J. Humphreys, Introduction to Lie Algebras and Representation Theory, Grad. Texts in Math. 9, Springer-Verlag, New York, 1972.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....( n Gammai 1 )ffl i Definition 2. 5 We define the following lattice e Q in h , which will contain all the weights we consider: e Q = a i ffl i j a i 2 Delta Z(1 i n) and a i Gamma a i 1 2 Z(1 i n Gamma 1) Unlike the usual weight lattice for sp(2n; C ) see, e.g. [Hum]) which is just the one freely generated by the ffl i s, here we also need to include those half integral linear combinations for which every coefficient a i is not an integer. Definition 2.6 We define two cones in the lattice e Q as follows C 0 = f a 1 ffl 1 a 2 ffl 2 : a n ffl n 2 ....

J.E. Humphreys, Introduction to Lie algebras and representation theory, Springer-Verlag, New York, 1972.

....polynomials and the Bessel functions. Here we will complete it studying the limit relations involving the q ultraspherical polynomials, the q Hermite and the Hermite polynomials. For an introduction of a q special functions see e.g. 7, 12] and for a review of the theory of root systems see [10]. The connection of root system with hypergeometric functions was studied in details in [9, 8] see also the survey [13] for a detailed introduction) The special functions associated to a root system have an algebraic interpretation [1, 2, 3] e mail:ran us.es, www:http: merlin.us.es renato in ....

J. E. Humphreys, Introduction to Lie Algebras and Representation Theory. SpringerVerlag N.Y. 1972.

....particles in their spectrum can be thought of in terms of a simple universal Lie algebraic structure. The formulation is based on an arbitrary simply laced Lie algebra g (possibly with a subalgebra h) with rank together with its associated Dynkin diagram (for more details see for instance [9]) To each node one attaches a simply laced Lie algebra g i and to each link between the nodes i and j a resonance parameter ij , as depicted in the following g h coset Dynkin diagram g 1 jk g lm g l mn gm g n Besides the usual rules for Dynkin diagrams, ....

J.E. Humphreys, Introduction to Lie Algebras and Representation Theory (Springer, Berlin, 1972).

....I omit some technical details of proof (leaving enough information so that an experienced reader always can fill the gaps) However, I do assume that the reader is familiar with (finite dimensional and affine) root systems and Weyl groups. A short introduction to these notions can be found in [Hu1, Chapter III]; see also a recent survey of Koornwinder [Ko1] to see how these notions appear in the theory of orthogonal polynomials. For more detailed expositions can we refer the reader to [B, Hu2, V] These lectures are of expository nature; they do not contain any new results. I include the reference to ....

Humphreys, J.E., Introduction to Lie algebras and representation theory, Springer-Verlag, New York, 1972.

....1 ffl n : Gamma ( n Gammai 1 )ffl i Definition 2. 5 We define the following lattice Q in h , which will contain all weights we consider: a i ffl i j a i 2 Z=2 (1 i n) and a i Gamma a i 1 2 Z (1 i n Gamma 1) g Unlike the usual weight lattice for sp(2n; C) see, e.g. [Hum]) which is just the one freely generated by the ffl i s, here we also need to include those half integral linear combinations for which every coefficient is purely half integral. Definition 2.6 We define two cones in the lattice e Q as follows C 0 = f a 1 ffl 1 a 2 ffl 2 : a n ffl n ....

J.E. Humphreys, Introduction to Lie algebras and representation theory, SpringerVerlag, New York, 1972.

....the signed permutations in one line notation with a bar over an element with a negative sign. The group B n is generated by the simple reflections oe i for 1 i n Gamma 1 as well as oe 0 where woe 0 is w 1 w 2 : w n . Note, we have chosen a different set of simple roots from that found in [8]. For any sequence a 1 ; a k of distinct non zero real numbers we define fl(a 1 ; a k ) to be the element in B k with signed numbers in the same positions and same relative order of the underlying permutation. Using this notation, we state the main theorem of this article. ....

