N. Aizawa
A new non-standard deformation of all types of classical Lie algebras is constructed by means of Drinfel'd twist based on a six dimensional subalgebra. This is an extension of extended twists introduced by Kulish et al. For the algebra M_3 = so(3,2), a relation to a known non-standard deformation is discussed.
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Chains of extended jordanian twists are studied for the universal enveloping algebras U(so(M)). The carrier subalgebra of a canonical chain F cannot cover the maximal nilpotent subalgebra N(so(M)). We demonstrate that there exist other types of Frobenius subalgebras in so(M) that can be large enough to include N(so(M)). The problem is that the canonical chains F do not preserve the primitivity on these new carrier spaces. We show that this difficulty can be overcome and the primitivity can be restored if one changes the basis and passes to the deformed carrier spaces. Finally the twisting elem…
We study analogues of the Yangian of the Lie algebra $gl_N$ for the other classical Lie algebras $so_N$ and $sp_N$. We call them twisted Yangians. They are coideal subalgebras in the Yangian $Y(gl_N)$ of $gl_N$ and admit homomorphisms onto the universal enveloping algebras $U(so_N)$ and $U(sp_N)$ respectively. In every twisted Yangian we construct a family of maximal commutative subalgebras parametrized by the regular semisimple elements of the corresponding classical Lie algebra. The images in $U(so_N)$ and $U(sp_N)$ of these subalgebras are also maximal commutative.
We classify anti-involutions of Lie superalgebra $\hsd$ preserving the principal gradation, where $\hsd$ is the central extension of the Lie superalgebra of differential operators on the super circle $S^{1|1}$. We clarify the relations between the corresponding subalgebras fixed by these anti-involutions and subalgebras of $\hat{gl}_{\infty|\infty}$ of types $OSP$ and $P$. We obtain a criterion for an irreducible highest weight module over these subalgebras to be quasifinite and construct free field realizations of a distinguished class of these modules. We further establish dualities between …
Serious difficulties arise in the construction of chains of twists for symplectic Lie algebras. Applying the canonical chains of extended twists to deform the Hopf algebras U(sp(N)) one is forced to deal only with improper chains (induced by the U(sl(N)) subalgebras). In the present paper this problem is solved. For chains of regular injections the sets of maximal extended jordanian twists F_{E,k} are considered. We prove that there exists for U(sp(N)) the twist F_{B,k} composed of the factors F_{E,k}. It is demonstrated that the twisting procedure deforms the space of the primitive subalgebra…
A well-known Peterson's theorem says that the number of abelian ideals in a Borel subalgebra of a rank-$r$ finite dimensional simple Lie algebra is exactly $2^r$. In this paper, we determine the dimensional distribution of abelian ideals in a Borel subalgebra of finite dimensional simple Lie algebras, which is a refinement of the Peterson's theorem capturing more Lie algebra invariants.
We solve a functional version of the problem of twist quantization of a coboundary Lie bialgebra (g,r,Z). We derive from this the following results: (a) the formal Poisson manifolds g^* and G^* are isomorphic; (b) we construct a subalgebra of U(g^*), isomorphic to S(g^*)^g. When g can be quantized, we construct a deformation of the morphism S(g^*)^g subset U(g^*). When g is quasitriangular and nondegenerate, we compare our construction with Semenov-Tian-Shansky's construction of a commutative subalgebra of U(g^*). We also show that the canonical derivation of the function ring of G^* is Hamilt…
We study maximal Poisson-commutative subalgebras in the Poisson algebra $S(\mathfrak{g})$ of a semisimple Lie algebra $\mathfrak{g}$ constructed by Mischenko and Fomenko with the help of the argument shift method. We prove that these subalgebras are Poisson centralizers of certain quadratic elements of $S(\mathfrak{g})$. We deduce from this that there is a unique quantization of Mischenko--Fomenko subalgebras, i.e. there is a unique way to lift Mischenko--Fomenko subalgebras to commutative subalgebras of the universal enveloping algebra $U(\mathfrak{g})$.
The twist deformations for simple Lie algebras U(g) whose twisting elements F are known explicitly are usually defined on the carrier subspace injected in the Borel subalgebra B^+(g). We solve the problem of creating the parabolic twist F_P whose carrier algebra P not only covers B^+(g) but also intersects nontrivially with B^-(g). This algebra P is the parabplic subalgebra in sl(3) and has the structure of the algebra of two-dimensional motions. The parabolic twist is explicitly constructed as a composition of the well known extended jordanian twist F_EJ and the new factor F_D. The latter can…
Given a Lie bialgebra (g,g*), we present an explicit procedure to construct coisotropic subalgebras, i.e. Lie subalgebras of g whose annihilator is a Lie subalgebra of g*. We write down families of examples for the case that g is a classical complex simple Lie algebra.
