Lie Groups: An Approach through Invariants and by Claudio Procesi PDF

Lie Groups: An Approach through Invariants and by Claudio Procesi PDF

By Claudio Procesi

Lie teams has been an expanding region of concentration and wealthy study because the heart of the twentieth century. Procesi's masterful method of Lie teams via invariants and representations offers the reader a complete remedy of the classical teams in addition to an in depth creation to quite a lot of subject matters linked to Lie teams: symmetric services, conception of algebraic types, Lie algebras, tensor algebra and symmetry, semisimple Lie algebras, algebraic teams, staff representations, invariants, Hilbert concept, and binary varieties with fields starting from natural algebra to sensible analysis.

Key to this detailed exposition is the big volume of history fabric awarded so the booklet is on the market to a reader with particularly modest mathematical historical past. historic info, examples, workouts are all woven into the text.

Lie teams: An procedure via Invariants and Representations will have interaction a wide viewers, together with complicated undergraduates, graduates, mathematicians in quite a few components from natural algebra to sensible research and mathematical physics.

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Extra resources for Lie Groups: An Approach through Invariants and Representations

Example text

7 Commuting Actions We need another generality. , assume that we have two groups G and H acting on the same set X. We say that the two actions commute if gh{x) = hg{x) for allx e X, g e G and he H. This means that every element of G gives rise to an H equivariant map (or we can reverse the roles of G and / / ) . It also means that we really have an action of the product group G x H onX given by (g, h)x = ghx. In this case, we easily see that if a function / is G-invariant and h e H, then hf is also G-invariant.

We have seen that: Proposition 2. The space A of antisymmetric polynomials is a free rank 1 module over the ring S of symmetric polynomials generated by V{x) or A = V(x)S. In particular, any integral basis of A gives, dividing by V(x), an integral basis of S. In this way we will presently obtain the Schur functions. To understand the construction, let us make a fairly general discussion. , JC„], let US consider the basis given by the monomials (which are permuted by 5„). Recall that the orbits of monomials are indexed by non-increasing sequences of nonnegative integers.

In general, a direct sum of vector spaces U = ^f^QUk is called a graded vector space. A subspace W of L^ is called homogeneous, if, setting Wj := WH Ui, we have The space of polynomials is thus a graded vector space P[V] = 0 ^ o ^ ^ [ ^ ] One has immediately {gf){otv) ~ f{ag~^v) = a^{gf){v), which has an important consequence: Theorem. If a polynomial f is an invariant (under some linear group action), then its homogeneous components are also invariant. Proof. Let f = Yl fi ^^ ^^^ decomposition of / into homogeneous components, gf = J2 Sfi is the decomposition into homogeneous components of gf If / is invariant / = gf then // = gfi for each / since the decomposition into homogeneous D components is unique.

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