James E. Humphreys. Introduction to Lie Algebras and Representation Theory. Number 9 in GTM. Springer-Verlag, New York, 1972.

....2n f matrices acting on 2n dimensional linear f space V with basis B 2n = p 1 ; pn ; q 1 ; q n ) by the standard way, preserving the symplectic 2 form = p 1 q 1 Delta Delta Delta pn q n : 2. 15) Recall some fundamental facts about the representations of Sp(2n) see [20]. The word representation will mean below a polynomial finite dimensional linear representation over f . Every representation of Sp(2n) is equivalent to a sum of irreducible representations. All classes of equivalence of the irreducible representations are parametrized by partitions of length n. ....

J. E. Humphreys, Introduction to Lie algebras and representation theory, Springer-Verlag, New York (1972).

.... and [Hum1] For basic facts on Schubert varieties of type A, one may refer to [Fulton] The theory of root systems and Weyl groups, independent from their corresponding Lie groups, is contained in [Hum2] The relationship between Lie algebras, root systems and Weyl groups is explained in [Hum3]. 2. Generalities on G=B and G=Q Let G be a semisimple and simply connected algebraic group defined over an algebraically closed field of arbitrary characteristic. Let T; B;W etc. be as in the Introduction. Let R be the system of roots of G relative to T . Let S (resp.R ) be the set of simple ....

James E. Humphreys. Introduction to Lie Algebras and Representation Theory. Number 9 in GTM. Springer-Verlag, New York, 1972.

....was independently conceived by Demazure [4] around the same time. We will cite just one source for simplicity. Given a semi simple Lie group and a Cartan subgroup there is a root system Delta contained in some ambient vector space V with a positive definite symmetric bilinear form (ff; fi) See [10] for details on Lie groups and root systems. Let ff 1 ; ff 2 ; ff n be a choice of simple roots in the root system. Let Delta (respectively Delta Gamma ) be the positive roots (negative roots) with respect to this choice of simple roots. Let R be the ring of polynomials in the simple ....

J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, no. 9 in GTM, Springer-Verlag, New York, 1972.

....Distributivity, and the Lack Thereof 4. The Mobius Function Throughout this paper, Phi ae R shall denote a (reduced) crystallographic root system with positive roots Phi , simple roots ff 1 ; ff n , inner product h ; i, and Weyl group W . Standard references are [B1] and [H]. For each ff 2 Phi, ff = 2ff=hff; ffi denotes the co root corresponding to ff. We let = f 2 R : h; ff i 2 Z for all ff 2 Phig denote the weight lattice, and 1 ; n the fundamental weights (i.e. h i ; ff j i = ffi ij ) The set of dominant weights (i.e. the ....

J. E. Humphreys, "Introduction to Lie Algebras and Representation Theory," Springer-Verlag, Berlin-New York, 1972.

....L U a U;k U , where U runs over irreducible representations. We may let V be a virtual representation (a formal difference of representations) We study the case of a complex reductive group such as GL n (C ) a compact real Lie group such as U(n) or O(n) or the Lie algebra of such a group. See [1, 3, 6]. For convenience, we will discuss g, a reductive Lie algebra over C , and a finite dimensional virtual representation V . We recall some standard facts [6] We know g possesses a maximal abelian subalgebra, its Cartan subalgebra h; a root system, a certain finite set in h (the linear ....

....a complex reductive group such as GL n (C ) a compact real Lie group such as U(n) or O(n) or the Lie algebra of such a group. See [1, 3, 6] For convenience, we will discuss g, a reductive Lie algebra over C , and a finite dimensional virtual representation V . We recall some standard facts [6]. We know g possesses a maximal abelian subalgebra, its Cartan subalgebra h; a root system, a certain finite set in h (the linear functionals on h) and a weight lattice in h . A Weyl group W defined by the root system acts on h and h . We choose a fundamental Weyl chamber of dominant ....