In his paper "A Construction for Coisotropic Subalgebras of Lie Bialgebras", Marco Zambon gave a way to use a long root of a complex semisimple Lie biaglebra $\mathfrak{g}$ to construct a coisotropic subalgebra of $\mathfrak{g}$. In this paper, we generalize Zambon's construction. Our construction is based on the theory of Lagrangian subalgebras of the double $\mathfrak{g}\oplus\mathfrak{g}$ of $\mathfrak{g}$, and our coisotropic subalgebras correspond to torus fixed points in the variety $\mathcal{L}(\mathfrak{g}\oplus\mathfrak{g})$ of Lagrangian subalgebras of $\mathfrak{g}\oplus\mathfrak{g}…
It is shown that the fixed point subalgebra of an EALA under a finite order automorphism (satisfying certain properties) is a sum of EALA's, an abelian subalgebra, and a subspace which is contained in the centralizer of the core.
The aim of this article is to give a quantization of some coisotropic subalgebras in complex semisimple Lie bialgebras. The coisotropic subalgebras that will be quantized are those given by Zambon in his paper "`A Construction for coisotropic subalgebras of Lie Bialgebras"' [Zam08]. We will also extend the construction for the exceptional complex semisimple Lie bialgebras.
We present methods and explicit formulas for describing simple weight modules over twisted generalized Weyl algebras. When a certain commutative subalgebra is finitely generated over an algebraically closed field we obtain a classification of a class of locally finite simple weight modules as those induced from simple modules over a subalgebra isomorphic to a tensor product of noncommutative tori. As an application we describe simple weight modules over the quantized Weyl algebra.
We show that for some Hopf subalgebras in U_F(so(M)) nontrivially deformed by a twist F it is possible to find the nonlinear primitive copies. This enlarges the possibilities to construct chains of twists. For orthogonal algebra U(so(M)) we present a method to compose the full chains with carrier space as large as the Borel subalgebra B(so(M)). These chains can be used to construct the new deformed Yangians.
A new twisted deformation, U_z(so(4,2)), of the conformal algebra of the (3+1)-dimensional Minkowskian spacetime is presented. This construction is provided by a classical r-matrix spanned by ten Weyl-Poincare generators, which generalizes non-standard quantum deformations previously obtained for so(2,2) and so(3,2). However, by introducing a conformal null-plane basis it is found that the twist can indeed be supported by an eight-dimensional carrier subalgebra. By construction the Weyl-Poincare subalgebra remains as a Hopf subalgebra after deformation. Non-relativistic limits of U_z(so(4,2)) …
New solutions of twist equations for universal enveloping algebras U(A_{n-1}) are found. They can be presented as products of full chains F_c of extended Jordanian twists, Abelian factors (rotations) F^R and sets of quasi-Jordanian twists F^J. The latter are the generalizations of Jordanian twists (with 2-dimensional Borel carrier b^2) for special deformed extensions of the Hopf algebra U(b^2). The carrier subalgebra g_P for the composition of these three twists is a nonminimal parabolic subalgebra in sl(n). The parabolic twisting elements F_P are obtained in the explicit form. The details of …
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. In our previous work (arXiv:1807.11126) we studied the analogous limit of shift of argument subalgebras for the Lie algebras $\mathfrak{g}=\mathfrak{sp}_{2n}$ and $\mathfrak{g}=\mathfrak{o}_{2n+1}$. We described the li…
We are continuing our study of ADE-orbifold subalgebras of the triplet vertex algebra W(p). This part deals with the dihedral series. First, subject to a certain constant term identity, we classify all irreducible modules for the vertex algebra $\bar{M(1)} ^+$, the $\Z_2$--orbifold of the singlet vertex algebra $\bar{M(1)}$. Then we classify irreducible modules and determine Zhu's and $C_2$--algebra for the vertex algebra $\triplet ^{D_2}$. A general method for construction of twisted $\triplet$--modules is also introduced. We also discuss classification of twisted $\bar{M(1)}$--modules includ…
In the present note we suggest an affinization of a theorem by Kostrikin et.al. about the decomposition of some complex simple Lie algebras ${\cal G}$ into the algebraic sum of pairwise orthogonal Cartan subalgebras. We point out that the untwisted affine Kac-Moody algebras of types $A_{p^m-1}$ ($p$ prime, $m\geq 1$), $B_r, \, C_{2^m}, D_r,\, G_2,\, E_7,\, E_8$ can be decomposed into the algebraic sum of pairwise or\-tho\-go\-nal Heisenberg subalgebras. The $A_{p^m-1}$ and $G_2$ cases are discussed in great detail. Some possible applications of such decompositions are also discussed.
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