By Shun-jen Cheng, Weiqiang Wang

This booklet provides a scientific account of the constitution and illustration conception of finite-dimensional advanced Lie superalgebras of classical style and serves as an exceptional advent to illustration conception of Lie superalgebras. a number of folklore effects are carefully proved (and sometimes corrected in detail), occasionally with new proofs. 3 very important dualities are offered within the ebook, with the unifying topic of deciding upon irreducible characters of Lie superalgebras. so as of accelerating sophistication, they're Schur duality, Howe duality, and large duality. The combinatorics of symmetric features is built as wanted in connections to Harish-Chandra homomorphism in addition to irreducible characters for Lie superalgebras. Schur-Sergeev duality for the queer Lie superalgebra is gifted from scratch with whole aspect. Howe duality for Lie superalgebras is gifted in ebook shape for the 1st time. large duality is a brand new procedure constructed some time past few years towards realizing the Bernstein-Gelfand-Gelfand type of modules for classical Lie superalgebras. great duality relates the illustration idea of classical Lie superalgebras on to the illustration concept of classical Lie algebras and therefore supplies an answer to the irreducible personality challenge of Lie superalgebras through the Kazhdan-Lusztig polynomials of classical Lie algebras.

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25. 2) ⇒ 3) The identity component of Γ0 is a Cartan subalgebra, and it is contained in the identity component of α Γ0 . 4], we may assume that the identity component of Γ contains h. Let Q = ηΓ (G). Then any element of Q restricts to identity on h and hence is contained in Stab(Γ0 ). As shown above, Stab(Γ0 ) = T . 35. Let L be a semisimple Lie algebra. Then any grading on L by a torsion-free abelian group is induced from a Cartan decomposition. Toral gradings and methods of refining them to obtain non-toral gradings play an important role in [DM06, DM09, DMV10, DV12].

31), but, for n ≥ 2, it admits a proper refinement in the class of semigroup gradings: namely, take the 1-dimensional subspaces span {Eij } as the components. 12, the notions of fine semigroup grading, fine group grading and fine abelian group grading are all equivalent for simple Lie algebras. 24 is uniquely determined by s ∈ S, so s → t defines a mapping π : S → T . Clearly, this mapping is surjective, and we have Wt = s∈S : π(s)=t Vs . If G is a group, Γ : A = g∈G Ag is a G-grading on an algebra A, and α : G → H is a homomorphism of groups, then the H-grading α Γ is a coarsening of Γ (not necessarily proper).

It is easily seen that D is a crossed product Δ ∗ T where Δ is a division algebra and T is a group (the support of the grading on D). If D is finite-dimensional and the ground field is algebraically closed, then Δ is just the ground field and T is finite. Assuming further that T is abelian, we classify the graded division algebras with support T that are simple as ungraded algebras. The classification is in terms of nondegenerate alternating bicharacters on T with values in the ground field. An explicit construction of the graded division algebra corresponding to a bicharacter β is given in terms of Kronecker products of generalized Pauli matrices where the roots of unity are certain values of β.

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