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J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York, 1972.

....de Lie, Chapitres I VIII , Masson, Paris, 1972. Dix] J. Dixmier, Enveloping algebras , Amer. Math. Soc. 1994) originally published in French by Gauthier Villars, Paris 1974 and in English by North Holland, Amsterdam 1977. The following are standard (and very useful) texts in Lie theory. [Hu] J. Humphreys, Introduction to Lie algebras and representation theory , Graduate Texts in Mathematics 9, Springer Verlag, New York Berlin, 3rd printing) 1980. K] V. Kac, Infinite dimensional Lie algebras , Birkhauser, Boston, 1983. The most comprehensive reference for modern deformation ....

J. Humphreys, "Introduction to Lie algebras and representation theory", Graduate Texts in Mathematics 9, Springer-Verlag, New York-Berlin (3rd printing) 1980.

....2.6. If ; 2 and 2 Z J , then c( c( Proof. For 2 Z , let P ( denote the number of (unordered) partitions of into a sum of positive roots the coecient of e in the formal series Q 0 (1 e ) 1 . By Steinberg s Formula (e.g. [H, x24]) one knows that c( X w;w 0 2W sgn(w) sgn(w 0 )P ( w) w 0 ) 2.1) where (w) w( Now consider that (s i w) w) hw( i i i = h ; w 1 i i i : Furthermore, s i w) w) implies that w 1 i is a ....

....c( X w;w 0 2W sgn(w) sgn(w 0 )P ( w) w 0 ) 2. 1) where (w) w( Now consider that (s i w) w) hw( i i i = h ; w 1 i i i : Furthermore, s i w) w) implies that w 1 i is a positive root (e.g. [H, x10]) so in this case, s i w) w) is a positive (integer) multiple of i . Proceeding by induction with respect to length, we deduce that (w) is a positive Z linear combination of the simple roots j such that s j occurs in some (equivalently, every) reduced expression for w. 11 It ....

J. E. Humphreys, \Introduction to Lie Algebras and Representation Theory," Springer-Verlag, Berlin-New York, 1972.

....) The xed point set of a (t) on C is the tritangent divisor D t which parametrises cubic surfaces for which the three lines in t meet in one point, called an Eckart point ( N] x8) 2. The toric variety 2.1. For general facts on toroidal compacti cations we refer to [Fu] for root systems see [Hu]. 2.2. The torus. The D 4 adjoint torus T = C ) 4 ; t 7 ( t) t) t) t) comes with a natural identi cation of its character group Hom(T;C ) Z 4 with the sublattice M : h e 1 e 2 ; e 2 e 3 ; e 3 e 4 ; e 3 e 4 i 4 i=1 Ze i : The lattice M , with the ....

....of the torus T in the identity element e. Since e is xed by W (D 4 ) we get an induced action of W (D 4 ) on P 3 w , and we will see that this action extends to a linear action of W (F 4 ) The root lattice Q(F 4 ) of F 4 is the lattice in R 4 generated by the 48 roots of F 4 which are (cf. [Hu], III 12.1) the 24 roots e i e j of D 4 (these roots have length 2 and are called the long roots of F 4 ) and the 24 vectors e i , e 1 e 2 e 3 e 4 ) 2 which have length 1, the short roots of F 4 . Q(F 4 ) h e i e j ; e 1 e 2 e 3 e 4 ) 2i Z ( R 4 ) 6.5. Theorem. Any ....

J. E. Humphreys, Introduction to Lie algebras and representation theory. Graduate Texts in Mathematics, Vol. 9. Springer-Verlag, New York-Berlin, (1972).

....K group of finite Morley rank. Then G #(G) is isomorphic to a direct sum of simple algebraic groups over algebraically closed fields. In Section 5, we will need the following facts about algebraic groups over algebraically closed fields. Apart from Fact 2. 38, these facts are found in [25] 14] or [13], which are our main references for the theory of algebraic groups and related subjects. Definition 2.34 ( 14] Section 7.5) Let G be an algebraic group over an algebraically closed field and M an arbitrary subset of G. A(M) the group closure of M) is the intersection of all closed subgroups ....

