Fine Gradings of Low-Rank Complex Lie Algebras and of Their Real Forms
Milena Svobodova
备注：
This is a contribution to the Proc. of the Seventh International Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007, Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/
In this review paper, we treat the topic of fine gradings of Lie algebras. This concept is important not only for investigating the structural properties of the algebras, but, on top of that, the fine gradings are often used as the starting point for studying graded contractions or deformations of the algebras. One basic question tackled in the work is the relation between the terms 'grading' and 'group grading'. Although these terms have originally been claimed to coincide for simple Lie algebras, it was revealed later that the proof of this assertion was incorrect. Therefore, the crucial statements about one-to-one correspondence between fine gradings and MAD-groups had to be revised and re-formulated for fine group gradings instead. However, there is still a hypothesis that the terms 'grading' and 'group grading' coincide for simple complex Lie algebras. We use the MAD-groups as the main tool for finding fine group gradings of the complex Lie algebras $A_3 \cong D_3$, $B_2 \cong C_2$, and $D_2$. Besides, we develop also other methods for finding the fine (group) gradings. They are useful especially for the real forms of the complex algebras, on which they deliver richer results than the MAD-groups. Systematic use is made of the faithful representations of the three Lie algebras by $4\times 4$ matrices: $A_3 = sl(4,\mathbb C)$, $C_2 = sp(4,\mathbb C)$, $D_2 = o(4,\mathbb C)$. The inclusions $sl(4,\mathbb C)\supset sp(4,\mathbb C)$ and $sl(4,\mathbb C) \supset o(4,\mathbb C)$ are important in our presentation, since they allow to employ one of the methods which considerably simplifies the calculations when finding the fine group gradings of the subalgebras $sp(4,\mathbb C)$ and $o(4,\mathbb C)$.
We describe all the fine group gradings, up to equivalence, on the Lie algebra $\mathfrak d_4$. This problem is equivalent to finding the maximal abelian diagonalizable subgroups of the automorphism group of $\mathfrak d_4$. We prove that there are fourteen by using two different viewpoints. The first approach is computational: we get a full description of the gradings by using a particular implementation of the automorphism group of the Dynkin diagram of $\mathfrak d_4$ and some algebraic groups stuff. The second approach, more qualitative, emphasizes some algebraic aspects, as triality, and …
The Z_2^n gradings of the classical Lie algebras are described. To elucidate the grading, the classical Lie algebras are expressed in terms of matrix algebras over one of eight fields or Clifford algebras which carry gradings ranging from zero to Z_2^3. When the matrix dimension is divisible by a power of two, the Lie algebras have higher gradings which may be expressed in terms of higher Clifford algebras: these are also described.
We study and give a complete classification of good $\ZZ$-gradings of all simple finite-dimensional Lie algebras. This problem arose in the quantum Hamiltonian reduction for affine Lie algebras.
The paper extends existing Lie algebra representation theory related to Lie algebra gradings. The notion of a representation compatible with a given grading is defined and applied to finite-dimensional representations of $sl(n,\mathbb{C})$ in relation with its $\mathbb{Z}_2$-gradings. For representation theory of $sl(n,\mathbb{C})$ the Gel'fand-Tseitlin method turned out very effective.
A Lax operator algebra is constructed for an arbitrary semi-simple Lie algebra over $\mathbb C$ equipped with a $\mathbb Z$-grading, and arbitrary compact Riemann surface with marked points. In this set-up, a treatment of almost graded structures, and classification of the central extensions of Lax operator algebras are given. A relation to the earlier approach based on the Tyurin parameters is established.
We present an explicit description of the 'fine group gradings' (i.e. group gradings which cannot be further refined) of the real forms of the semisimple Lie algebras $sl(4,\C)$, $sp(4,\C)$, and $o(4,\C)$. All together 12 real Lie algebras are considered, and the total of 44 of their fine group gradings are listed. The inclusions $sl(4,\C)\supset sp(4,\C)\supset o(4,\C)$ are an important tool in our presentation. Systematic use is made of the faithful representations of the three Lie algebras by $4\times 4$ matrices.
Classically important examples of Lie superalgebras have been constructed starting from the Witt and Virasoro algebra. In this article we consider Lie superalgebras of Krichever-Novikov type. These algebras are multi-point and higher genus equivalents. The grading in the classical case is replaced by an almost-grading. The almost-grading is determined by a splitting of the set of points were poles are allowed into two disjoint subsets. With respect to a fixed splitting, or equivalently with respect to an almost-grading, it is shown that there is up to rescaling and equivalence a unique non-tri…
Generalized quantum statistics (GQS) associated to a Lie algebra or Lie superalgebra extends the notion of para-Bose or para-Fermi statistics. Such GQS have been classified for all classical simple Lie algebras and basic classical Lie superalgebras. In the current paper we finalize this classification for all exceptional Lie algebras and superalgebras. Since the definition of GQS is closely related to a certain Z-grading of the Lie (super)algebra G, our classification reproduces some known Z-gradings of exceptional Lie algebras. For exceptional Lie superalgebras such a classification of Z-grad…
We continue the study of Exceptional Periodicity and Magic Star algebras, which provide non-Lie, countably infinite chains of finite dimensional generalizations of exceptional Lie algebras. We analyze the graded algebraic structures arising in the Magic Star projection, as well as the Hermitian part of rank-3 Vinberg's matrix algebras (which we dub HT-algebras), occurring on each vertex of the Magic Star.
