures of differential complexes on two dual exterior Hopf algebras: 1) an exterior extension of A and 2) an exterior extension of the dual algebra A^*. The Heisenberg double of these two exterior Hopf algebras defines the differential algebra for the Cartan differential calculus on A. The first differential complex is an analog of the de Rham complex. In the situation when A^* is a universal enveloping of a Lie (super)algebra the second complex coincides with the standard complex. The differential is realized as an (anti)commutator with a BRST- operator…
We introduce twists by Cartan elements of conformal blocks on a curve X, corresponding to a Lie algebra g. We show that these twists define holomorphic functions, with theta-like behaviour, on a product of copies of its Jacobian J(X)^r. We then parametrise the conformal blocks by their twisted correlation functions of nilpotent currents, in the spirit of Feigin-Stoyanovsky. We compute the action of the Sugawara tensor in terms of these correlation functions. The result is a family of operators acting on forms on the product J(X)^r \times \prod_{i} S^{n_i}X of the Jacobian of X with its symmetr…
Let $A$ be an arbitrary symmetrizable Cartan matrix of rank $r$, and ${\bf n}={\bf n_+}$ be the standard maximal nilpotent subalgebra in the Kac-Moody algebra associated with $A$ (thus, ${\bf n}$ is generated by $E_1,\ldots,E_r$ subject to the Serre relations). Let $\hat U_q({\bf n})$ be the completion (with respect to the natural grading) of the quantized enveloping algebra of ${\bf n}$. For a sequence ${\bf i}=(i_1,\ldots,i_m)$ with $1\le i_k\le r$, let $P_{\bf i}$ be a skew polynomial algebra generated by $t_1,\ldots,t_m$ subject to the relations $t_lt_k=q^{C_{i_k,i_l}}t_kt_l$ ($1\le k<l\le…