....i# j# (i, j # Z) arranged in some fixed order, and where the c ij are integers depending on #, # and the chosen ordering, but not on t and u. Fact 2.40 ( 25] Example (a) on page 24) If # # is not a root, then the right side of the commutator formula in Fact 2.39 reduces to 1. Fact 2. 41 ([13], Lemma 10.1) Let # be a root system. If # is a base of #, then (#, #) # 0 for # #= # in #, and # # is not a root. Fact 2.42 ( 14] Theorem 27.3) Let G be a reductive algebraic group, T a fixed maximal torus and # = #(G, T ) Let # be a base of #. Let Z# denote CG (T # ) where T# = ker#) ....

J. E. Humphreys. Introduction to Lie Algebras and Representation Theory. Springer Verlag, Berlin-New York, third edition, 1980.

....i 2 H; X 2 = 1 2 (E Gamma F ) X 3 = Gamma i 2 (E F ) The Killing form on sl(2) is (8) E H F E 0 0 4 H 0 8 0 F 4 0 0 The quaternion algebra H can be recovered from sl(2) and the Killing form. Before explaining this we recall some of the representation theory of sl(2) References are [H], Section 7, and [FH] Lecture 11. Let V (n) denote the (unique) simple highest weight sl(2) module with highest weight n, where n is any nonnegative integer. This module has basis v 0 , vn , and the sl(2) action is (9) Ev i = n 1 Gamma i)v i Gamma1 ; Hv i = n Gamma 2i)v i ; Fv i = ....

J. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York, 1972.

....and R(G) #(G) # m (G) for some natural number m. 6.1.3. Corollary (Zorn) A finite Engel group is nilpotent. 6.1. 3 was the first result to be proved about Engel groups; it is a grouptheoretic version of Engel s theorem about ad nilpotence in finite dimensional Lie Algebras (see Humphreys [14] Section 3.2) It follows from 6.1.3 that if G is a finite group with every 2 generator subgroup nilpotent, then G is nilpotent. Linear groups have a nice Engel structure. More precisely, 6.1.4. Theorem (Gruenberg, 9] Let G be a linear group. Then L(G) #(G) and R(G) #(G) In moving up to ....

J. E. Humphreys, Introduction to Lie algebras and representation theory, Springer, 1972.

....x 2 j ) Y ff 0 ff(x) 1 The analogous result for type Cn is proved in detail in [15] this is indicated there. the class of the diagonal in flag bundles 483 This verifies Fulton s formulas for types A and D; types B and C are checked similarly. For type G 2 , we use the description of [17]: we realize the maximal torus of G 2 as the subset of C 3 where the coordinates add up to 0, and write S(h ) Omega C S(h ) C [x 1 ; x 2 ; x 3 ; y 1 ; y 2 ; y 3 ] modulo the relations P x i = P y i = 0. The 6 positive roots are x i Gamma x j (i j) x i x j Gamma 2x k . Note that ....

J. E. Humphreys, Introduction to Lie algebras and representation theory, Springer, Berlin, 1972.

....are uniquely determined by N b and by its decomposition N b M i2 A Phin i i Phi M j0 D Phim j j 4 Phi 8 M k=6 E Phil k k in indecomposable sublattices. Proof. The assertion ii) follows from i) and from the well known uniqueness of the decomposition of N b (see for example [9], Proposition 11.3) For i) let [D] 2 NS(X b ) be a representative of a class ff 2 N , with (D:D) Gamma2. Since KX b is numerically equivalent to a Delta F , for some a 2 Q , one obtains from 16 KEIJI OGUISO AND ECKART VIEHWEG the Riemann Roch formula (OX b (D) D: D Gamma KX b ) 2 ....

Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory. Graduate Texts in Math. 9 (1972), Springer Verlag, Berlin-Heidelberg-New York

....t ; Then [D(X) D(Y ) D( X; Y ] see ( 54] p. 12) Fix a basis fX 1 ; X d g, de ne D i : D(X i ) and for any multi integer n = n 1 ; n d ) 2 N d de ne the monomial D n : D n1 1 : D nd d (where D 0 i : I) The Poincar e Birkho Witt theorem, 8] 25] [26], 28] 54] 56] implies the monomials form a basis for U . De ne 6 WAYNE LAWTON jnj : P k n k and for p 2 N de ne U p : f X jnj p a(n)D n : a : N d Cg : Then U p is independent of the choice of basis and consists of all analytic di erential operators of degree p. For N ....

J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York, 1972.

....space, we will only brie y recapitulate the operators that span the open string algebra. The reader is referred to Refs. 12] 9] and [13] for more complete discussions. Also, the reader can nd an account of some basic notions of the representation theory of Lie algebras to be used below in Ref. [14]. BASIC FORMALISM An open matrix chain is a matrix product of an N dimensional row vector, a (possibly empty) series of N N square matrices and an N dimensional column vector. It can be abstractly written as 1 s K 2 ; where 1 is a positive integer, K a nite integer ....

J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, 2nd. ed., Springer-Verlag, New York, 1978.

....langen Wurzelelemente in Chevalley Gruppen. Es sei Phi ein reduziertes irreduzibles Wurzelsystem und P ein Gitter zwischen dem Gewichtegitter P ( Phi) und dem Wurzelgitter Q( Phi) Weiter sei K ein Korper. Diesen Daten entspricht eine Chevalley Gruppe G = GP ( Phi; K) siehe [B] C1] [Hu], St] Sei T = TP ( Phi; K) ein zerfallender maximaler Torus in G und B = BP ( Phi; K) eine Borel Untergruppe, die T enthalt. Im folgenden spielt P eigentlich keine Rolle und wird in den Bezeichnungen unterdruckt. Wir konnen alles auf den Fall von einfach zusammenhangenden G = G sc (d.h. auf den ....

Humphreys J.E. Introduction to Lie algebras and representation theory, 3rd Printing. -- Springer: New York et al. -- 1980. -- 171P.

....determined for all , one can then use the formula to compute K ; Of course, there would be obvious difficulties if it happened that c = c . However, it is easily shown that c c if is dominant, and replacing with a non dominant member of its orbit can only decrease the value of c [H1, x13]. Without further refinements, what we have just described is too unwieldy to be useful for computing weight multiplicities in all but the smallest cases. Indeed, if Gamma = 28 c 1 ff 1 Delta Delta Delta c n ff n , then the above scheme would require (c 1 1) Delta Delta Delta (c n ....

....to see algorithms based on Kashiwara s crystal bases [K] or Littelmann s path model [Li] it seems that a widely used strategy for computing tensor product multiplicities is based on the following result. It is often attributed to Klimyk (see [Kl] although in the notes for Chapter 24 in [H1], Humphreys traces it back to a 1937 paper of R. Brauer [B] Theorem 7.1 (Brauer Klimyk) For all ; 2 , we have ( X 2 K ; 7.1) 31 Proof. Since ( Delta(ae) Delta( ae) we find ( Delta(ae) Delta( ae) X 2 K ; e X w2W sgn(w)e w( ae) X 2; w2W ....

J. E. Humphreys, "Introduction to Lie Algebras and Representation Theory," Springer Verlag, Berlin-New York, 1972.

....the direct product W 1 W 2 of the Weyl groups of g 1 and g 2 . 3.2 Klimyk s algorithm. There is a useful and explicit algorithm for the decomposition of the tensor product of two irreducible representations of a simple Lie algebra g into irreducible components, based on the Klimyk formula (see [H], Sec.24, Ex.9) For any weight 2 h ; let f g denote the dominant weight lying on the orbit of under the Weyl group. If f g 2 C; then there is the unique w 2 W such that f g = w . Let t( be equal to the sign of w in this case and zero otherwise. Suppose moreover that we know the list ....