The Lie algebra $sl(3,\C)$ is considered in the basis of generalized Pauli matrices. Corresponding grading is the Pauli grading here. It is one of the four gradings of the algebra which cannot be further refined. The set $\es$ of 48 contraction equations for 24 contraction parameters is solved. Our main tools are the symmetry group of the Pauli grading of $sl(3,\C)$, which is essentially the finite group $SL(2,\Z_3)$, and the induced symmetry of the system $\es$. A list of all equivalence classes of solutions of the contraction equations is provided. Among the solutions, 175 equivalence classe…
We consider the grading of $sl(n,\mathbb{C})$ by the group $Π_n$ of generalized Pauli matrices. The grading decomposes the Lie algebra into $n^2-1$ one--dimensional subspaces. In the article we demonstrate that the normalizer of grading decomposition of $sl(n,\mathbb{C})$ in $Π_n$ is the group $SL(2, \mathbb{Z}_n)$, where $\mathbb{Z}_n$ is the cyclic group of order $n$. As an example we consider $sl(3,\mathbb{C})$ graded by $Π_3$ and all contractions preserving that grading. We show that the set of 48 quadratic equations for grading parameters splits into just two orbits of the normalizer of t…
On the construction of polynomial Poisson algebras: a novel grading approach
Instituto de Matemática Interdisciplinar and Dpto. Geometría y Topología, UCM（马德里康普顿斯大学交叉数学研究所与几何与拓扑系）；Università degli Studi di Milano（米兰大学）；INFN Sezione di Milano（意大利国家核物理研究所米兰分部）；La Trobe University（拉筹伯大学）；The University of Queensland（昆士兰大学）
The Gell-Mann grading, one of the four gradings of sl(3,C) that cannot be further refined, is considered as the initial grading for the graded contraction procedure. Using the symmetries of the Gell-Mann grading, the system of contraction equations is reduced and solved. Each non-trivial solution of this system determines a Lie algebra which is not isomorphic to the original algebra sl(3,C). The resulting 53 contracted algebras are divided into two classes - the first is represented by the algebras which are also continuous Inonu-Wigner contractions, the second is formed by the discrete graded…
For a given Lie superalgebra, two ways of constructing color superalgebras are presented. One of them is based on the color superalgebraic nature of the Clifford algebras. The method is applicable to any Lie superalgebras and results in color superalgebra of $ {\mathbb Z}_2^{\otimes N} $ grading. The other is discussed with an example, a superalgebra of boson and fermion operators. By treating the boson operators as "second" fermionic sector we obtain a color superalgebra of ${\mathbb Z}_2 \otimes {\mathbb Z}_2$ grading. A vector field representation of the color superalgebra obtaind from the …
We demonstrate advantages of non-standard grading for computing cohomology of restricted Hamiltonian and Poisson algebras. These algebras contain the inner grading element in the properly defined symmetric grading compatible with the symplectic structure. Using modulo p analog of the theorem on the structure of cohomology of Lie algebra with inner grading element, we show that all nontrivial cohomology classes are located in the grades which are the multiples of the characteristic p. Besides, this grading implies another symmetries in the structure of cohomology. These symmetries are based on …
Exactly integrable systems connected to semisimple algebras of second rank with an arbitrary choice of grading are presented in explicit form. General solutions of these systems are expressed in terms of matrix elements of two fundamental representations of the corresponding semisimple groups.
A Toda equation is specified by a choice of a Lie group and a $\mathbb Z$-gradation of its Lie algebra. The Toda equations associated with loop groups of complex classical Lie groups, whose Lie algebras are endowed with integrable $\mathbb Z$-gradations with finite dimensional grading subspaces, are described in an explicit form.
The grading operators for all nonequivalent Z-gradations of classical Lie algebras are represented in the explicit block matrix form. The explicit form of the corresponding nonabelian Toda equations is given.
All exactly integrable systems connected with the semisimple algebras of the second rank with an arbitrary choice of the grading in them are presented in explicit form. General solution of such systems are expressed in terms of the matrix elements of two fundamental representations of corresponding semisimple groups.
We define the twisted loop Lie algebra of a finite dimensional Lie algebra $\mathfrak g$ as the Fréchet space of all twisted periodic smooth mappings from $\mathbb R$ to $\mathfrak g$. Here the Lie algebra operation is continuous. We call such Lie algebras Fréchet Lie algebras. We introduce the notion of an integrable $\mathbb Z$-gradation of a Fréchet Lie algebra, and find all inequivalent integrable $\mathbb Z$-gradations with finite dimensional grading subspaces of twisted loop Lie algebras of complex simple Lie algebras.
The exactly integrable systems connected with semisimple series $A$ for arbitrary grading are presented in explicit form. Their general solutions are expressed in terms of the matrix elements of various fundamental representations of $A_n$ groups. The simplest example of such systems is the generalized Toda chain with the matrices of arbitrary dimensions in each point of the lattice.
