Root Systems Lie Algebras

n 112 _,x ll< (2) We use induction on n. For n = 1 our assertion is obvious. We will show that if it holds for all values less than n, then it holds for n. Suppose ~ = {xl,...,xn} is an arbitrary L-system. For i = i, . . ., n we put A i = { ] ~ { t . . . . . n}: ] • i , ( x ~ , x j ) ~ 0 } . (3 ) We c o n s i d e r two c a s e s . The f i r s t c a s e : f o r e a c h i = 1 , . . . , n t h e c a r d i n a l i t y o f A i i s a t m o s t n 2 / 3 . T h e n n 2 n I ] ~ I Xi[t = 2 i=1 (Zt, Xji ) = ~ = i ( xi' Xi) + ~'i~j( xi' Xj)= = n J r ~ . # j , (xi, xp§ (xt, xr ---n + ~ = 1 2 J E A i (xi' Xj) = n n =n + 21=1(xi, ~,j~AiXj)<n + y,i=xil2i~AxjH. (4)

In [10] Patera and Sharp conceived a new relation, subjoining, between semisimple Lie algebras. Our objective in this paper is twofold. Firstly, to lay down a mathematical formalization of this concept for arbitrary Lie algebras. Secondly, to give a complete classification of all maximal subjoinings between Lie algebras of the same rank, of which many examples were already known to the above authors.The notion of subjoining is a generalization of the subalgebra relation between Lie algebras. To give an intuitive idea of what is involved we take a simple example. Suppose is a complex simple Lie algebra of type B2. Let be a Cartan subalgebra of and Δ the corresponding root system. We have the standard root diagramInside B2 there lies the subalgebra A1 × A1 which can be identified with the sum of and the root spaces corresponding to the long roots of B2.

Let R be an irreducible root system. A Lie Algebra L is called graded by R if L is graded with grading group the root lattice of R such that the only nonzero homogeneous subspaces of L have degree 0 or a root in R , the grading is induced by the adjoint action of a split Cartan subalgebra of a finite-dimensional simple Lie subalgebra of L with root system R , and L is generated by the homogeneous subspaces of nonzero degree. This class of Lie algebras was introduced and studied by S. Berman and R. Moody in Invent. Math. 108 (1992), where, in particular, a classification up to central equivalence is given in the simply-laced case. The doubly-laced cases have recently been classified by G. Benkart and E. Zelmanov. Let R be a 3-graded root system, i.e., R is not of type E_8, F_4 or G_2. In this paper, Lie algebras graded by R are described in a unified way, without case-by-case considerations. Namely, it is shown that a Lie algebra L is 3-graded if and only if L is a central extension of the Tits-Kantor-Koecher algebra of a Jordan pair covered by a grid whose associated 3-graded root system is isomorphic to R . This result is then used to classify Lie algebras graded by R : we give the classification of Jordan pairs covered by a grid and describe their Tits-Kantor-Koecher algebras. One of the advantages of this approach is that it works over rings containing 1/2 and 1/3, and also for infinite root systems. Another application is the description of Slodowy's intersection matrix algebras arising from multiply-affinized Cartan matrices.

An associative unitary (respectively, Lie) algebra is said to be proper in case the intersection of all of its cofinite two-sided (respectively, Lie) ideals is zero.Using the Hopf algebra structure of UL, it is shown that over a field of characteristic zero a Lie algebra is proper if and only if its universal enveloping algebra is proper.(In the finite-dimensional case this provides a new proof of a result of Harish-Chandra.)The analogous result for Lie palgebras and their restricted universal enveloping algebras holds and is proved by the same technique.Notational conventions.Throughout this paper, p will denote a prime number.Barring mention to the contrary, we work over an arbitrary ground field K except that when Lie p-algebras (^restricted Lie algebras of characteristic p) are discussed, the ground field must have characteristic p (cf. [1, p. 78, Exercise 20] or [7, p. 187, Definition 4]).The unmodified words "algebra" and "coalgebra" shall, respectively, mean "associative unitary algebra" and "associative counitary coalgebra."If A is an algebra, (A) shall denote the Lie algebra whose underlying vector space coincides with that of A, and whose bracket is defined for all elements x and y of A by lx>y]c(A) = x y -y x where "" denotes the multiplication of A; while P(A) shall denote the Lie p-algebra whose underlying Lie algebra is (A) and whose p-mapping, x h-> z'pl, is the pth power mapping, x >-> xp, of the algebra A. For any algebra map f: A-* B, (f): (A) -> (B) (respectively, P(F): P(A) -P(B)) shall denote the map of Lie algebras (respectively, Lie p-algebras) which coincides with / as a vector space map.For any Lie algebra L, UL shall denote the universal enveloping algebra of L; while for any Lie p-algebra L, VL shall denote the restricted universal enveloping algebra of L; furthermore, i and j shall denote, respectively, the canonical injections i: L -> (UL) and j: L - P(VL).Occasionally, the vector space whose only element is zero shall be denoted by 0, and, occasionally, the end of proof shall be landmarked by the symbol D. DEFINITION 1.(a) If A is an algebra (respectively, a Lie algebra), denote by C.I.(A) the collection of all cofinite two-sided ideals (respectively, all cofinite Lie