....basis of g 0 with respect to the form ( We may choose it in such a way that Y 0 = E 2 a and fY a 0 g, a 0 0 is an orthonormal basis for g s 0 . For any representation V of g s 0 , the Casimir operator C(V) is de ned by C(V) P a 0 0 Y a 0 Y a 0 . It is well known (see [H]) that if Vis an irreducible representation with a highest weight , then C(V) 2 ) 1=2 X 2 (g s 0 ) As we have noticed already, our algebras g s 0 are irreducible in all cases except the sl(n; C ) series, but even then the formula C(V ) 2 ) 1 ; 2 ....

Humphreys, J.E., Introduction to Lie algebras and representation theory, Springer-Verlag, Berlin Heidelberg New York, 1972.

....dimension and different signature. The numeration of Lie algebras as used here was first introduced by Bianchi [5] now it is the standard classification within relativity theory. For the general background the reader is referred to the following basic monographs: 6] for homogeneous structures, [7] for Lie algebras, 8] for homogeneous cosmological models and explanation of ref. 5] 9] for General topology. Acknowledgements The authors would like to thank for financial support by Deutsche Forschungsgemeinschaft (M.R. and by the Wissenschaftler Integrations Programm (H. J.S. THE ....

J. Humphreys, Introduction to Lie algebras and representation theory, Springer New York 1972.

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J. Humphreys, Introduction to Lie Algebras and Representation Theory, Grad. Texts in Math. 9, Springer-Verlag, New York, 1972.

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J.E. Humphreys, Introduction to Lie Algebras and Representation Theory Graduate Texts vol. 9, Springer-Verlag, New York, NY, 1972.

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Humphreys, J.E., "Introduction to Lie Algebras and Representation Theory," Springer-Verlag GTM, 9, Berlin New York, 1980.

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Humphreys J. E., Introduction to Lie algebras and representation theory , Graduate Texts in Mathematics, 9. Springer-Verlag, New York-Berlin (1978).

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J.E. Humphreys, Introduction to Lie Algebras and Representation Theory Graduate Texts vol. 9, Springer-Verlag, New York, NY, 1972.

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J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York 1972.

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J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York, 1972.

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J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, 2nd ed. (Springer-Verlag, New York, 1978).

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J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, New York, 1972.

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J.E. Humphreys, "Introduction to Lie algebras and representation theory", SpringerVerlag, New York, 1972. 44

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J. Humphreys, "Introduction to Lie Algebras and Representation Theory", Springer-Verlag 1972

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J. Humphreys, Introduction to Lie algebras and Representation Theory, Springer, Graduate Texts in Mathematics, vol. 9, Springer, 1972, New York.

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J.E. Humphreys, Introduction to Lie Algebras and Representation Theory (Springer, Berlin, 1972).

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J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York, 1972.

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J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York 1972.

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J.E. Humphreys, Introduction to Lie Algebras and Representation Theory (Springer, Berlin, 1972).

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J. Humphreys, Introduction to Lie algebras and Representation Theory, Springer, Graduate Texts in Mathematics, vol. 9, Springer, 1972, New York.

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J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York 1972.

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J.Humphreys, Introduction to Lie algebras and representation theory. Springer (1972).

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J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York, 1972.

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J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York, 1972.

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J. E. Humphreys, \Introduction to Lie Algebras and Representation Theory," Springer-Verlag, Berlin-New York, 1972.

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J. E. HUMPHREYS, Introduction to Lie Algebras and Representation Theory (Springer Verlag, New York-Heidelberg-Berlin, 1972). 3, 4, 4, 5

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James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, vol. 9, Springer-Verlag, 1972.

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J. E. Humphreys, Introduction to Lie algebras and representation theory, Springer-Verlag GTM 9, New York 1972.

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