# Algebra. Rings, modules and categories by Carl Faith

By Carl Faith

VI of Oregon lectures in 1962, Bass gave simplified proofs of a few "Morita Theorems", incorporating rules of Chase and Schanuel. one of many Morita theorems characterizes while there's an equivalence of different types mod-A R::! mod-B for 2 earrings A and B. Morita's resolution organizes rules so successfully that the classical Wedderburn-Artin theorem is an easy outcome, and in addition, a similarity type [AJ within the Brauer staff Br(k) of Azumaya algebras over a commutative ring okay includes all algebras B such that the corresponding different types mod-A and mod-B such as k-linear morphisms are an identical by means of a k-linear functor. (For fields, Br(k) comprises similarity periods of easy important algebras, and for arbitrary commutative ok, this can be subsumed lower than the Azumaya [51]1 and Auslander-Goldman [60J Brauer team. ) a number of different circumstances of a marriage of ring conception and classification (albeit a shot­ gun wedding!) are inside the textual content. additionally, in. my try and additional simplify proofs, particularly to dispose of the necessity for tensor items in Bass's exposition, I exposed a vein of principles and new theorems mendacity wholely inside ring thought. This constitutes a lot of bankruptcy four -the Morita theorem is Theorem four. 29-and the root for it's a corre­ spondence theorem for projective modules (Theorem four. 7) recommended by means of the Morita context. As a derivative, this gives beginning for a slightly entire thought of straightforward Noetherian rings-but extra approximately this within the creation.

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5 (Binomial Formula For Schur Functions). sλ (1 + z1 , . . , 1 + zn ) = sλ (1, . . , 1) where s∗m m1 ≥···≥mn ≥0 δ! s∗ (λ)sm (z), (m + δ)! m is the shifted Schur function s∗m (λ) = det [λi + δi ]mj +δj . det [λi + δi ]δj 34 18 JACQUES FARAUT Proof. 4) with ∞ [λi + δi ]m m fi (w) = (1 + w)λi +δi = w . m! m=0 A function f deﬁned on the set of signatures is said to be shifted symmetric if f (. . , λi , λi+1 , . ) = f (. . , λi+1 − 1, λi + 1, . . ). The algebra of the shifted symmetric functions is denoted by Λ∗ .

26, No 3, 570–574. D. VOICULESCU (1976) Repr´esentations factorielles de type II1 de U (∞), J. Math. , 55, 1–20. A. WOLF (2007). Harmonic analysis on commutative spaces. Amer. Math. Soc.. fr/∼faraut/ This page intentionally left blank Contemporary Mathematics Volume 544, 2011 Restriction of Discrete Series of a semisimple Lie group to reductive subgroups Jorge Vargas Abstract. In this note, for a square integrable representation of a semisimple Lie group, we analyze the continuous spectrum of its restriction to a semisimple subgroup, we also write explicit examples of representations so that its restriction to some particular reductive subgroup have non empty discrete as well as continuous spectrum.

8, δ! lim sm (λ(n) ) = sm (ω). n→∞ (m + δ)! 7 with δ! sm (λ(n) )sm (ix), ψn (x) = ϕn (λ(n) ; x) = (m + δ)! m1 ≥···≥mk ≥0 and sm (ω)sm (ix). ψ(x) = det Φ(ω; x) = m1 ≥···≥mk ≥0 This ﬁnishes the proof of (i). 8. 3, part (ii). We assume that lim ϕn (λ(n) ; x) = ϕ(x), n→∞ uniformly on compact sets in Herm(∞; C). e. the sequence Tn (λ(n) ) converges in Ω. The function ψn deﬁned on R by ψn (τ ) = ϕn (λ(n) ; x) with x = diag(τ, 0, . e. the restriction of x → ϕ(λ(n) ; x) to Herm(1, C) R, being a function of positive type, is the Fourier transform of a probability measure μn on R, by Bochner’s theorem: ψn (τ ) = eitτ μn (dt).