Our aim here is to study this property. The exclusion of the vector 0 is not essential in what follows. It turns out that many of the other properties of root systems of Lie algebras depend only on property P and are thus shared by a much wider class of vector systems. In the statements to follow it is assumed that all vectors considered come from a real Euclidean space Vof finite dimension n. 1.1 Let S be finite and have property P, and let A be a subset with property P. For each x in S, assume that at most one (resp. exactly one, at least one) of x and -x is in A. Then there is an ordering of the space V such that A is contained in (resp. A is, A contains) the set of positive elements of S. Harish-Chandra [6, Lemma 4] proves the second part of this result for Lie algebra root systems, however, using nontrivial properties of Lie algebras. Borel and Hirzebruch [2, pp. 471-473] give a geometric proof of the second and third parts, but then revert to Lie algebra techniques to prove the first part, all for Lie algebra root systems. All of these authors make a somewhat stronger assumption on A than property P, namely, if x, y e A and x + y e S, then x + y e A. Our proof of 1.1 depends on a preliminary result which may have some independent interest. 1.2. Let B be finite and have property P, and assume that x e B implies -x q B. Then ther-e is an ordering of the space V such that all elements of B are positive. The real numbers of the form k 112 (k, I positive integers) show that the assumption of finiteness in 1.1 (or 1.2) can not be dropped. It can, however, be weakened thus. 1.1'. In 1.1 replace the assumption of finiteness by the assumption that 0 is not a point of accumulation of S. In 1.2 we can go further, in terms of a weakening of property P.

The universal Lie algebra associatedto a root system I

In [2], we considered a class of Lie algebras generalizing the classical simple Lie algebras. Using a field Φ of characteristic zero and a square matrix (Aij) of integers with the properties (1) Aii = 2, (2) Aij ≦ 0 if i ≠ j, (3) Aij = 0 if and only if Ajt = 0, and (4) is symmetric for some appropriate non-zero rational a Lie algebra E = E((Aij)) over Φ can be constructed, together with the usual accoutrements: a root system, invariant bilinear form, and Weyl group.For indecomposable (A ij), E is simple except when (Aij) is singular and removal of any row and corresponding column of (Aij) leaves a Cartan matrix. The non-simple Es, Euclidean Lie algebras, were our object of study in [3] as well as in the present paper. They are infinite-dimensional, have ascending chain condition on ideals, and proper ideals are of finite codimension.

The first part of this chapter treats Lie algebras, beginning with definitions and many examples. The notions of solvable, nilpotent, radical, semisimple, and simple are introduced, and these notions are followed by a discussion of the effect of a change of the underlying field. The idea of a semidirect product begins the development of the main structural theorems for real Lie algebras—the iterated construction of all solvable Lie algebras from derivations and semidirect products, Lie's Theorem for solvable Lie algebras, Engel's Theorem in connection with nilpotent Lie algebras, and Cartan's criteria for solvability and semisimplicity in terms of the Killing form. From Cartan's Criterion for Semisimplicity, it follows that semisimple Lie algebras are direct sums of simple Lie algebras. Cartan's Criterion for Semisimplicity is used also to provide a long list of classical examples of semisimple Lie algebras. Some of these examples are defined in terms of quaternion matrices. Quaternion matrices of size n-by-n may be related to complex matrices of size 2n-by-2n. The treatment of Lie algebras concludes with a study of the finite-dimensional complex-linear representations of sl[(2, ℂ). There is a classification theorem for the irreducible representations of this kind, and the general representations are direct sums of irreducible ones. Section 10 contains a review of the elementary theory of Lie groups and their Lie algebras. The abstract theory as in Chevalley [1946] is summarized, and the correspondence is made with the concrete theory of closed linear groups, where the Lie algebra is obtained as the space of derivatives at t = 0 of smooth curves in the group passing through the identity at t = 0. The section ends with a discussion of the adjoint representation. The remainder of the chapter explores some aspects of the connection between Lie groups and Lie algebras. One aspect is the relationship between automorphisms and derivations. The derivations of a semisimple Lie algebra are inner, and consequently the identity component of the group of automorphisms of a semisimple Lie algebra consists of inner automorphisms. In addition, simply connected solvable Lie groups may be built one dimension at a time as semidirect products with ℝ1, and consequently they are diffeomorphic to Euclidean space. For simply connected nilpotent groups the exponential map is itself a diffeomorphism. The earlier long list of classical semisimple Lie algebras corresponds to a list of the classical semisimple Lie groups. The issue that needs attention for these groups is their connectedness, and this is proved by using the polar decomposition of matrices.

In this talk we introduce generalised Calogero–Moser models and demonstrate their integrability by constructing universal Lax pair operators. These include models based on non-crystallographic root systems, that is the root systems of the finite reflection groups, H 3, H 4, and the dihedral group I 2(m), besides the well-known ones based on crystallographic root systems, namely those associated with Lie algebras. Universal Lax pair operators for all of the generalised Calogero–Moser models and for any choices of the potentials are linear combinations of the reflection operators. The equivalence of the Lax pair with the equations of motion is proved by decomposing the root system into a sum of two-dimensional sub-root systems, A 2, B 2, G 2, and I 2(m). The root type and the minimal type Lax pairs, derived in our previous papers, are given as the simplest representations. The spectral parameter dependence plays an important role in the Lax pair operators, which bear a strong resemblance to the Dunkl operators.

Extended affine Lie algebras are higher nullity generalizations of finite dimensional simple Lie algebras and affine Kac Moody Lie algebras. In this paper we completely describe the structure of the core modulo its centre and the root system for extended affine Lie algebras of type \( B_l (l\ge 3) \), \( C_l (l \ge 2)\), F4 and G2.

A complex \omega -Lie algebra is a vector space L over the complex field, equipped with a skew-symmetric bracket [-,-] and a bilinear form \omega such that [[x,y],z]+[[y,z],x]+ [[z,x],y]=\omega(x,y)z+\omega(y,z)x+\omega(z,x)y for all x,y,z\in L . The notion of \omega -Lie algebras, as a generalization of Lie algebras, was introduced in Nurowski [3]. Fundamental results about finite-dimensional \omega -Lie algebras were developed by Zusmanovich [5]. In [3], all three-dimensional non-Lie real \omega -Lie algebras were classified. The purpose of this note is to provide an approach to classify all three-dimensional non-Lie complex \omega -Lie algebras. Our method also gives a new proof of the classification in Nurowski [3].

For every finite-dimensional nilpotent complex Lie algebra or superalgebran, we offer three algorithms for realizing it in terms of creation and annihilation operators. We use these algorithms to realize Lie algebras with a maximal subalgebra of finite codimension. For a simple finite-dimensionalg whose maximal nilpotent subalgebra isn, this gives its realization in terms of first-order differential operators on the big open cell of the flag manifold corresponding to the negative roots ofg. For several examples, we executed the algorithms using the MATHEMATICA-based package SUPERLie. These realizations form a preparatory step in an explicit construction and description of an interesting new class of simple Lie (super) algebras of polynomial growth, generalizations of the Lie algebra of matrices of complex size.

We introduce countably infinite series of finite dimensional generalizations of the exceptional Lie algebras: in fact, each exceptional Lie algebra (but g2) is the first element of an infinite series of finite dimensional algebras, which we name Magic Star algebras. All these algebras (but the first elements of the infinite series) are not Lie algebras, but nevertheless they have remarkable similarities with many characterizing features of the exceptional Lie algebras; they also enjoy a kind of periodicity (inherited by Bott periodicity), which we name Exceptional Periodicity. We analyze the graded algebraic structures arising in a certain projection (named Magic Star projection) of the generalized root systems pertaining to Magic Star algebras, and we highlight the occurrence of a class of rank-3, Hermitian matrix (special Vinberg T)-algebras (which we call H algebras) on each vertex of such a projection. We then focus on the Magic Star algebra f4(n), which generalizes the non-simply laced exceptional Lie algebra f4, and deserves a treatment apart. Finally, we compute the Lie algebra of the inner derivations of the H algebras, pointing out the enhancements occurring for each first element of the series of Magic Star algebras, thus retrieving the result known for the derivations of cubic simple Jordan algebras.

We study possible cases of complex simple graded Lie algebras of depth 2, which are the Tanaka prolongations of pseudo $H$-type Lie algebras arising through representation of Clifford algebras. We show that the complex simple Lie algebras of type $B_n$ with $|2|$-grading do not contain non-Heisenberg pseudo $H$-type Lie algebras as their negative nilpotent part, while the complex simple Lie algebras of types $A_n$, $C_n$ and $D_n$ provide such a possibility. Among exceptional algebras only $F_4$ and $E_6$ contain non-Heisenberg pseudo $H$-type Lie algebras as their negative part of $|2|$-grading. An analogous question addressed to real simple graded Lie algebras is more difficult, and we give results revealing the main differences with the complex situation.

Splint is a decomposition of root system into union of root systems. Splint of root system for simple Lie algebra appears naturally in studies of (regular) embeddings of reductive subalgebras. Splint can be used to construct branching rules. We consider special embedding of Lie subalgebra to Lie algebra. We classify projections of algebra root systems using extended Dynkin diagrams and single out the conditions of splint appearance and coincidence of branching coefficients with weight multiplicities. While such a coincidence is not very common it is connected with Gelfand-Tsetlin basis.